Tsu-2011 - ieeetsu

omar furTuxia 

albaToba da statistika 

magaliTebsa da amocanebSi 

(Teoria, magaliTebis amoxsnis nimuSebi, 

amocanebi) 

Tsu-2011

s a r C e v i 

albaTobis Teoria 

Tavi I. albaTobis Teoriis elementebi ............................................................................ 

elementarul xdomilebaTa sivrce. operaciebi xdomilobebze. albaTobis klasikuri, 

statistikuri da geometriuli ganmarteba. kombinatorikis elementebi. albaTobis 

gamoTvla kombinatorikis gamoyenebiT. 

Tavi II. rTuli xdomilebis albaTobebi ......................................................................... 

albaTobaTa Sekrebis kanoni. sxvaobis albaTobis formula. pirobiTi albaToba. 

namravlis albaToba. xdomilebaTa damoukidebeloba. sruli albaTobis formula. 

baiesis formula. ganmeorebiTi cdebi. bernulisa da puasonis formulebi. 

Tavi III. SemTxveviT sidideTa maxasiaTeblebi .......................................................... 

SemTxveviTi sidide. ganawilebis kanoni. ganawilebis funqcia. ganawilebis simkvrive. 

zogierTi mniSvnelovani ganawileba. kvantili, mediana, moda. maTematikuri 

lodini, dispersia. organzomilebiani SemTxveviTi sidide. regresiis funqcia. 

momentebi, asimetria, eqscesi. kovariacia. korelaciis koeficienti. 

Tavi IV. diskretul ganawilebaTa gamoyenebebi ...................................................... 

binomialuri da puasonis ganawilebebis gamoyenebebi 

Tavi V. uwyveti tipis ganawilebebi........................................................................................ 

ganawilebis simkvrive. kvantili, moda, mediana, qveda da zeda kvartili, zeda 

-kritikuli wertili, momentebi (lodini, dispersia, asimetria, eqscesi). 

Tavi VI. normaluri ganawileba ............................................................................................... 70 

Tavi VII. albaTobis Teoriis zRvariTi Teoremebi ............................................. 84 

CebiSevis utoloba. did ricxvTa kanoni. CebiSevis Teorema. bernulis Teorema. 

centraluri zRvariTi Teorema. liapunovis Teorema. muavr-laplasis lokaluri 

da integraluri Teoremebi. 

statistikuri daskvnebis Teoria 

Tavi I. wertilovani da intervaluri Sefasebebi. ndobis intervali 

saSualosaTvis ........................................................................................................................................... 

wertilovan SefasebaTa saxeebi (Caunacvlebeli, Zalmosili da efeqturi Sefasebebi). 

SerCeviTi momentebi (saSualo, dispersia, Sesworebuli dispersia, standartuli 

gadaxra, asimetria, eqscesi). SerCeviTi korelacia. maqsimaluri dasajerobis 

meTodi. momentTa meTodi. ndobis intervali populaciis saSualosaTvis. 

Tavi II. ndobis intervali bernulis sqemaSi ............................................................. 

asimptoturi ndobis intervali da SerCevis moculoba populaciis proporciisaTvis. 

ndobis intervali puasonis ganawilebis ucnobi parametrisaTvis. 

Tavi III. ndobis intervali dispersiisaTvis ............................................................... 

ndobis intervali normaluri populaciis dispersiisa da standartuli gadaxrisaTvis 

cnobili da ucnobi saSualoebisaTvis. didi SerCevebis SemTxveva. 

2 

Tavi IV. hipoTezaTa Semowmeba saSualosaTvis ( cnobilia) ................ 

ZiriTadi da alternatiuli hipoTezebi. mniSvnelovnebis done. kriteriumis statistika. 

kritikuli mniSvneloba. kritikuli are. I da II gvaris Secdomebi. kriteriumis 

simZlavre. hipoTezaTa Semowmeba normaluri populaciis saSualosaTvis 

(dipersia cnobilia). SerCevis minimaluri moculoba. P -mniSvnelobis meTodi. 

2 

Tavi V. hipoTezaTa Semowmeba saSualosaTvis ( ucnobia) ...................... 

hipoTezaTa Semowmeba normaluri populaciis saSualosaTvis ucnobi dipersiis 

SemTxvevaSi. Z da T kriteriumebis gamoyenebis SesaZlo SemTxvevebi. 

Tavi VI. hipoTezaTa Semowmeba dispersiisaTvis ...................................................... 129 

1 

3 

15 

33 

50 

61 

94 

104 

108 

114 

123

statistikuri hipoTezis Semowmeba normaluri ganawilebis dispersiis Sesaxeb 

cnobili da ucnobi saSualoebis SemTxvevaSi. P -mniSvnelobis meTodi. 

Tavi VII. hipoTezaTa Semowmeba bernulis sqemaSi ................................................. 

hipoTezaTa Semowmeba bernulis sqemaSi ucnobi albaTobisaTvis (didi da mcire 

SerCevebi). hipoTezaTa Semowmeba puasonis ganawilebis ucnobi parametrisaTvis 

(mcire SerCevebi). P -mniSvnelobis meTodi. 

Tavi VIII. ndobis intervali da hipoTezaTa Semowmeba .................................... 140 

Tavi IX. oramokrefiani amocanebi populaciaTa saSualoebisaTvis ... 147 

ori damoukidebeli normaluri populaciis saSualoTa Soris gansxvavebis 

hipoTezaTa Semowmeba cnobili da ucnobi dispersiebis SemTxvevaSi. hipoTezaTa 

Semowmeba aranormaluri populaciebisaTvis (didi moculobis mqone SerCevebi). 

ndobis intervalebi saSualoTa sxvaobisaTvis. 

Tavi X. oramokrefiani amocanebi populaciaTa saSualoebisaTvis 

(mcire SerCevebi). satertvaitis meTodi ............................................................................ 

ori damoukidebeli aranormaluri populaciis saSualos Soris gansxvavebis hipoTezaTa 

Semowmeba mcire SerCevebis SemTxvevaSi. ndobis intervali saSualoTa 

sxvaobisaTvis. satertvaitis meTodi. 

Tavi XI. oramokrefiani amocanebi populaciaTa dispersiebisaTvis ... 

hipoTezaTa Semowmeba ori damoukidebeli normaluri populaciis dispersiebis 

Sesaxeb. dispersiaTa tolobis Sesaxeb hipoTezaTa Semowmebis gamartivebuli 

procedura. ndobis intervali dispersiaTa fardobisaTvis. 

Tavi XII. statistikuri daskvnebi dawyvilebuli monacemebisaTvis .... 

statistikuri daskvnebi dawyvilebul monacemTa saSualoebis sxvaobebisaTvis 

(damokidebuli SerCevebi). P -mniSvnelobis meTodi. ndobis intervali. 

Tavi XIII. oramokrefiani amocanebi bernulis sqemaSi ...................................... 

hipoTezaTa Semowmeba warmatebaTa albaTobebisaTvis bernulis cdaTa ori damoukidebeli 

mimdevrobisaTvis. ndobis intervali p 1 

p 2 

sxvaobisaTvis. 

Tavi XIV. Tanxmobis kriteriumebi ......................................................................................... 

xi-kvadrat Tanxmobis kriteriumis formula. normalurobis hipoTezis Semowmeba. 

Tavi XV. damoukideblobis Semowmebis xi-kvadrat kriteriumi .............. 193 

Tavi XVI. erTgvarovnebis Semowmebis xi-kvadrat kriteriumi .................. 201 

Tavi XVII. dawyvilebuli monacemebi, korelacia ................................................... 206 

gabnevis diagrama. misadagebis wiri. regresiis wrfis gantoleba. Sefasebis standartuli 

Secdoma. prognozi da saprogrozo intervali. daskvnebi korelaciis 

koeficientis Sesaxeb (hipoTezaTa Semowmeba). 

Tavi XVIII. regresia da korelacia ...................................................................................... 

martivi wrfivi regresia. regresiis wrfe. umcires kvadratTa meTodi. regresiis 

wrfis Sefaseba. determinaciis koeficienti. statistikuri daskvnebi regresiis 

koeficientebis Sesaxeb. mravlobiTi regresia. modelis vargisianobis Semowmeba. 

statistikuri daskvnebi mravlobiTi modelis parametrebis Sesaxeb. rangobrivi 

korelacia. spirmenis rangobrivi korelaciis koeficienti. 

Tavi XIX. dispersiuli analizi (ANOVA) ...................................................................... 

hipoTezaTa Semowmeba sami an meti populaciis saSualoTa tolobis Sesaxeb. 

Tavi XX. sakontrolo werebisa da Sualeduri da saboloo 

gamocdebis bileTebis nimuSebi 2007—2009 wlebSi .............................................. 249 

danarTi (statistikuri cxrilebi) ....................................................................................... 254 

damateba (sagamocdo amocanebi xdomilebebisaTvis)............................................. 263 

literatura .................................................................................................................................................. 272 

135 

157 

164 

171 

178 

186 

222 

238 

2

a l b a T o b a 

Tavi I 

albaTobis Teoriis elementebi 

eqsperimentis calkeul SesaZlo Sedegs elementaruli xdomileba 

ewodeba, xolo maT erTobliobas – elementarul xdomilebaTa sivrce da 

aRiniSneba asoTi: { 

1, 2,...} 

. 

savarjiSoebi: I. monetis erTxel agdebisas -- ={g, s}; II. monetis 

orjer agdebisas, an ori monetis erTdroulad agdebisas -- ={gg, gs, sg, 

ss}; III. monetis samjer agdebisas, an sami monetis erTdroulad agdebisas 

-- = {ggg, ggs, gsg, sgg, gss, sgs, ssg, sss}; IV. monetis n-jer agdebisas 

{ : ( a1,..., 

a n 

), a g an s} 

da Sedegebis saerTo raodenoba tolia 

i 

n 

2 -is; V. erTi saTamaSo kamaTlis gagorebisas -- ={1, 2, 3, 4, 5, 6}; VI. 

vTqvaT, Tavidan vagdebT monetas. Tu mova gerbi, maSin vagorebT saTamaSo 

kamaTels; xolo Tu mova safasuri, maSin kidev erTjer vagdebT monetas. 

am SemTxvevaSi { g1, 

g2, 

g3, 

g4,g5, g6,sg, ss} 

; VII. ori saTamaSo kamaTlis gagorebisas 

-- ={(1,1); (1,2); . . . ; (1,6); (2,1); (2,2); . . . ; (2,6); . . . ; (6,1); . . . ; 

(6,6)} anu {( i , j) : i, 

j 1,2,..., 

6} 

; VIII. produqciis vargisinobis dadgenisas -- 

={“vargisi”, “uvargisi”}; IX. satelefono sadgurSi gamoZaxebaTa raodenoba 

-- = {0, 1, 2, . . . }; X. Zabva qselSi -- ={[0, 220]}. 

diskretul elementarul xdomilebaTa sivrcis nebismier qvesimravles 

xdomileba ewodeba. Tu eqsperimentis konkretuli Sedegi ekuTvnis 

raime xdomilebas, maSin amboben rom es xdomileba moxda, xolo 

romelsac ar ekuTvnis – is xdomileba ar moxda. xdomilebebi aRiniSneba 

didi laTinuri asoebiT: A , B, 

C, 

D,... 

. xdomilebas A uwodeben 

aucilebel xdomilebas, xolo Ø -- SeuZlebel xdomilebas. A da B 

xdomilebis gaerTianeba (an jami) ewodeba iseT xdomilebas, romelic 

xdeba maSin, roca am xdomilebebidan erTi mainc xdeba da aRiniSneba 

simboloTi A B (an A B). A da B xdomilebis TanakveTa (an namravli) 

ewodeba iseT xdomilebas, romelic xdeba maSin, roca es xdomilebebi 

erTdroulad xdeba da aRiniSneba simboloTi A B (an AB). A 

xdomilebis sawinaaRmdego xdomileba ewodeba iseT xdomilebas, romelic 

xdeba maSin, roca A ar xdeba da aRiniSneba simboloTi A . A da B 

xdomilebis sxvaoba ewodeba iseT xdomilebas, romelic xdeba maSin, 

roca xdeba A magram ar xdeba B da aRiniSneba simboloTi A\ B. A da 

B xdomilebas ewodeba uTavsebadi Tu A B=Ø. 

Tu -s nebismier elementarul xdomilebas i Seesabameba garkveuli 

ricxvebi p i 

P( i 

) , romlebic akmayofileben pirobebs: 0 p i 

1 

da 

p i 

1, maSin am ricxvebs ewodebaT 

i i elementaruli xdomilebebis alb- 

3

. Tu P( ) 

aTobebi. P( A): P( ) 

A 

i 

i 

i 

const da | | 

, vRebulobT albaTobis 

klasikur ganmartebas: P( A) | A| / | | . 

xdomilebis statistikur albaTobad iTvleba am xdomilebis fardobiTi 

sixSire WN 

( A) M / N (sadac N – cdaTa saerTo ricxvia, М ki -- A 

xdomilebis moxdenaTa ricxvi) an masTan axlos myofi ricxvi (maTematikurad 

zusti formulireba aseTia: P( A) lim W ( A) 

). 

geometriuli albaToba. Tu L monakveTze SemTxveviT agdeben wertils, 

maSin albaToba imisa, rom agdebuli wertili daecema l L monakve- 

Tze: P | l | / | L | . analogiuri ganmarteba gvaqvs sibrtyeze da sivrceSi. 

N 

kombinatorikis elementebi. 

gamravlebis principi: Tu asarCevia m obieqti da arsebobs pirveli 

obieqtis arCevis n 1 

SesaZlebloba, pirveli obieqtis arCevis Semdeg arsebobs 

meore obieqtis arCevis n 2 

SesaZlebloba, da a. S., m 1 obieqtis ar- 

Cevis Semdeg arsebobs m-uri obieqtis arCevis n m 

SesaZlebloba, maSin 

arsebobs am m obieqtis am mimdevrobiT arCevis n 1 

n 2 

n m 

SesaZlebloba. 

magaliTad, Tu mamakacs aqvs 4 perangi da 2 pijaki, maSin am mamakacs 

aqvs perangisa da pijakis SerCevis 42 8 SesaZlebloba (varianti). 

namravlis principTan dakavSirebiT xSirad sasargebloa xis msgavsi 

(xisebri) diagramis anu dendrogramis gamoyeneba. 

magaliTad, dendrogramiT gamosaxuli monetis orjer agdebis Sesabamisi 

Sedegebis simravle iqneba: 

N 

gadanacvlebebi – es aris kombinaciebi, romlebic Sedgenilia mocemuli 

п elementiani simravlis yvela п elementisagan da erTamaneTisagan 

ganxsvavdeba mxolod elementebis ganlagebis rigiT. Р п = п!. 

wyobebi – es aris т elementiani kombinaciebi п gansxvavebuli elementis 

mqone simravlidan, romelebic erTmaneTisagan gansxvavdebian an elementebis 

SemadgenlobiT an elementebis ganlagebis rigiT. А т n!/( n m)! 

. 

п 

4

jufdebebi – es aris п elementiani simravlis daulagebeli т elementiani 

qvesimravleebi. т 

m m 

С п!/( т!( п т)!) 

. cxadia, rom C m! 

A . 

п 

n 

n 

albaTobis gamoTvla kombinatorikis gamoyenebiT. 

amorCeva dabrunebiT: eqsperimentis yovel nabijze yuTidan amoRebuli 

burTi ukan brundeba. vigulisxmoT, rom burTebi gadanomrilia ricxvebiT 

1-dan M -mde. ganasxvaveben ori tipis amorCevebs: dalagebuli amorCevebi 

da daulagebeli amorCevebi. dalagebuli amorCevebi aRiniSneba simboloTi 

( a 

1 

,...,a 

n ), xolo daulagebeli amorCevebi – [ a 

1 

,...,a 

n ]. 

dabrunebiT dalagebuli amorCevis SemTxvevaSi: 

n 

{ : ( a1,..., 

an) : ai 

1,..., 

M; 

i 1,..., 

n} 

da | | M . 

daulagebeli amorCevebi dabrunebiT: 

n 

{ : [ 

a1,..., 

a ]: a 1,..., 

M; 

i 1,..., 

n} 

, | . 

n 

i 

| CM 

n1 

n M da amoRebuli burTi ukan ar 

amorCeva dabrunebis gareSe: 

brundeba. 

dabrunebis gareSe dalagebuli SerCevis SemTxvevaSi: 

{ : ( a1,..., 

a ) : a a , k l; 

a 1,..., 

M; 

i 1,..., 

n} 

da | | 

n 

k 

daulagebeli amorCevebi dabrunebis gareSe: 

{ : [ 

a1,..., 

an]: 

ak 

al 

, k l; 

ai 

1,..., 

M; 

i 1,..., 

n} 

, 

saboloo suraTi aseTia: 

l 

i 

. 

n 

| | C M 

n 

A M 

dabrunebiT 

dabrunebis 

gareSe 

dalagebuli daulagebeli 

n 

M C n 

M n 1 

n 

n 

A 

M 

C 

M 

magaliTad, M 3 da n 2 -is SemTxvevaSi Sesabamis elementarul 

xdomilebaTa sivrceebs eqnebaT Semdegi saxis struqturebi: 

erToblioba 

savarjiSoebi: 

(1,1)(1,2)(1,3) 

(2,1)(2,2)(2,3) 

(3,1)(3,2)(3,3) 

(1,2)(1,3) 

(2,1)(2,3) 

(3,1)(3,2) 

[1,1][2,2][3,3] 

[1,2][1,3][2,3] 

[1,2] [1,3] [2,3] 

dalagebuli daulagebeli 

SerCeva 

dabrunebiT 

dabrunebis 

gareSe 

5

I. kursze, romelzec sami jgufia, jgufxelebis arCevis yvela SesaZlo 

variantebis ricxvia n1 n2 

n3, sadac n i 

-- i -ur jgufSi studentebis 

raodenobaa (viyenebT namravlis princips); 

II. raodenoba yvela SesaZlo kombinaciebis, ramdennairadac SesaZlebelia 

m mgzavri ganvaTavsoT n vagonSi tolia n m (dalagebuli amorCeva 

dabrunebiT, sadac M n da n m); 

III. m adamianis dabadebis dReebis yvela SesaZlo kombinacia (im pirobiT, 

rom TiTeuli dabadebis dRe aris romelime 365 dRidan) tolia 

m 

365 (saqme gvaqvs amorCevasTan dabrunebiT, amasTanave amorCevebi iTvleba 

dalagebulad, sadac M 365 da n m); 

IV. raodenoba yvela SesaZlo kombinaciebis, ramdennairadac SesaZlebelia 

5 burTi ganvaTavsoT 5 yuTSi, ise rom erT yuTSi iyos erTi bur- 

Ti, tolia 5! (namravlis principis Tanaxmad); 

V. partiebis raodenoba, romelic unda Sedges n monawilisagan Semdgar 

wriul saWadrako turnirSi tolia C n 

2 

n( 

n 1)/ 

2 (daulagebeli amorCeva 

dabrunebis gareSe). 

magaliTebis amoxsnis nimuSebi: 

magaliTi 1. vipovoT albaToba imisa, rom simetriuli monetis ori 

damoukidebeli agdebisas erTjer mainc mova gerbi. 

amoxsna. am SemTxvevaSi ={gg, gs, sg, ss}; P {gg} P {gs} 

= P {sg} 

= P { ss} 

1/ 

4; A={gg, gs, sg} da P ( A) 

3/ 4. 

magaliTi 2 (“bednier” bileTebze). 25 sagamocdo bileTidan 5 “bednieria”, 

xolo danarCeni 20 – “ara bednieri”. romel students aqvs “bednieri” 

bileTis aRebis meti albaToba: vinc pirveli iRebs bileTs, Tu 

vinc meore iRebs bileTs? 

amoxsna. avRniSnoT pirveli studentis mier aRebuli bileTis nomeri 

i asoTi, xolo meore studentis mier aRebuli bileTis nomeri j aso- 

Ti. davuSvaT, rom “bednieri” bileTebis nomrebia: 1, 2, 3, 4, 5. maSin cxadia, 

rom {( i, 

j) : i, 

j 1,2,..., 

25, i j} 

, | | 2524 

600 da bunebrivia CavTvaloT, 

rom yvela elementaruli xdomileba tolalbaTuria: P ( i, 

j) 

1/ 

600 . 

SemoviRoT xdomilebebi: 

A{ pirvelma studentma aiRo kargi bileTi }, 

B { meore studentma aiRo kargi bileTi }, 

maSin am xdomilebebs aqvT Semdegi saxe: 

A {( i. 

j) : i 1,..., 

5; j 1,..., 

25; i j} 

da B {( i, 

j) : i 1,..., 

25; j 1,..., 

5; i j} 

. 

advili dasanaxia, rom | A | | 

B | 524 

120 

. amitom albaTobis klasikuri 

ganmartebis Tanaxmad: 

120 1 

P( A) P( B) 

. 

600 5 

e. i. orive students aqvs kargi bileTis aRebis erTi da igive albaToba. 

6

davaleba. wina amocanaSi vipovoT albaToba imisa, rom mesame studenti 

amoiRebs “bednier” bileTs. 

magaliTi 3 (galtonis dafa). gvaqvs dafaze samkuTxedis formiT dalagebuli 

rgolebi, ise rom wveroSi erTi rgolia, meore rigSi winasgan 

Tanabr manZilebze ori rgoli, mesame rigSi zeda ori rgolidan Tanabar 

manZilebze sami rgoli da a.S. boloSi aris eqvsi rgoli. me-7 rigSi 

ki aris bolo 6 rgolidan Tanabar manZilebze 7 Rrmuli. zeda rgolze 

agdeben burTs da mas SeuZlia igoraos Tanabari albaTobiT an marjvniv 

an marcxniv rgolidan rgolze, rac sabolood sruldeba romelime RrmulSi 

CavardniT. rogoria albaToba imisa, rom burTi Cavardeba mexuTe 

RrmulSi? 

● 

O 

O 

O O O 

O O O O 

O O O O O 

O O O O O O 

U U U U U U U 

1 2 3 4 5 6 7 

amoxsna. rogorc vxedavT arsebobs pirvel da me-7 RrmulebSi burTis 

Cavardnis erTaderTi gza (traeqtoria), meore da me-6 RrmulebSi bur- 

Tis Cavardnis -- eqvs-eqvsi gza, mesame da mexuTe RrmulebSi – TxuTmet- 

TxuTmeti gza da bolos, meoTxe RrmulSi – burTis Cavardnis 20 gza. 

gzebis (Sedegebis) sruli raodenobaa 1+6+15+20+15+6+1=64 da yvela es Sedegi 

Tanabrad mosalodnelia, vinaidan TiToeuli traeqtoriis gavlisas 

burTi ganicdis eqvs dajaxebas rgolebze da yoveli dajaxebisas is Tanabari 

albaTobebiT gadaadgildeba an marjvniv an marcxniv. TanabaralbaTuri 

64 Sedegidan mexuTe RrmulSi Cavardnas xels uwyobs 15 Sedegi da 

Sesabamisad, saZebni albaToba iqneba 15/64. 

qvemoT moyvanilia galtonis dafaze rgolebis 7 rigis SemTxvevaSi 

TiToeul poziciaze burTis moxvedris SesaZlo gzebis ricxvi. 

O 

7

magaliTi 4. vipovoT albaToba imisa, rom wreSi SemTxveviT Cagdebuli 

wertili ar Cavardeba am wreSi Caxazul wesier eqvskuTxedSi. 

amoxsna. davuSvaT, rom wris radiusia R , maSin masSi Caxazuli wesieri 

eqvskuTxedis gverdic iqneba R . amasTanave, wris farTobia | S | R 

2 

, 

3 3 2 

xolo eqvskuTxedis farTobia -- | s | R . amitom saZiebeli albaToba 

2 

iqneba: 

2 3 3 2 

R 

R 

| S | | s | 2 2 

3 3 

P 

0,174. 

2 

| S | R 

2 

magaliTi 5. AB monakveTze SemTxveviT agdeben sam wertils C , D 

da M . vipovoT albaToba imisa, rom AC, AD da AM monakveTebisagan Sei- 

Zleba aigos samkuTxedi? 

amoxsna. avRniSnoT AC, AD da AM monakveTebis sigrZeebi Sesabamisad 

x, y da z -iT da elementarul xdomilebaTa sivrcis rolSi ganvixiloT 

sivrcis wertilTa simravle koordinatebiT ( x , y, 

z) 

. Tu CavTvliT, 

rom AB monakveTis sigrZe tolia 1-is, maSin elementarul xdomilebaTa 

sivrce iqneba kubi, romlis wiboa erTi. amave dros, xelSemwyob elementarul 

xdomilebaTa simravle (samkuTxedis aqsiomis Tanaxmad )Sedgeba im 

wertilebisagan, romelTa koordinatebisaTvis sruldeba samkuTxedis utolobebi: 

x +y > z, x + z > y, y + z > x. es ki warmoadgens kubis nawils, romelic 

moWrilia misgan sibrtyeebiT: x + y = z, x + z = y, y + z = x 

z 

х 

y 

(erT-erTi am sibrtyidan, kerZod x + y = z, moyvanilia naxazze). yoveli ase- 

Ti sibrtye kubidan moWris piramidas, romlis moculoba tolia 

1 1 1 

1 

. 

3 2 6 

Sesabamisad, kubis darCenili nawilis moculoba iqneba 

1 1 

| v | 13 

. 

6 2 

amitom saZiebeli albaToba, ganmartebis Tanaxmad, iqneba 

| v | 1 1 

| P | :1 

. 

| V | 2 2 

magaliTi 6 (Sexvedris amocana). ori piri SeTanxmda garkveul adgilze 

Sexvdes erTmaneTs 6-dan 7 saaTamde. TiToeuli maTgani SemTxveviT 

8

momentSi midis daTqmul adgilas da meores elodeba 20 wuTis manZilze 

(Semdeg ki midis). vipovoT albaToba imisa, rom es pirebi Sexvdebian erTmaneTs? 

amoxsna. avRniSnoT, erT-erTi piris daTqmul adgilze misvlis dro 

6 x -iT, xolo meore piris -- 6 

y -iT (sadac x da y gamosaxulia saaTeb- 

Si). elementarul xdomilebaTa sivrcis rolSi SegviZlia aviRoT im ( x , y) 

wertilTa simravle, romlebic ekuTvnian erTeulovan kvadrats. am SemTxvevaSi 

CvenTvis saintereso wertilTa (xelSemwyob elementarul xdomilebaTa) 

simravle iqneba erTeulovani kvadratis im wertilTa simravle, romelTa 

koordinatebi erTmaneTisagan daSorebulia araumetes 20/60 

1/ 

3 - 

iT: 

A {( x, 

y);| 

x y | 1/3} 

. 

vinaidan, | x y | 1/3 

1/3 

y x 1/3 

x 1/3 

y x 1/ 

3. amitom advili 

gasagebia, rom A simravle iqneba erTeulovani kvadratis SigniT 

y x 1/3 da y x 1/ 3 wrfeebs Soris moqceuli gaferadebuli are (roca 

0 x 1/3, maSin y x 1/ 3 wrfis nacvlad qveda sazRvris rolSi iqneba x 

RerZi: 0 y x 1/ 3, xolo roca 2/3 

x 1, maSin zeda sazRvari y x 1/ 3 

wrfis nacvlad iqneba y 1 

wrfe: x 1/3 

y 1). 

y 

1 

 

1/3 

0 1/3 2/3 1 x 

amitom saZiebeli albaToba iqneba: 

| | 1 

2/32/3 

5 

P ( A) 

A . 

| | 

1 9 

magaliTi 7. ramdegi gansxvavebuli sia SeiZleba Sedgenil iqnes 7 

sxvadasxva gvarisagan? 

amoxsna. 7 

7! 1234567 5040 . 

magaliTi 8. SejibrebaSi monawile 10 sportsmenidan pirvel sam adgilze 

gasuli sami gamarjvebuli ramden sxvadasxvanairad SeiZleba ganTavsdes 

dasajildoebel kvarcxlbekze? 

3 

amoxsna. А 10 1098 

720. 

magaliTi 9. SesarCev SejibrebaSi monawileobs 10 adamiani, romelTagan 

finalSi gadis sami. finalistebis ramdeni gansxvavebuli sameuli 

SeiZleba gamovlindes? 

amoxsna. wina magaliTisagan gansxvavebiT, aq finalistebis rigs (dalagebas) 

mniSvneloba ara aqvs. amitom viyenebT jufdebis formulas: 

9

3 10! 8910 

С 

10 

120. 

3!7! 6 

magaliTi 10 (damTxvevebze). vipovoT albaToba imisa, rom: a). m Sem- 

TxveviT arCeuli adamianis dabadebis dReebi ar daemTxveva erTmaneTs (im 

pirobiT, rom yvela dRe tolalbaTuria); b). m SemTxveviT arCeul adamianSi 

moiZebneba ori mainc iseTi, romelTa dabadebis dReebi daemTxveva 

erTmaneTs. 

amoxsna. a). elementarul xdomilebaTa sivrce Seesabameba dalagebul 

amorCevebs dabrunebiT, sadac M 365 da n m, anu | | 365 m ; xolo 

xelSemwyobi elementaruli xdomilebebi ki dalagebul amorCevebs dabrunebis 

gareSe, sadac agreTve M 365 da n m, amitom maTi raodenobaa -- 

A . Sesabamisad, albaTobis klasikuri ganmartebis Tanaxmad, gvaqvs 

m 

365 

m 

A365 1 2 m1 

P( m) 

(1 )(1 ) (1 

). 

m 

365 365 365 365 

b). sawinaaRmdego xdomilebis albaTobis gamoTvlis wesis Tanaxmad 

ki gvaqvs, rom 

m 

A365 

Q( m) 

1 

. 

m 

365 

moviyvanoT am albaTobis mniSvnelobebis cxrili zogierTi m-is 

SemTxvevaSi: 

m 4 16 22 23 40 64 70 

Q(m) 0.01636 0.28360 0.47569 0.50730 0.89123 0.99711 0.99916 

sainteresoa aRiniSnos, rom (molodinis sawinaaRmdegod!) klasis 

moswavleTa raodenoba, romelSic 1/2-is toli albaTobiT moiZebneba ori 

moswavle mainc erTi da igive dabadebis dReebiT, arc ise didia: igi tolia 

mxolod 23-is. 

magaliTi 11 (mogeba latareaSi). gvaqvs M bileTi gadanomrili ricxvebiT 

erTidan M -mde, romelTagan n bileTi nomrebiT erTidan n-mde 

momgebiania ( M 2n 

). vyidulobT n cal bileTs. ras udris albaToba imisa, 

rom am n bileTidan erTi mainc iqneba momgebiani (avRniSnoT es xdomileba 

A asoTi)? 

amoxsna. vinaidan bileTebis amoRebis (yidvis) Tanmimdevrobas ara 

aqvs mniSvneloba nayid bileTebSi momgebiani bileTebis arsebobis an ar 

arsebobis TvalsazrisiT, amitom elementarul xdomilebaTa sivrces eqneba 

Semdegi stuqtura: 

{ : [ 

a1,..., 

a ]: a a , k l; 

a 1,..., 

M} 

. 

n 

n 

Sesabamisad, Cvens mier zemoTmoyvanili cxrilis Tanaxmad | | C M . 

xdomilebas (avRniSnoT igi B 0 -iT), rom nayid bileTebSi ar aris momgebiani 

bileTebi, eqneba Semdegi struqtura: 

n 

B0 { : [ 

a1,..., 

an]: 

ak 

al 

, k l; 

ai 

n 1,..., 

M} 

da | B0 | C M n 

. 

amitom 

k 

l 

i 

10

n n 

CM 

n 

AM 

n 

n n 

n 

P( B 0 

) (1 )(1 ) (1 

) . 

n n 

CM 

AM 

M M 1 

M n 1 

Sesabamisad, saZebni albaToba iqneba: 

n n n 

P( B) 1 P( B0 

) 1 (1 )(1 ) (1 ) 

M M 1 M n 1 

. 

2 

1 

Tu magaliTad, M n da n , maSin P ( B0) 

e (aq e neperis ricxvia) 

da P ( B) 

1e 

1 

0, 632, sadac krebadobis siCqare sakmaod didia: ukve 

roca n 10 

-- P ( B) 

0, 670. 

davaleba. wina amocanis pirobebSi vipovoT albaToba imisa, rom nayidi 

n bileTidan zustad m ( m n) iqneba momgebiani. 

magaliTi 12 (urTierTobis upiratesobaze). davuSvaT klasSi, romelSic 

10 moswavlea tardeba gamokiTxva, sadac TiToeulma moswavlem anketaSi 

unda miuTiTos is sami amxanagi, romelsac aZlevs upiratesobas 

cxra amxanagidan. A iyos xdomileba, rom erTerTi moswavle dasaxelebul 

iqna yvela SesaZlo cxra anketaSi. ras udris misi albaToba, Tu anketis 

Sevseba iyo SemTxveviTi, anu anketis Sevsebis nebismieri kombinacia 

tolalbaTuria. 

amoxsna. calkeuli moswavlisaTvis anketis Sevsebis sxvadasxva kombinaciaTa 

raodenoba tolia C -is, xolo 10 anketis Sevsebis variantebis 

3 

9 

3 10 

raodenoba pirveli magaliTis analogiurad tolia (C9 ) -is. vinaidan er- 

Ti anketa SeiZleba Sevsebul iqnes nebismierad, xolo danarCen cxra anketaSi 

erTi pasuxi dafiqsirebulia da danarCeni ori pasuxi ki SeiZleba 

nebismierad amoirCes rva SesaZlebeli pasuxidan, amitom elementaruli 

3 2 9 

xdomilebebis raodenoba A-Si tolia | A| C9 ( C8) 

(Tu wyvilis pirvel 

komponents SeuZlia miiRos m gansxvavebuli mniSvneloba, xolo meore 

komponents ki pirvelisgan damoukideblad -- n gansxvavebuli mniSvneloba, 

maSin aseTi wyvilebis raodenoba namravlis principis Tanaxmad iqneba 

-- mn ). aqedan gvaqvs 

3 2 9 2 

C9 

( 

C8 

) C8 

9 1 

P ( A) 

( ) . 

3 10 

3 9 

( C9 

) C9 

3 

davaleba. wina amocanaSi ipoveT albaToba imisa, rom erTerTi studenti 

dasaxelebuli iqneba k -jer ( k 9). 

magaliTi 13 (or “tuzze”). ganvixiloT preferansis TamaSi, rodesac 

kartis Sekvris maRali 32 karti SemTxveviT nawildeba (rigdeba) sam moTamaSes 

Soris, ise rom TiTeuli Rebulobs 10 karts da ori karti inaxeba 

“sayidlebSi”. rogoria albaToba imisa, rom “sayidlebSi” aRmoCndeba ori 

“tuzi”? 

amoxsna. ori kartis sxvadasxva kombinaciebis raodenoba, romelic 

2 

SeiZleba aRmoCndes “sayidlebSi” tolia C 

32 

496. kartis SekvraSi oTxi 

“tuzia”Dda sxvadasxva kombinaciebis raodenoba, romelic mogvcemda or 

2 

“tuzs”Atolia C 6 . Sesabamisad, saZiebeli albaToba tolia 

4 

 

11

C 

2 

4 6 

2 

496 

 

C32 

0,012 . 

davaleba. davuSvaT wina amocanaSi erTerTma moTamaSem, naxa ra Tavisi 

kartebi, icis, rom mas “tuzi” ara aqvs. Seicvleba Tu ara maSin alba- 

Toba imisa, rom “sayidlebSi” ori “tuzia”? gamoTvaleT es albaToba. 

magaliTi 14 (mxedvelobiT moZebnaze). davuSvaT gvaqvs N cali Sem- 

TxveviT dalagebuli geometriuli figura, romelTa Soris M cali mar- 

TkuTxedia ( M N ). moiTxoveba moiZebnos yvela marTkuTxedi, Tu Zebna warmoebs 

elementebis (figurebis) saTiTaod skanirebiT fiqsaciis moculobiT 

erTi elementi, amasTanave xdeba dakvirvebuli elementis poziciis 

damaxsovreba da mas Tavidan aRar vubrundebiT. vipovoT albaToba imisa, 

rom damkvirvebeli SeZlebs aRmoaCinos yvela M marTkuTxedi araumetes 

n dakvirvebisas ( n M,..., 

N )? 

amoxsna. yvela SesaZlo elementarul xdomilebaTa raodenoba iqneba 

| | C N 

n 

. xelSemwyobi elementaruli xdomilebebi iseTi [ a 

1 

,...,a 

n ] erTobliobebia, 

romlebSic M adgilas ganTavsebulia marTkuTxedebi (amis 

SesaZleblobaTa raodenobaa C ), xolo danarCen N M adgilas ki ara 

marTkuTxedebi (amis SesaZleblobaTa raodenobaa 

). amitom xelSemwyobi 

elementaruli xdomilebebis raodenoba iqneba 

ad, saZiebeli albaToba tolia 

M nM 

nM 

M 

CM 

CNM 

CNM 

Cn 

PM 

( n) 

. 

n 

n M 

C C C 

M 

M 

N 

amocanebi 

N 

N 

n M 

C N 

M 

M 

C M 

n M 

C 

. Sesabamis- 

1. saTamaSo “ruleti” Sedgeba 38 toli farTobis mqone seqtorisagan, 

romelTagan 18 Savia, 18 wiTelia da 2 ki mwvane. agdeben burTs, romelic 

sabolood Cerdeba erT-erT seqtorSi. ipoveT albaToba imisa, rom burTi: 

a). gaCerdeba wiTel seqtorSi; b). ar gaCerdeba Sav seqtorSi. 

2. ipoveT albaToba imisa, rom SemTxveviT SerCeuli erTi cifri iqneba 

3-is jeradi. 

3. ipoveT albaToba imisa, rom SemTxveviT SerCeuli ori cifri: a). 

ar daemTxveva erTmaneTs; b). daemTxveva erTmaneTs. 

4. ipoveT albaToba imisa, rom SemTxveviT SerCeuli sami cifridan: 

a). samive gansxvavebulia; b). samive daemTxveva erTmaneTs; g). ori mainc daemTxveva 

erTmaneTs; d). mxolod ori daemTxveva erTmaneTs. 

5. vipovoT albaToba imisa, rom ori saTamaSo kamaTlis gagorebisas 

qulebis jami iqneba: a). 9 qula; b). 10 qula. 

6. vipovoT albaToba imisa, rom sami saTamaSo kamaTlis gagorebisas 

qulebis jami iqneba: a). 9 qula; b). 10 qula. 

7. 60 sagamocdo sakiTxidan studentma icis 50 sakiTxi. ipoveT alba- 

Toba imisa, rom studenti upasuxebs orsakiTxiani bileTis orive sakiTxs. 

N 

M 

12

8. yuTSi aris 11 wiTeli, 12 yviTeli, 10 TeTri da 17 Savi burTi. ipoveT 

albaToba imisa, rom SemTxveviT amoRebuli burTi iqneba wiTeli an 

Savi. 

9. 1000 dadebiTi mTeli ricxvidan SemTxveviT irCeven erTs. ipoveT 

albaToba imisa, rom es ricxvi iqneba: a). 4-is jeradi; b). erTdroulad 4- 

isa da 6-is jeradi. 

10. kalaTaSi devs 7 wiTeli da 4 TeTri diski. dabrunebis gareSe 

irCeven or disks. ipoveT albaToba imisa, rom: a). orive diski wiTelia; b). 

orive diski erTi da igive ferisaa; g). diskebi sxvadasxva ferisaa; d). meore 

diski wiTelia; e). meore diski TeTria. 

11. CanTaSi devs 6 wiTeli da 4 mwvane kalkulatori. iReben or kalkulators 

dabrunebis gareSe. ipoveT albaToba imisa, rom: a). orive kalkulatori 

wiTelia; b). orive mwvanea; g). zustad erTi kalkulatori wiTelia; 

d). erTi mainc kalkulatori wiTelia; e). meore kalkulatori wiTelia. 

12. 52 kartidan iReben or karts dabrunebis gareSe. ipoveT albaToba 

imisa, rom: a). orive karti suraTebiania ( K, Q, 

J ); b). arcerTi ar aris 

suraTebiani; g). erTi mainc suraTebiania; d). erTi mainc wiTelia. 

13. CanTaSi devs 6 wiTeli da 4 mwvane kalkulatori. SemTxveviT iReben 

erT kalkulators, mis fers iniSnaven da mas abruneben CanTaSi. Semdeg 

SemTxveviT iReben meore kalkulators. ipoveT albaToba imisa, rom: 

a). orive kalkulatori wiTelia; b). orive mwvanea; g). zustad erTi wiTelia; 

d). erTi mainc wiTelia; e). meore wiTelia. 

14. A yuTi Seicavs 1 wiTel da 1 Sav burTs, B yuTi Seicavs 2 wiTel 

burTs. A yuTidan SemTxveviT iReben 1 burTs da gadaaqvT B yuTSi. 

Semdeg B yuTidan SemTxveviT iReben 1 burTs da gadaaqvT A yuTSi. ipoveT 

albaToba imisa, rom am ori gadatanis Semdeg Savi burTi iqneba A yu- 

TSi. 

15. A yuTi Seicavs 6 wiTel da 2 mwvane burTs, B yuTi Seicavs 4 wi- 

Tel da 3 mwvane burTs. agoreben wesier kamaTels. Tu movida luwi qula, 

maSin A yuTidan SemTxveviT iReben 1 burTs, xolo winaaRmdeg SemTxveva- 

Si B yuTidan SemTxveviT iReben 1 burTs. ipoveT albaToba imisa, rom: a). 

amoRebuli burTi wiTelia; b). burTi amoRebuli iyo B yuTidan, Tu cnobilia, 

rom amoRebuli burTi wiTelia. 

16. ipoveT albaToba imisa, rom erTeulovan gverdian kvadratSi Sem- 

TxveviT SerCeuli wertili am kvadratis uaxloesi gverdidan daSorebuli 

iqneba araumetes 0.15-iT. 

17. ori signali mimReb mowyobilobaze T drois manZilze SemTxveviT 

momentSi miiReba. mowyobiloba maT ganarCevs, Tu isini t droiT mainc 

arian dacilebuli erTmaneTs. ipoveT albaToba imisa, rom orive signali 

miRebuli iqneba? 

18. centris garSemo mbrunavi wre gayofilia sami toli farTobis 

mqone seqtorad da gadanomrilia cifrebiT 1, 2 da 3. mas atrialeben orjer 

da yoveli gaCerebisas iniSnaven im seqtoris nomers, romelSic moxvdeba 

am mowyobilebaze uZravad mimagrebuli isari. ipoveT albaToba imi- 

13

sa, rom: a). orive nomeri iqneba erTidaigive; b). arcerTi nomeri ar iqneba 

2; g). erTi nomeri mainc iqneba 3; d). arcerTi nomeri araris 2 da orive 

erTidaigivea; e). arcerTi nomeri araris 2 an orive erTidaigivea. 

19. 36 kartidan A moTamaSes aqvs ori dedofali da erTi mefe. A 

moTamaSisagan B moTamaSe SemTxveviT iRebs erT karts da ubrunebs isev 

mas. A moTamSe erTmaneTSi urevs am sam karts da Semdeg B moTamaSe 

kvlav iRebs erT karts. B moTamaSe igebs Tu mis mier amoRebuli orive 

karti iqneba mefe. ipoveT B moTamaSis mogebis albaToba. 

20. wesieri tetraedris waxnagebi gadanomrilia cifrebiT 1, 2, 3 da 

4. mas agdeben orjer da iweren im cifrs, romliTac tetraedri daecema 

iatakze. ipoveT albaToba imisa, rom: a). orive cifri iqneba erTidaigive; 

b). arcerTi cifri ar iqneba 3; g). erTi cifri mainc iqneba 3; d). arcerTi 

cifri ar aris 4 da orive erTidaigivea; e). arcerTi cifri ar aris 4 an 

orive erTidaigivea. 

21. yuTSi devs 10 burTi gadanomrili 1-dan 10-mde ricxvebiT. ipoveT 

albaToba imisa, rom SemTxveviT amoRebul 6 burTSi aRmoCndeba: a). burTi 

# 9; b). burTi # 9 da #10. 

22. 200 detalis Semowmebisas standartulebis fardobiTi sixSire 

aRmoCnda 0.9. ipoveT standartuli detalebis raodenoba. 

23. sibrtyeze, romelic dafarulia a gverdis mqone kvadrtTa badiT, 

SemTxveviT agdeben monetas radiusiT r 

a/2. ipoveT albaToba imisa, 

rom moneta ar gadakveTs arcerTi kvadratis gverds. 

24. sibrtyeze, romelic dafarulia 6 sm-iT daSorebuli paraleluri 

wrfeebiT, SemTxveviT agdeben wres radiusiT 1 sm. ipoveT albaToba 

imisa, rom wre ar gadakveTs arcerT wrfes. 

25. SemTxveviT iReben or dadebiT ricxvs, romelTagan TiToeuli ar 

aRemateba 2-s. ipoveT albaToba imisa, rom maTi namravli ar aRemateba 1-s, 

xolo ganayofi ki ar aRemateba 2-s. 

pasuxebi: 

1. 9/19; 10/19. 2. 3/10. 3. 9/10; 1/10. 4. 18/25; 1/100; 14/50; 27/100. 5. 1/9; 1/12. 6. 25/216; 

1/8. 7. 245/354. 8. 14/25. 9. 1/4; 83/1000. 10. 21/55; 27/55; 28/55; 7/11; 4/11. 11. 1/3; 2/15; 

8/15; 13/15; 3/5. 12. 11/221; 10/17; 7/17; 77/102. 13. 9/25; 4/25; 12/25; 21/25; 3/5. 14. 2/3. 15. 

37/56; 16/37. 18. 1/3; 4/9; 5/9; 2/9; 5/9. 19. 1/9. 20. 1/4; 9/16; 7/16; 3/16; 5/8. 21. 0.6; 1/3. 

2 2 

22. 180. 23. ( a 2 r) / a . 24. 2/3. 25. (1 3ln2)/8 0.38 . 

14

Tavi II 

rTuli xdomilebis albaTobebi 

albaTobaTa Sekrebis kanoni: Tu AB 

Ø , maSin 

P( A B) 

P( 

A) 

P( 

B) 

. 

sawinaaRmdego xdomilebis albaToba: P( A) 

1 

P( 

A) 

. 

sxvaobis albaTobis formula: Tu B A, maSin P( A\ 

B) 

P( 

A) 

P( 

B) 

. 

Tu 

A A Ø, roca i j , maSin: 

i 

j 

n 

n 

1 i 

 

i 

i1 

i 

. 

P ( A ) P ( A ) 

roca xdomilebebi Tavsebadia: P( A B) 

P( 

A) 

P( 

B) 

P( 

A 

B) 

. 

P( ABC) P( A) P( B) P( C) P( AB) P( AC) P( BC) P( AB 

C) 

sazogadod: 

( 

n 

) n 

( ) ( ) ( ) ( 1) 1 

( n 

n ) 

i i i j i j k i 

i1 i1 

i1 

i j i jk 

. 

P A P A P A A P A A A P A 

pirobiTi albaTobis formula 

A xdomilebis pirobiTi albaToba pirobaSi, rom adgili hqonda B 

xdomilebas, aRiniSneba P ( A| 

B) 

(an PB 

( A )) simboloTi da 

P( A| B): P( A B)/ P( B) 

, Tu P ( B) 

0. 

0 P ( A| 

B) 

1; A B=Ø P ( A| 

B) 

0; B A P ( A| 

B) 

1; 

Tu 

A A Ø, roca i j , maSin: 

i 

j 

n 

n 

1 i 

 

i 

i1 

i 

. 

P [( A )| B ] P ( A | B ) 

namravlis albaTobis formula 

P( AB) P( A) P( B| A) P( B) P( A| B) 

. 

P( ABC) P( A) P( B| A) P[ C | ( A B)] 

. 

sazogadod: P( A1 A2 An ) P( A1 ) P( A2 | A1 ) P( An | A1 An 

1) 

. 

damokidebuli da damoukidebeli xdomilebebi 

A xdomilebas ewodeba B xdomilebisagan damoukidebeli, Tu 

P( A| 

B) 

P( 

A) 

an rac igivea P( A B) 

P( 

A) 

P( 

B) 

. Tu P( A| 

B) 

P( 

A) 

, maSin 

gvaqvs damokidebuli xdomilebebi. Tu A da B xdomilebebi damoukidebelia, 

maSin xdomilebebi A da B agreTve damoukidebelia. 

xdomilebaTa erTobliobas A 

1 

, A2 

,..., An 

ewodeba wyvil-wyvilad damoukidebeli 

Tu: P( A A ) P( 

A ) P( 

A ), i 

j . 

i 

j 

i 

j 


1 

, A2 

,..., An 

ewodeba erToblivad 

damoukidebeli Tu k n , i1 i2 ik 

: 

P ( Ai 

A ) ( ) ( ) ( ) 

1 i 

A 

2 

i 

P A 

k i 

P A 

1 i 

 

P A 

2 

i 

. 

k 

sruli albaTobis formula 


1 

, A2 

,..., An 

ewodeba xdomilebaTa sruli 

sistema, Tu 

i 

j 

n 

 

i 1 

A A Ø, roca i j da A (magaliTad, A da A ). 

i 

15

Tu 

A ,..., 

1 

, A2 

An 

xdomilebaTa sruli sistemaa ( ( 

i) 0, 

n 

16 

PA i 1,2,..., n), 

maSin adgili aqvs sruli albaTobis formulas: P( 

B) 

P( 

A ) P( 

B | A ) . 

baiesis formula 

Tu A 

1 

, A2 

,..., An 

xdomilebaTa sruli sistemaa, P( A i 

) 0, 

maSin adgili aqvs baiesis formulas: 

P( Ai) P( B| Ai) 

P( Ai 

| B) 

. 

n 

P( A ) P( B| A ) 

j1 

j 

j 

 

i1 

i 

i 1,2 ,..., n , 

ganmeorebiTi cdebi. bernulis formula 

ganvixiloT erTi da igive eqsperimentebis seria, romlebic tardeba 

erTi da igive pirobebSi erTmaneTisagan damoukideblad. amasTanave, yovel 

konkretul eqsperimentSi Cven ganvasxvavebT mxolod or Sedegs: garkveuli 

A xdomilebis moxdena (romelsac pirobiTad “warmatebas” uwodeben) 

da misi ar moxdena -- A (romelsac “marcxs” uwodeben). A xdomilebis 

moxdenis albaToba nebismieri eqsperimentisaTvis mudmivia da tolia 

P( A) 

p , sadac 0 p 1. Sesabamisad, P( A) 

1 

P( 

A) 

1 

p: 

q ( p q 1). 

albaTobas imisa, rom n eqsperimentSi A xdomileba moxda k -jer 

gamoiTvleba e. w. bernulis formulT: 

k k nk n! 

k nk 

Pn() 

k Cnp q p q . 

k!( nk)! 

( nk) 

p 

amasTanave, adgili aqvs Tanafardobas: Pn( k 1) Pn( k) 

. 

( k1) 

q 

albaTobebis erTobliobas Pn 

( k ), roca k 0,1,..., n ewodeba albaTobebis 

binomialuri ganawileba. 

iseT k 0 

ricxvs, romlis Sesabamisi albaToba Pn( k 

0) 

udidesia P n(0) 

, 

P n(1) ,..., P n 

(n) 

albaTobebs Soris ualbaTesi ricxvi ewodeba. ualbaTesi ricxvi 

gviCvenebs n damoukidebel cdaSi warmatebaTa ra raodenobaa yvelaze 

ufro mosalodneli. ualbaTesi ricxvi warmoadgens Semdegi utolobis 

mTel amonaxsns: np q k0 

np p . 

puasonis formula 

Tanafardobas lim ( ) k 

p k e / k! 

puasonis formula ewodeba. igi sa- 

np 

n 

Sualebas iZleva vipovoT n damoukidebel cdaSi A xdomilebis k -jer 

moxdenis albaToba (roca n sakmaod didia, xolo p sakmaod mcire, amas- 

Tanave np 

15 ) puasonis miaxloebiTi formuliT: p ( k) k e / k! 

. 


magaliTi 1. ganvixiloT ojaxebi, sadac or-ori bavSvia. rogoria 

albaToba imisa, rom ojaxSi orive bavSvi vaJia Tu cnobilia, rom: a). 

ufrosi bavSvi – vaJia; b). erTi bavSvi mainc – vaJia? 

n 

i

amoxsna. aq elementarul xdomilebaTa sivrce aseTia 

{ vv, vq, qv, qq }, 

sadac “v” aRniSnavs vaJs, xolo “q” – qals. CavTvaloT, rom oTxive 

Sedegi tolalbaTuria. SemoviRoT xdomilebebi: A -- iyos xdomileba, 

rom ufrosi bavSvi -- vaJia, xolo B -- iyos xdomileba, rom umcrosi 

bavSvi – vaJia. maSin A B -- iqneba xdomileba, rom orive bavSvi vaJia, 

xolo A B -- ki iqneba xdomileba, rom erTi bavSvi mainc vaJia. 

Sesabamisad, saZiebeli albaTobebi iqneba: a). P( A B | A) 

da b). 

P( A 

B | A 

B) 

. advili dasanaxia, rom: 

P[( 

A 

B) 

A] 

P( 

A 

B) 

1/ 4 1 

P ( A 

B | A) 

 

, 

P( 

A) 

P( 

A) 

1/ 2 2 

P[( 

A 

B) 

( 

A 

B)] 

P( 

A 

B) 

1/ 4 1 

P ( A 

B | A 

B) 

 

. 

P( 

A 

B) 

P( 

A 

B) 

3/ 4 3 

magaliTi 2 (saukeTesos SerCevaze). mocemulia m obieqti gadanomrili 

ricxvebiT 1 ,2,..., m , amasTanave ise, rom vTqvaT, obieqti #1 klasificirdeba 

rogorc “saukeTeso”, . . . , obieqti # m ki rogorc “yvelaze uaresi”. 

igulisxmeba rom obieqtebi Semodian drois momentebSi 1 ,2,..., m SemTxveviTi 

rigiT (anu yvela SesaZlo m! gadanacvleba tolalbaTuria). damkvirvebels 

SeuZlia ori maTganis SedarebiT Tqvas romelia ukeTesi da 

romeli uaresi. saWiroa saukeTesos SerCeva im pirobiT rom obieqtebi warmoidgineba 

saTiTaod da ukugdebulis damaxsovreba xdeba damkvirveblis 

mier. ar SeiZleba saukeTesod miCneul iqnes is obieqti, romelic dakvirvebuli 

obieqtebidan erTze mainc uaresia an romelic ukve iqna ukugdebuli. 

vTqvaT, damkvirvebelma obieqti SearCia k -ur nabijze ( k m), anu 

daTvalierebuli obieqtebidan ukanaskneli aRmoCnda yvela winaze ukeTesi 

da amitom moxda misi SerCeva. rogoria albaToba imisa, rom amorCeuli 

obieqti iqneba saukeTeso mTeli erTobliobidan rogorc ukve ganx-ilul, 

ise jer kidev ganuxilav obieqtebs Soris? 

amoxsna. SemoviRoT xdomilebebi: A iyos xdomileba, rom k -uri obieqti 

saukeTeoa yvela arsebul m obieqts Soris da B iyos xdomileba, 

rom k -uri obieqti saukeTeoa dakvirvebul k obieqts Soris. gasagebia, 

rom mosaZebnia pirobiTi albaToba P ( A| 

B) 

. 

vinaidan A B, amitom A B A da P( A 

B) 

P( 

A) 

. Sesabamisad, pirobiTi 

albaTobis ganmartebis Tanaxmad 

P( A| 

B) 

P( 

A)/ 

P( 

B) 

vinaidan obieqtebis yvela SesaZlo gadanacvlebebi tolalbaTuria, 

amitom albaTobis klasikuri ganmartebis Tanaxmad advili dasanaxia, rom 

( k 1)! 

1 

( m1)! 

1 

P( B) 

da P( A) 

. 

k! 

k 

m! 

m 

Sesabamisad, P ( A| 

B) 

k / m . 

strategia. SeiZleba damtkicdes, rom saukeTeso obieqtis amorCevis 

optimaluri strategia mowyobilia Semdegnairad. avRniSnoT simboloTi 

* 

m iseTi naturaluri ricxvi, romlisTvisac samarTliania utoloba 

17

1 1 1 1 

1 

. 

* 

* 

m m1 

m 1 

m1 

saukeTeso obieqtis arCevis optimaluri strategia mdgomareobs ima- 

Si, rom davakvirdeT da ukuvagdoT pirveli m * 1 

obieqti da Semdeg gavagrZeloT 

dakvirveba iseT * momentamde, roca pirvelad gamoCndeba yvela 

winamorbedze ukeTes obieqti. 

magaliTad, Tu m 1,..., 10, maSin m * -is Sesabamisi mniSvnelobebia: 

m 1 2 3 4 5 6 7 8 9 10 

m-optimaluri 1 1 1 1 2 2 2 3 3 4 

sakmaod didi m-isaTvis m * m/ 

e (sadac e -- neperis ricxvia, 

e 2.718 ) da saukeTeso obieqtis arCevis albaToba daaxloebiT tolia 

1/ e 0.368 (Tumca, erTi SexedviT bunebrivia mogvCveneboda, rom gansaxilveli 

obieqtebis m raodenobis zrdasTan erTad, saukeTeso obieqtis arCevis 

albaToba unda wasuliyo nulisaken). e. i. saukeTeso obieqtis arCevis 

optimaluri strategia mdgomareobs imaSi, rom unda ukuvagdoT obieqtebis 

saerTo raodenobis daaxloebiT mesamedi da Semdeg avirCioT pirveli 

iseTi obieqti, romelic yvela winaze ukeTesia. 

magaliTi 3 (“bednier” bileTebze). 25 sagamocdo bileTidan 5 “bednieria”, 

xolo danarCeni 20 – “ara bednieri”. romel students aqvs “bednieri” 

bileTis aRebis meti albaToba: vinc pirveli iRebs bileTs, Tu vinc 

meore iRebs bileTs? 

amoxsna. es amocana Cven ukve amovxseniT albaTobis klasikuri ganmartebis 

gamoyenebiT. amovxsnaT axla igi elementarul xdomilebaTa sivrcis 

axleburi SemotaniTa da namravlis albaTobis formulis gamoyenebiT. 

winaswar SemoviRoT Semdegi xdomilebebi: A iyos xdomileba, rom pirvelma 

studentma aiRo bednieri bileTi, xolo B iyos xdomileba, rom 

meore studentma aiRo bednieri bileTi. maSin cxadia, rom elementarul 

xdomilebaTa sivrce Sedgeba oTxi xdomilebisagan 

{ AB, 

AB, 

AB, 

AB} 

. 

albaTobis klasikuri ganmartebis Tanaxmad P ( A) 

5/ 25 1/ 

5, xolo 

PA ( ) 20/25 4/5. meores mxriv, amocanis Sinaarsidan gamomdinare, Tu 

cnobilia, rom pirvelma studentma aiRo bednieri bileTi, maSin albaToba 

imisa rom meore studenti aiRebs bednier bileTs isev SeiZleba gamoviTvaloT 

albaTobis klasikuri ganmartebiT: am SemTxvevaSi yvela SesaZlo 

SedegTa raodenobaa 24, xolo xelSemwyob elementarul xdomilebaTa 

raodenoba ki mxolod 4 (radgan erTi bednieri bileTi ukve aRebulia) da 

Sesabamisad, 

P ( B | A) 

4/ 24 1/6. 

analogiurad, P ( B | A) 

20/24 5/ 6, P ( B| 

A) 

5/24 5/ 24 da 

P( B| A) 19/24 . amitom ori xdomilebis namravlis albaTobis formulis 

Tanaxmad gvaqvs: 

P ( A 

B) 

P( 

A) 

P( 

B | A) 

1/51/6 

1/30; P ( AB) 

1/55/6 

1/ 

6; 

18

P ( AB) 

4/55/24 

1/6 

da P ( AB) 

4/519/24 

19/ 

30 

(SevniSnavT, rom Cven aq mxolod sisrulisaTvis gamovTvaleT yvela 

SesaZlo pirobiTi da namravlis albaTobebi). 

cxadia, rom B ( AB) 

( 

AB) 

. amitom albaTobaTa Sekrebis kanonis 

Tanaxmad 

P( B) 

P( 

AB) 

P( 

AB) 

1/301/6 

1/5 ( P( 

A)) 

. 

magaliTi 4. yuTSi m burTia, maT Soris n TeTria. vipovoT albaToba 

imisa, rom yuTidan ori burTis mimdevrobiT dabrunebis gareSe amoRebisas: 

a). pirveli burTi TeTria; b). meore burTi TeTria; g). orive bur- 

Ti TeTria. 

amoxsna. A i 

iyos xdomileba, rom i -uri burTi TeTria ( i 1, 2 ). maSin 

albaTobis klasikuri ganmartebis Tanaxmad: 

a). P ( A 1 

) n/ 

m. 

garda amisa, 

P ( A2 | A1 

) ( n1)/( 

m1) 

da P ( A2 | A1 

) n/( 

m1) 

. 

amitom namravlis albaTobis formulis Tanaxmad: 

g). P ( A1 A2 

) P( 

A1 

) P( 

A2 

| A1 

) n( 

n1)/ 

m( 

m1) 

. 

analogiurad, 

P ( A1 A2 

) P( 

A1 

) P( 

A2 

| A1 

) n( 

mn)/ 

m( 

m1) 

. 

amitom: 

b). P ( A2 ) P[( 

A1 

A2 

) ( 

A1 

A2 

)] P( 

A1 

A2 

) P( 

A1 

A2 

) n/ 

m. 

magaliTi 5. yuTs aqvs n ujra. albaToba imisa rom burTi aris am 

ujrebidan erT-erTSi tolia p -si. ipoveT albaToba imisa, rom burTi aris 

i -ur ujraSi, Tu cnobilia, rom burTi TiTeul ujraSi SesaZlebelia 

iyos Tanabari albaTobebiT? 

amoxsna. A iyos xdomileba, rom burTi aris i -ur ujraSi. A iyos 

am xdomilebebis gaerTianeba 

i 

A 

n 

A i 

i1 

19 

. pirobis Tanaxmad 

P( A) 

p da P( A i 

| A) 

1/ 

n. 

amitom 

P( Ai ) P( 

Ai 

A) 

P( 

A) 

P( 

Ai 

| A) 

p1/ 

n p/ 

n. 

magaliTi 6. kartebis nakrebidan (romelSic 36 kartia) SemTxveviT 

iReben erT karts. ganvixiloT xdomilebebi: A iyos xdomileba, rom amo- 

Rebuli karti “agurisa”, xolo B iyos xdomileba, rom amoRebuli karti 

“mefea”. arian Tu ara es xdomilebebi damoukidebeli? 

cxadia, rom am SemTxvevaSi 

| | 36, P ( A) 

9/36 1/ 

4, P ( B) 

4/36 1/ 

9 da 

P( A B) 

1/36 

1/ 41/9 

P( 

A) 

P( 

B) 

. 

e.i. es xdomilebebi damoukidebelia. 

magaliTi 7. davuSvaT, rom vagorebT or saTamaSo kamaTels. ganvixiloT 

xdomilebebi: A -- pirvel kamaTelze movida kenti qula, B -- meore

kamaTelze movida kenti qula, C -- orive kamaTelze mosul qulaTa jami 

kentia. gavarkvioT am xdomilebebis damoukideblobis sakiTxi. 

cxadia, rom P ( A) 

P( 

B) 

3/6 1/ 

2, xolo P ( A B) 

33/36 

1/ 

4. 

amitom A da B xdomilebebi damoukideblia. 

davaleba. SeamowmeT, rom P ( C) 

1/ 

2. 

SevniSnoT, rom A da B xdomilebidan erT-erTis moxdenis 

pirobaSi C xdomileba xdeba maSin da mxolod maSin, roca Sesabamisad, 

an pirvel an meore kamaTelze movida luwi qula, anu gvaqvs 

Tanafardobebi: 

A C AB 

da B C AB. 

A da B xdomilebebis damoukideblobidan gamomdinare xdomilebebi 

A da B da A da B agreTve damoukideblebia, amitom 

P ( AB) 

P( 

A) 

P( 

B) 

1/4 

da P ( AB) 

1/ 

4. 

Sesabamisad, 

P ( AC) 

P( 

AB) 

1/4 

da P ( BC) 

P( 

AB) 

1/ 

4 . 

es Tanafardobebi ki, P ( C) 

1/ 

2 albaTobis gaTvaliswinebiT, niSnavs, 

rom damoukideblebia A da C da B da C xdomilebaTa wyvilebic. 

davaleba. SeamowmeT, rom xdomilebebi magaliTi 7-dan ar arian erToblivad 

damoukidebeli. 

magaliTi 8. davuSvaT, vagdebT sam monetas. SemoviRoT xdomilebebi: 

A 

1 

-- pirveli da meore moneta daeca erTi da igive mxareze; 

A 

2 

-- meore da mesame moneta daeca erTi da igive mxareze; 

A 

3 -- pirveli da mesame moneta daeca erTi da igive mxareze. 

advili Sesamowmebelia, rom aqedan nebismieri ori xdomileba 

damoukidebelia, xolo samive erTad damokidebulia, vinaidan Tu Cven 

gvecodineba rom magaliTad, A 1 

da A 2 

moxda, maSin Cven zustad viciT, 

rom A 3 agreTve moxda. 

davaleba. SeamowmeT, rom xdomilebebi A 1 

, A 2 

da A 3 wyvil-wyvilad 

damoukidebelia. 

magaliTi 9 (dabadebis dReebze). vipovoT albaToba imisa, konkretuli 

skolis rom 150 moswavlidan erTi mainc dabadebulia mocemul fiqsirebul 

dRes, magaliTad pirvel seqtembers? 

amoxsna. SemoviRoT xdomilebebi: 

A i 

{ i uristudentidabadebulia1.09} 

, i 1,2,..., 150; 

A { erTimainc150studentida ndabadebuli a1.09} . 

naTelia, rom A xdomilebebi erToblivad damoukidebelia da 

i 

A i 

A 150 i1 

P ( ) 1/365. garda amisa, Ai 

da maSasadame, sapovnelia damoukidebel 

xdomilebaTa gaerTianebis albaToba. gadavideT sawinaaRmdego 

xdomilebaze da visargebloT de-morganis kanoniT. maSin gvaqvs: 

20

amdenadac 365, Tu visargeblebT niutonis binomis formuliT 

150 

150 

( 

i 

i 

1 

i1 

i1 

P ( A i 

) 11/ 

P A) 

1 

P( 

A) 

1 

P( 

A ) 1 

P( 

A ) 1 

P( 

A ) 

P( 

A ). 

n 

 

n 

j j 

( 1 

x) 

C x , miviRebT, rom: 

j1 

n 

150 2 1 2 3 1 3 4 1 4 

P ( A) 

C150 ( ) C150( 

) C150( 

) . 

365 365 365 365 

vinaidan, 150/365 0, 41, xolo 

4 

4 1 4 150 4 1 (0,41) 

C150( ) ( ) 0.005 , 

365 365 24 24 

amitom mwkrivis niSancvladobis gamo, Tu gadavagdebT mwkrivis 

wevrebs dawyebuli me-5 wevridan (mwkrivebis zogadi Teoriidan 

gamomdinare), SesaZlebelia vamtkicoT, rom 

2 3 

(0.41) (0.41) 

PA ( ) 0.41 0.41 0.08 0.01 0.34. 

2 6 

magaliTi 10 (asoebis amocnoba). gvaqvs asoebis ori erToblioba: 

I={К, И, З, Н} da II={З, Ч, Н}. 

SemTxveviT virCevT erT erTobliobas da arCeuli erTobliobidan 

erT asos. amorCeul asos Zalian mcire drois ganmavlobaSi vuCvenebT 

damkvirvebels (ise rom mas ar SeuZlia mTlianad aRiqvas aso). rogoria 

asos sworad gamocnobis albaToba, Tu damkvirvebelis pasuxia “Н”, roca 

is asos gamosaxulebaSi dainaxavs vertikalur xazs da pasuxia “З”, roca 

asos gamosaxulebaSi vertikaluri xazi ar aris? 

amoxsna. SemoviRoT xdomilebebi: 

A i 

{ amorCeulia i urierToblioba 

, i 1,2} 

; 

“К”, “И”, “З”, “Н” da “Ч” – iyos xdomileba, rom warmodgenilia Sesabamisad 

К, И, З, Н da Ч asoebi; 

B { damkvirveb elmasworad upasuxa }. 

amocanis pirobebSi cxadia: 

P ( A1 ) P( 

A2 

) 1/2; 

P ( ―К‖| A 

1)= P ( ―И‖| A 

1)= P ( ―З‖| A 

1)= P ( ―Н ‖| A 

1)=1/4, P ( ―Ч‖| A 

1)=0; 

P ( ―З‖| A 

2)= P ( ―Ч‖| A 

2)= P ( ―Н‖| A 

2 

)=1/3, P ( ―К‖| A 

2 

)= P ( ―И‖| A 

2 

)=0. 

vinaidan A 1 

da A 2 

qmnian xdomilebaTa srul sistemas, amitom TiToeuli 

asos sworad amocnobis albaToba SegviZlia gamovTvaloT sruli 

albaTobis formuliT: 

P ( ―К‖)= P ( A 1 

) P ( ―К‖| A 

1)+ P ( A 2 

) P ( ―К‖| A 

2)=1/8; 

P ( ―И‖)= P ( A 1 

) P ( ―И‖| A 

1)+ P ( A 2 

) P ( ―И‖| A 

2 

)=1/8; 

P ( ―З‖)= P ( A 1 

) P ( ―З‖| A 

1)+ P ( A 2 

) P ( ―З‖| A 

2)=7/24; 

P ( ―Н‖)= P ( A 1 

) P ( ―Н‖| A 

1)+ P ( A 2 

) P ( ―Н‖| A 

2 

)=7/24; 

P ( ―Ч‖)= P ( A 1 

) P ( ―Ч‖| A 

1)+ P ( A 2 

) P ( ―Ч‖| A 

2 

)=1/6. 

n 

21

amocanis pirobebSi cxadia agreTve, rom swori pasuxis pirobiTi 

albaTobebi sxvadasxva asoebis warmodgenis SemTxvevaSi Sesabamisad iqneba: 

P( 

B |―К‖)= P( 

B |―И‖)= P( 

B |―Ч‖)=0 da P( 

B |―З‖)= P( 

B |―Н‖)=1. 

vinaidan, “К”, “И”, “З”, “Н” da “Ч” agreTve xdomilebaTa sruli sistemaa, 

amitom sruli albaTobis formula gvaZlevs swori pasuxis albaTobas: 

P (B) P ( ―К‖) P( 

B |―К‖)+ P ( ―Н‖) P( 

B |―Н‖)+ P ( ―И‖) P( 

B |―И‖)+ 

+ P ( ―З‖) P( 

B |―З‖)+ P ( ―Ч‖) P( 

B |―Ч‖)= P ( ―Н‖)+ P ( ―З‖)=7/12. 

magaliTi 11 (moTamaSis gakotrebaze). ganvixiloT e. w. “gerbi-safasuris” 

TamaSi: Tu monetis agdebisas mova moTamaSis mier winaswar dasaxelebuli 

monetis mxare, maSin igi igebs 1 lars, winaaRmdeg SemTxvevaSi ki 

agebs 1 lars. vTqvaT, moTamaSis sawyisi kapitali Seadgens x lars da misi 

mizania miiyvanos es Tanxa a laramde. TamaSi grZeldeba manam sanam 

moTamaSe ar miiyvans Tavis Tanxas winaswar gansazRrul a laramde, an 

igi ar gakotrdeba (anu waagebs mis xelT arsebul mTel x lars). rogoria 

albaToba imisa, rom moTamaSe gakotrdeba? 

amoxsna. es albaToba damokidebuli iqneba sawyis x kapitalze. aRvniSnoT 

igi p (x) 

simboloTi. cxadia, rom igi ganmartebulia nebismieri 

0 x a da amasTanave, P ( 0) 1 

da P ( a) 

0. SemoviRoT xdomilebebi: 

A 

1 

{ 

moTamaSem moigo pirvel nabijze} 

, 

B { moTamaSe, romelsacgaaCniasawyisikapitali x, 

gakotrdeba }. 

amocanis pirobebSi gvaqvs: 

P ( A1 ) P( 

A1 

) 1/2 

, P ( B| 

A1 ) p( 

x 1) 

da P ( B| 

A1 ) p( 

x 1) 

( 1 

x a 1). 

vinaidan, A 1 

da A 1 

xdomilebaTa sruli sistemaa, amitom sruli albaTobis 

formula p (x) 

albaTobisaTvis gvaZlevs Semdeg gantolebas: 

1 1 

p ( x) 

p( 

x 1) 

p( 

x 1) 

, 1 

x a 1 

2 2 

(am tipis gantolebebs maTematikaSi rekurentul gantolebebs 

uwodeben). SeiZleba Semowmdes, rom am gantolebis amoxsnas aqvs saxe: 

p( x) 

bx c , 

sadac b da c -- nebismieri mudmivebia. am koeficientebis mosaZebnad unda 

visargebloT sasazRvro pirobebiT P ( 0) 1 

da P ( a) 

0. maSin miviRebT, 

rom 

c 1 da ab c 0, 

saidanac, b 1/ 

a da sabolood p ( x) 

1 

x/ 

a , 0 x a . 

davaleba. davuSvaT, rom Sesamowmebeli jgufis 1% avadmyofia, xolo 

danarCeni 99% ki janmrTelia. adamianebis SerCeva xdeba SemTxveviT 

da amitom 

P( avadmyofi ) 1% 0.01 

da P( janmrTeli ) 99% 0.99 

. 

22

vigulisxmoT, rom im SemTxvevaSi, roca testireba utardeba 

adamians, romelsac ara aqvs avadmyofoba, maSin 1%-ia albaToba imisa, 

rom miviRoT mcdari dadebiTi Sedegi, e.i. 

P ( dadebiTi| janmrTeli ) 1% 

da P ( uaryofiTi| janmrTeli ) 99% 

. 

dabolos, davuSvaT, rom im SemTxvevaSi, roca testireba utardeba 

avadmyof adamians, maSin 1%-ia albaToba imisa, rom miviRoT mcdari 

uaryofiTi Sedegi, e.i. 

P ( uaryofiTi| avadmyofi ) 1% 

da P ( dadebiTi| avadmyofi ) 99% 

. 

gamoTvaleT albaToba imisa, rom: a). adamiani janmrTelia, xolo 

testma aCvena uaryofiTi Sedegi; b). adamiani avadmyofia, xolo testma 

aCvena dadebiTi Sedegi; g). adamiani janmrTelia, xolo testma aCvena 

dadebiTi Sedegi; g). adamiani avadmyofia, xolo testma aCvena uaryofiTi 

Sedegi. 

ganvixiloT realuri situacia, romelic gviCvenebs erTi SexedviT 

moulodnel gansxvavebas P ( A| 

B) 

da P ( B | A) 

pirobiT albaTobebs Soris. 

imisaTvis, rom gamovavlinoT seriozuli avadmyofobis mqone adamianebi 

adreul stadiaze, xdeba adamianebis didi jgufis testireba. miuxedavad 

winaswari Semowmebis sargeblobisa, am midgomas gaaCnia uaryofiTi mxare: 

Tu adamians sinamdvileSi ar gaaCnia avadmyofoba da sawyisma testma aCvena 

dadebiTi Sedegi (daudgina avadmyofoba), is iqneba stresul mdgomareobaSi 

(rac Tavis mxriv uaryofiTad moqmedebs mis cxovrebaze) sanam ufro 

warmatebuli testi ar aCvenebs, rom is janmrTelia. am problemis mni- 

Svneloba SesaZlebelia kargad gavigoT pirobiTi albaTobebis termineb- 

Si. 

wina davalebis monacemebSi gamovTvaloT albaToba imisa, rom testi 

aCvenebs dadebiT Sedegs. sruli albaTobis formulis Tanaxmad: 

P( dadebiTi ) P ( janmrTeliP ) ( dadebiTi| janmrTeli ) 

P( avadmyofi ) P( dadebiTi| avadmyofi ) 0.99 0.01 0.01 0.99 0.0198 

. 

rogorc cnobilia, magaliTis pirobebSi 

P ( dadebiTi| avadmyofi ) 99% . 

gamovTvaloT axla Sebrunebuli pirobiTi albaToba, risTvisac visargebloT 

pirobiTi albaTobis ganmartebiTa da namravlis albaTobis 

formulebiT. maSin zemoT miRebuli 

P( dadebiTi ) 0.0198 1.98% 

Sedegis Tanaxmad: 

P( 

avadmyofi 

dadebiTi) 

P( 

avadmyofi| dadebiTi) 

 

 

PPP(dadebiTi) 

P( avadmyofi)P P( dadebiTi | avadmyofi ) 1% 99% 50% . 

PP1.98% 

1.98% 

rogorc vxedavT, pirobiTi albaToba imisa rom testi mogvcems dadebiT 

Sedegs, pirobaSi rom adamiani avadmyofia tolia 99%-is, maSin rodesac 

pirobiTi albaToba imisa rom adamiani avadmyofia, pirobaSi rom 

testma mogvca dadebiTi Sedegi aris mxolod 50%. aq SerCeuli monacemeb- 

23

is SemTxvevaSi ukanaskneli Sedegi SeiZleba CaiTvalos miuRebelad: naxevari 

adamianebis, romelTa testirebam aCvena dadebiTi Sedegi, faqtiurad 

aris janmrTeli. 

magaliTi 12. yuTSi moTavsebulia ori moneta: A 1 

-- simetriuli moneta 

gerbis mosvlis albaTobiT 1/2, da A 2 

-- arasimetriuli moneta gerbis 

mosvlis albaTobiT 1/3. SemTxveviT viRebT erT monetas da vagdebT. davu- 

SvaT, rom movida gerbi. rogoria albaToba imisa, rom amoRebuli moneta 

iyo simetriuli? 

amoxsna. am SemTxvevaSi elementarul xdomilebaTa sivrce iqneba: 

{( A1 , g),( 

A1 

, s),( 

A2 

. g),( 

A2 

, s)} 

, 

sadac magaliTad, ( A 1 

, g) 

-- niSnavs, rom amoviReT A 1 

moneta da misi agdebis 

Sedegad movida gerbi. amocanis pirobebSi gvaqvs: 

P ( A1 ) P( 

A2 

) 1/2, P ( g | A1 ) 1/ 

2 da P( g | A2 ) 1/ 

3. 

Sesabamisad, namravlis albaTobis formulis gamoyenebiT 

gamoviTvliT: 

P {( A 1 

, g)} 

1/ 

4, P {( A 1 

, s)} 

1/ 

4, P {( A 2 

, g)} 

1/ 

6 da P {( A 2 

, s)} 

1/ 

3. 

amitom baiesis formulis Tanaxmad 

P( 

A1 

) P( 

g | A1 

) 3 

P( 

A1 | g ) 

. 

P( 

A1 

) P( 

g | A1 

) P( 

A2 

) P( 

g | A2 

) 5 

cxadia, agreTve rom P ( A 2 

| g) 

2/ 5. 

magaliTi 13 (keTil gamomcdelze I). vTqvaT, Cven Casabarebeli 

gvaqvs gamocda da SegviZlia avirCioT nebismieri sami gamomcdelidan. 

davuSvaT, CvenTvis cnobilia, rom erTerTi sami gamomcdelidan (ucnobia 

romeli) -- “keTilia” da albaToba imisa, rom masTan Caabaro gamocda 

tolia 0.4-is, xolo danarCeni ori gamomcdeli “avia” da maTTan gamocdis 

Cabarebis albaToba tolia 0.1-is. Cven SemTxveviT avirCieT gamomcdeli da 

warmatebiT CavabareT gamocda. rogoria labaToba imisa, rom Cven 

avirCieT “keTili” gamomcdeli? 

amoxsna. SemoviRoT Semdegi xdomilebebi: A -- amorCeuli gamomcdeli 

“keTilia” (maSin A -- iqneba xdomileba, rom amorCeuli gamomcdeli 

“avia”) da B -- gamocda Cabarebulia (Sesabamisad, B -- gamocda araa Cabarebuli). 

amocanis pirobebSi gvaqvs: 

P ( A) 

1/3, P ( A) 

1 

P( 

A) 

2/ 3; 

P( B| A) 0.4 , P( B| A) 1 P( B| A) 0.6 ; 

P( B| A) 0.1 , P( B| A) 1 P( B| A) 0.9 . 

cnobilia, rom moxda B xdomileba da gamosaTvlelia pirobiTi 

albaToba P ( A| 

B) 

. vinaidan, A da A xdomilebebi qmnian srul sistemas, 

baiesis formulis Tanaxmad saZiebeli albaToba iqneba: 

P( 

A) 

P( 

B | A) 

2 

P ( A| 

B) 

 

. 

P( 

A) 

P( 

B | A) 

P( 

A) 

P( 

B | A) 

3 

24

davaleba. magaliT 13-is pirobebSi gamoTvaleT albaToba imisa, rom 

arCeul iqna “avi” gamomcdeli, Tu cnobilia, rom gamocda Cabarebul iqna 

warmatebiT? 

magaliTi 14 (keTil gamomcdelze II). davuSvaT, rom gamomcdelTan, 

romelTanac warmatebiT Caiara gamocdam (ix. magaliTi 13) gamosacdelad 

rig-rigobiT mivida kidev ori moswavle. jer gamocda ver Caabara meore 

moswavlem, Semdeg mivida mesame da manac ver Caabara gamocda. am faqtis 

Semdeg romeli hipoTezaa ufro dasajerebeli: es gamomcdeli “keTilia” 

Tu “avi”? 

amoxsna. avRniSnoT P i 

(A) 

(Sesabamisad, P i 

(A) 

) simboloTi albaToba 

(aposterioruli) imisa, rom es gamomcdeli “keTilia” (Sesabamisad, “avia”) 

mas Semdeg rac gamocdil iqna i -uri studenti, i 1,2, 3. Cven ukve davadgineT, 

rom P ( A) 

2/ 3. Sesabamisad, 

1 

 

P 

1( A) 

1 

P1 

( A) 

1/3. 

meore moswavlis TvalsazrisiT es albaTobebi warmoadgenen ori 

SesaZlo hipoTezis apriorul albaTobebs. amitom, baiesis formulis Tanaxmad, 

meore studentis CaWris Semdeg aposterioruli albaTobebi iqneba: 

P( 

B | A) 

P1 

( A) 

4 

3 

P 

2( 

A) 

 

da P 

2( A) 

1 

P2 

( A) 

. 

P( 

B | A) 

P ( ) ( | ) ( ) 7 

1 

A P B A P1 

A 

7 

analogiurad, axla miRebuli albaTobebi ukve iqneba aprioruli 

albaTobebi mesame moswavlisaTvis, da amitom saZiebeli aposterioruli 

albaTobebi, mas Semdeg rac mesame moswavlem ver Caabara gamocda, gamoi- 

Tvleba isev baiesis formuliT: 

P( 

B | A) 

P2 

( A) 

8 

9 

P 

3( 

A) 

 

da P3 ( A) 

1 

P3 

( A) 

P3 

( A) 

. 

P( 

B | A) 

P ( ) ( | ) ( ) 17 

2 

A P B A P2 

A 

17 

rogorc vxedavT, eqsperimentis (gamocdis) dawyebis win aprioruli 

albaToba imisa. rom arCeuli gamomcdeli “keTilia” toli iyo 1/3-is. eqsperimentebis 

Semdeg am xdomilebis aposterioruli albaToba gaizarda 

da gaxda 8/17. miuxedavad amisa, Tu sami eqsperimentis Semdeg misaRebia gadawyvetileba 

am gamomcdelis Sesaxeb, maSin ufro sarwmunoa CavTvaloT 

igi “avad” (vinaidan, P3 ( A) 

P3 

( A) 

). 

magaliTi 15. yuTSi 3 TeTri da 5 Savi burTia. yuTidan SemTxveviT 

iReben 4 burTs dabrunebiT. ipoveT albaToba imisa, rom amoRebuli bur- 

Tebidan TeTri burTebis raodenoba meti iqneba Savi burTebis raodenobaze? 

amoxsna. Tu warmatebad CavTvliT TeTri burTis amoRebas, maSin pirobis 

Tanaxmad erT cdaSi warmatebis albaToba iqneba p 3/ 8. saZiebeli 

xdomilebis xelSemwyob uTavsebad SemTxvevebs warmoadgens oTx cdaSi 3 

an 4 TeTri burTis amoReba, romelTa albaTobebi bernulis formulis 

Tanaxmad Sesabamisad tolia: 

3 3 

43 

P 

4(3) 

C4 

(3/8) 

(1 

3/8) 

135/1024 

da 

4 4 0 

P (4) C (3/8) 

(5/8) 

81/ 4096. 

4 4 

 

25

Sesabamisad, saZiebeli albaToba albaTobaTa Sekrebis kanonis 

Tanaxmad iqneba: 

135/1024 

81/ 4096 621/ 4096. 

magaliTi 16. davuSvaT vamowmebT defeqturobaze saqonlis partias, 

romelic Sedgeba 30 nawarmisagan. cnobilia, rom defeqturi produqciis 

wili Seadgens 5%-s. rogoria saqonlis am partiaSi defeqturi produqciis 

ama Tu im ricxvis aRmoCenis albaTobebi? 

amoxsna. am SemTxvevaSi eqsperimentebis ricxvia n 30, xolo albaTobis 

klasikuri ganmartebis Tanaxmad p 5/100 0.05 (Sesabamisad, q 0.95 ). 

Sesabamisad, gvaqvs: 

0 0 300 30 

P30 (0) C30 

0.05 0.95 0.95 0.2146 , 

garda amisa, 

30 k p 30 k 0.05 30 k 

P30 ( k 1) P30 ( k) P30 ( k) P30 

( k) 

, 

k 1 q k 1 0.95 19( k 1) 

30 0 

saidanac roca k 0 : P 30 

(01) 0.2146 0.3389 ; 

19 (01) 

30 1 

roca k 1: P 30 

(1 1) 0.3389 0.2586 da a. S. sabolood gveqneba 

19 (1 1) 

Semdegi cxrili: 

defeqturi nawa- albaToba kumulatiuri 

rmis ricxvi k Pn( k ) albaToba Pn() 

k 

0 0.2146 0.2146 

1 0.3389 0.5535 

2 0.2586 0.8122 

3 0.1270 0.9392 

4 0.0451 0.9844 

5 0.0124 0.9967 

6 0.0027 0.9994 

7 0.0005 0.9999 

8 0.0001 0.999998 

9 0.000001 0.999999 

am cxrilis Sesabamisi albaTobebis ganawilebis grafiki iqneba: 

26

0.4 

0.35 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

0 

0 2 4 6 8 10 

kumulatiuri albaTobebis Sesabamisi ganawilebis grafiki iqneba: 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 

0 2 4 6 8 10 

savarjiSo 1. eqsperimenti mdgomareobs sami saTamaSo kamaTlis gagorebaSi. 

ipoveT alabaToba imisa, rom eqsperimentis 10-jer gameorebisas, 

zustad 4 eqsperimentSi mova zustad or-ori “6”? 

savarjiSo 2. ramdeni SemTxveviTi cifri unda aviRoT, rom cifri 

“5” movides erTjer mainc aranakleb 0.9-is toli albaTobiT? 

27 

amocanebi 

1. A yuTi Seicavs 8 wiTel da 7 mwvane burTs, B yuTi Seicavs 6 wi- 

Tel da 4 mwvane burTs. agoreben wesier kamaTels. Tu movida 3-is jeradi 

qula, maSin A yuTidan SemTxveviT iReben 1 burTs, xolo winaaRmdeg Sem- 

TxvevaSi B yuTidan SemTxveviT iReben 1 burTs. ipoveT albaToba imisa, 

rom: a). amoRebuli burTi mwvanea; b). burTi amoRebuli iyo A yuTidan, Tu 

cnobilia, rom amoRebuli burTi mwvanea. 

2. PA ( ) 3/ 4 , P( B| A) 1/5, P( B| A) 4/7 . ipoveT: a). P( A B) 

; b). PB; ( ) g). 

P( A| B ) . 

3. studentma unda gaiaros testireba laboratoriaSi muSaobis dasawyebad. 

testirebis gavlis Semdeg studenti tests ukan ar abrunebs. albaToba 

imisa, rom SemTxveviT SerCeuli studenti testirebas gaivlis pirvel 

cdaze aris 1/3. ganmeorebiTi testirebis SemTxvevaSi CaWris albaT-

oba aris wina cdaze CaWris albaTobis naxevari. ipoveT albaToba imisa, 

rom studenti: a). gaivlis testirebas araumetes 3 mcdelobis Sedegad; b). 

gaivlis testirebas pirvel mcdelobaze, Tu cnobilia, rom man testireba 

gaiara araumetes 3 mcdelobiT. 

4. PA ( ) 13/ 25 , PB ( ) 9/ 25 , P( A| B) 5/9. ipoveT: a). P( A B) 

; b). P( B | A ) ; 

g). P( A B) 

; d). P( A| B ) ; e). P( A B) 

. 

5. 12 waxnagian wesier mravalwaxnagaze aRniSnulia ricxvebi 1, 2, . . . 

, 12. mocemulia 9 danayofiani firfita, romlis centrSi devs moneta. 

L ☻ R 

agdeben mravalwaxnagas: Tu movida 2, 3, 5, 7 an 11 qula, maSin monetas gadaadgileben 

TiTo danayofze R-is mimarTulebiT (marjvniv), xolo winaaRmdeg 

SemTxvevaSi L-is mimarTulebiT (marcxniv). TamaSi mTavrdeba, roca 

moneta miaRwevs an R–s an L (firfitis romelime bolos). ipoveT albaToba 

imisa, rom: a). TamaSi damTavrdeba 4 gadaadgilebis Semdeg R-ze; b). TamaSi 

damTavrdeba 4 gadaadgilebis Semdeg; g). TamaSi damTavrdeba 5 gadaadgilebis 

Semdeg; d). TamaSis damTavrebas dasWirdeba 6-ze meti gadaadgileba. 

6. moyvaruli sinoptikosis Teoriis Tanaxmad Tu erT wels iyo wyaldidoba, 

maSin albaToba imisa, rom igi ganmeordeba momdevno wels aris 

0.7, xolo Tu erT wels ar iyo wyaldidoba, maSin albaToba imisa, rom 

igi ar iqneba momdevno wels aris 0.6. gasul wels wyaldidoba ar yofila. 

ipoveT albaToba imisa, rom wyaldidoba iqneba: a). momdevno sam weliwads 

zedized; b). zustad erTjer momdevno sami wlis ganmavlobaSi. 

7. albaToba imisa, rom safexburTo klubis I gundis yvela moTamaSe 

formaSia aris 0.7. roca I gundis yvela moTamaSe formaSia, klubi sakuTar 

moedanze igebs albaTobiT 0.9. Tu I gundis yvela moTamaSe ar aris 

formaSi, maSin sakuTar moedanze mogebis albaTobaa 0.4. ipoveT albaToba 

imisa, rom: a). klubi moigebs sakuTar moedanze; b). I gundis yvela moTama- 

Se iyo formaSi, Tu klubma ukanaskneli Sexvedra ver moigo. 

8. CanTaSi devs 25 diski, romelTagan nawili TeTria da danarCeni 

Savi. CanTidan erTdroulad SemTxveviT iReben or disks. albaToba imisa, 

rom amoRebuli diskebi erTi da igive ferisaa emTxveva albaTobas imisa, 

rom es diskebi sxvadasxva ferisaa. ramdeni TeTri diskia CanTaSi? 

9. PA ( ) 0.6 , PB ( ) 0.8 , P( B| A) 0.45 , P( B C) 0.28 . ipoveT: a). P( A B) 

; 

b). P( C | B ); g). P( A| B ) . 

10. PA ( ) 0.75 , P( B| A) 0.8 , P( B| A) 0.6 . ipoveT PB ( ) da P( A| B ) . 

11. klasSi aris 7 biWi da 9 gogona. SemTxveviT SerCeulia am klasis 

ori moswavle. A {pirveli moswavle gogonaa}, B {meore moswavle gogonaa}. 

ipoveT: a). P( B | A ) ; b). P( B| A ); g). P( B| A ) ; d). P ( B | A) 

; e) PB. ( ) 

12. albaToba imisa, rom momavali wlis ivnisis pirveli dRe iqneba 

mSrali aris 0.4. Tu ivnisis romelime konkretuli dRe mSralia, maSin 

albaToba imisa, rom momdevno dRe iqneba mSrali aris 0.6. sxva SemTxveva- 

Si albaToba imisa, rom momdevno dRe iqneba mSrali aris 0.3. ipoveT albaToba 

imisa, rom: a). ivnisis pirveli ori dRe iqneba mSrali; b). ivnisis 

28

meore dRe iqneba mSrali g). sul cota erTi ivnisis pirveli sami dRidan 

iqneba mSrali. 

13. albaToba imisa, rom momavali wlis ivnisis I dRe iqneba mSrali 

aris 0.5. Tu ivnisis romelime konkretuli dRe mSralia, maSin albaToba 

imisa, rom momdevno dRe iqneba mSrali aris 0.4. sxva SemTxvevaSi albaToba 

imisa, rom momdevno dRe iqneba mSrali aris 0.6. ipoveT albaToba imisa, 

rom: a). ivnisis pirveli ori dRe iqneba sveli; b). ivnisis meore dRe 

iqneba sveli; g). sul cota erTi ivnisis pirveli sami dRidan iqneba sveli. 

14. 30 students dausves kiTxvebi: uWers is mxars argentinas, brazilias, 

Tu arcerTs da is caciaa Tu ara. 12 studentma upasuxa, rom is aris 

cacia da maTgan 4 uWers mxars argentinas, xolo 7 ki brazilias. ara 

cacia studentebidan 1 uWers mxars argentinas, xolo 10 ki brazilias. 

jgufidan SemTxveviT aarCies 1 studenti. ipoveT albaToba imisa, rom: a). 

studenti caciaa; b). studenti arcerT gunds ar uWers mxars; g). studenti 

caciaa, Tu cnobilia, rom studenti arcerT gunds ar uWers mxars. 

15. PA ( ) 0.3 . B xdomileba damoukidebelia A xdomilebisagan. 

PB ( ) 0.4 . C aris xdomileba rom arc A moxda da arc B . ipoveT: a). 

P( A\ B ) ; b). P( A B) 

; g). P ( C | A) 

. 

16. isvrian wiTel da lurj kamaTels da maTze mosuli qulebi ikribeba. 

vipovoT albaToba imisa, rom: a). qulaTa jami iqneba aranakleb 10- 

isa, Tu wiTel kamaTelze movida 6 qula; b). qulaTa jami iqneba aranakleb 

10-isa, Tu erT kamaTelze mainc movida 6 qula. 

17. kolejis studentebis naxevari swavlobs mecnierebas, xolo 30% 

ki maTematikas. maT Soris vinc swavlobs mecnierebas 40% swavlobs maTematikas. 

ipoveT albaToba imisa, rom SemTxveviT SerCeuli studenti: a). 

swavlobs orives: mTematikas da mecnierebas; b). swavlobs maTematikas, magram 

ar swavlobs mecnierebas. 

18. P( D| C) 1/5, P( C | D) 1/ 4, P( C D) 

p . ipoveT: a). PC ( ); b). PD ( ). 

19. P( D| C) 1/5, P( C | D) 1/ 4, P( C D) 1/5, P( C D) 

p . ipoveT p . 

20. stadionis 40 bileTidan 10 aris CrdiloeTis mxare, 14 aRmosavleTis 

da 16 ki dasavleTis mxare. SemTxveviT iReben erT bileTs da aZleven 

X -s (adamians saxelad X ). Semdgom, SemTxveviT iReben meore bileTs 

da aZleven Y -s da analogiurad mesame bileTs aZleven Z -s. ipoveT alba- 

Toba imisa, rom: a). X -s Sexvdeba CrdiloeTis mxare; b). X -s da Y -s Sexvdeba 

CrdiloeTis mxare; g). samives Sexvdeba erTi da igive mxare; d). ors 

Sexvdeba erTi mxare, xolo mesames sxva mxare. 

21. 24 prizidan 4 avtomanqanaa, 8 motocikleti da 12 saaTi. anam, ekam 

da elzam moiges TiTo prizi. ipoveT albaToba imisa, rom: a). anam moigo 

manqana, xolo ekam -- motocikleti an saaTi; b). anam da ekam (orivem) moigo 

manqana, Tu cnobilia, rom elzam moigo manqana; g). anasa da elzas Soris 

erTma mainc moigo manqana; d). anam moigo manqana, xolo ekam ki manqana 

an motocikleti; e). anam moigo manqana pirobaSi, rom ekam moigo 

manqana an motocikleti. 

29

22. ( A B) 

, P( A| B) 1/3, PA ( ) 6/7 . ipoveT PB. ( ) 

23. albaToba imisa, rom konkretuli dRe mSralia aris 3/10. albaToba 

imisa, rom “dinamo” moigebs mSral dRes aris 3/8, xolo – svel dRes 

3/11. a). “dinamom” Semdegi matCi orSabaTs unda Caataros. ras udris alba- 

Toba imisa, rom “dinamo” moigebs; b). gasul oTxSabaTs “dinamom” matCi 

moigo. ras udris albaToba imisa, rom oTxSabaTs mSrali amindi iyo; g). 

gasul SabaTs “dinamom” matCi waago. ras udris albaToba imisa, rom SabaTs 

sveli amindi iyo. 

24. dedamiwis garkveul nawilSi mSrali dReebi ufro metia, vidre 

sveli. Tu romelime dRe mSralia, maSin albaToba imisa, rom momdevno 

dRe iqneba isev mSrali aris 0.8. Tu romelime dRe svelia, maSin albaToba 

imisa, rom momdevno dRe iqneba isev sveli aris 0.6. cnobilia, rom or- 

SabaTi sveli dRe iyo. ipoveT albaToba imisa, rom imave kviris: a). samSabaTi 

da oTxSabaTi orive iqneba mSrali; b). oTxSabaTi iqneba mSrali. 

25. Tu romelime dRe mSralia, maSin albaToba imisa, rom momdevno 

dRe iqneba isev mSrali aris 0.8. Tu romelime dRe svelia, maSin albaToba 

imisa, rom momdevno dRe iqneba isev sveli aris 0.6. wlis ganmavlobaSi 

Catarda 44 kriketis matCi, TiToeuli grZeldeboda sami dRe mimdevrobiT, 

romelTagan I da III dRe iyo sveli. ipoveT albaToba imisa, rom II 

dRe iyo mSrali. 

26. kazinoSi dgas ori A da A ' saTamaSo magida, romlebic sruliad 

erTnairad gamoiyureba. A magidaze mogebis albaToba aris 1/3, xolo A 

' 

magidaze – 1/4. a). ipoveT SemTxveviT arCeul magidaze mogebis albaToba; 

b). Tqven SemTxveviT airCieT magida da moigeT. ras udris albaToba imisa, 

rom Tqven airCieT A magida; g). Tqven SemTxveviT airCieT magida da waageT. 

ras udris albaToba imisa, rom Tqven airCieT A ' magida. 

27. kazinoSi dgas sami saTamaSo magida A da A ' tipis, romlebic 

sruliad erTnairad gamoiyureba. amasTanave, A ' tipis magida ori calia. 

A tipis magidaze mogebis albaToba aris 1/3, xolo A ' tipis magidaze – 

1/4. a). ipoveT SemTxveviT arCeul magidaze mogebis albaToba; b). Tqven SemTxveviT 

airCieT magida da moigeT. ras udris albaToba imisa, rom Tqven 

airCieT A magida; g). Tqven SemTxveviT airCieT magida da waageT. ras udris 

albaToba imisa, rom Tqven airCieT A ' magida. 

28. albaToba imisa, rom gvanca adre gaiRviZebs aris 1/3. Tu gvanca 

adre gaiRviZebs, maSin albaToba imisa, rom skolaSi droulad miva aris 

3/4. Tu gvanca gvian gaiRviZebs, maSin skolaSi droulad misvlis albaToba 

aris 1/5. erT SemTxveviT arCeul dRes gvanca droulad mivida skola- 

Si. ipoveT albaToba imisa, rom man gvian gaiRviZa. 

29. sami erTnairi A, B da C yuTi Sedgeba or-ori ujrisagan. A yu- 

Tis TiToeul ujraSi devs TiTo lari, B yuTis TiTeul ujraSi devs 

xuT-xuTi lari, xolo C yuTis erT ujraSi devs 1 lari da meoreSi ki 5 

lari. SemTxveviT irCeven yuTs, xsnian mis erT ujras da pouloben masSi 

5 lars. ras udris albaToba imisa, rom am yuTis meore ujraSi agreTve 

devs 5 lari. 

30

30. ojaxSi aris ori biWi da ori gogo, romelTagan erTs hqvia eka. 

SemTxveviT arCeven or bavSvs. ipoveT albaToba imisa, rom: a). orive bavSvi 

gogoa, Tu erTi maTgani gogoa; b). orive gogia, Tu erTi maTgani ekaa. 

31. magidas aqvs sami ujra. I ujraSi devs 3 oqros moneta, II-Si – 2 

oqrosa da 1 vercxlis moneta, xolo III-Si – 3 oqrosa da 2 vercxlis moneta. 

SemTxveviT irCeven ujras da iqidan SemTxveviT iReben monetas. ipoveT 

albaToba imisa, rom: a). amoRebuli moneta iqneba oqrosi; b). amorCeuli 

iyo III ujra, Tu amoRebuli moneta oqrosia. 

32. kviris samuSao dReebSi juanSeri samsaxurSi dadis matarebliT. 

albaToba imisa, rom is miuswrebs 8 saaTian matarebels orSabaTs aris 

0.66, xolo sxva samuSao dReebSi ki es albaToba 0.75-ia. SemTxveviT virCevT 

kviris erT samuSao dRes. vipovoT albaToba imisa, rom: a). am dRes 

juanSeri miuswrebs 8 saaTian matarebels; b). arCeuli dRe iyo orSabaTi, 

Tu cnobilia, rom am dRes juanSerma miuswro 8 saaTian matarebels. 

33. albaToba imisa, rom zafxulis romelime dRes baTumSi iwvimebs 

aris 0.2. albaToba imisa, rom dRis maqsimaluri temperatura wvimian dRes 

25 graduss gadaaWarbebs aris 0.3, xolo Tu wvima ar aris – 0.6. vipovoT 

albaToba imisa, rom zafxulis SemTxveviT arCeul dRes baTumSi 

iwvimebs, Tu cnobilia, rom am dRis maqsimaluri temperatura 25 

gradusze metia. 

34. sakarnavalo TamaSSi monawile jer agdebs monetas, xolo 

Semdeg isvris kamaTels. is igebs prizs im SemTxvevaSi, Tu monetaze 

movida gerbi, xolo kamaTelze – samze naklebi qula. vipovoT albaToba 

imisa, rom is moigebs prizs. 

35. vipovoT albaToba imisa, rom kamaTlis samjer agdebisas orjer 

mova sami qula. 

36. vipovoT albaToba imisa, rom kamaTlis oTxjer agdebisas 

samjer mova oTxi qula. 

37. A da B damoukidebelia. PA ( ) 0.4 , P( A B) 0.7 . ipoveT: PB. ( ) 

38. agdeben erT wesier kamaTels da meore iseT kamaTels, romelzec 

6 qulis mosvlis albaTobaa 1/4. ipoveT albaToba imisa, rom: a). wesier 

kamaTelze mova 6 qula da arawesierze ar mova 6 qula; b). sul cota erT 

kamaTelze mainc mova 6 qula; g). zustad erT kamaTelze mova 6 qula, Tu 

cnobilia, rom sul cota erT kamaTelze mainc movida 6 qula. 

39. agdeben erT wesier monetas da meore iseT monetas, romelzec 

gerbis mosvlis albaTobaa 1/3. ipoveT albaToba imisa, rom: a). wesier monetaze 

mova gerbi da arawesierze ar mova gerbi; b). sul cota erT monetaze 

mainc mova gerbi; g). zustad erT monetaze mova gerbi, Tu cnobilia, 

rom sul cota erT monetaze mainc movida gerbi. 

40. A da B damoukidebeli xdomilebebia. PA ( ) 0.4 , P( A 

B) 0.88 . ipoveT: 

a). PB; ( ) b). albaToba imisa, rom romelime A da B xdomilebebidan 

moxdeba, magram ara orive. 

41. ipoveT alabaToba imisa, rom SemTxveviT SerCeuli naTura iqneba 

umaRlesi xarisxis, Tu cnobilia, rom naTurebis 4% arastandartulia da 

standartuli naTurebis 75% umaRlesi xarisxisaa. 

31

42. albaToba imisa, rom oTx damoukidebel cdSi A xdomileba erTjer 

mainc moxdeba aris 15/16. ipoveT calkeul cdaSi A xdomilebis moxdenis 

albaToba. 

43. erT yuTSi aris 18 TeTri da 2 wiTeli burTi, meore yuTSi – 9 

TeTri da 1 wiTeli burTi. II yuTidan SemTxveviT amoRebul burTs deben 

I yuTSi. ipoveT albaToba imisa, rom amis Semdeg I yuTidan amoRebuli burTi 

iqneba TeTri. 

44. erT yuTSi aris 3 TeTri da 2 lurji burTi, meoreSi ki 4 TeTri 

da 4 lurji burTi. I yuTidan SemTxveviT gadaitanes 2 burTi meoreSi. ipoveT 

albaToba imisa, rom amis Semdeg II yuTidan amoRebuli burTi iqneba 

TeTri. 

45. 10 yuTidan cxraSi devs 2 wiTeli da 2 mwvane kalkulatori, xolo 

erTSi ki 5 wiTeli da 1 mwvane kalkulatori. SemTxveviT SerCeuli 

yuTidan SemTxveviT SerCeuli kalkulatori aRmoCnda wiTeli. ipoveT albaToba 

imisa, rom kalkulatori amoRebuli iyo yuTidan, romelSic 5 wi- 

Teli kalkulatoria. 

pasuxebi: 

1. 19/45; 7/19. 2. 3/20; 9/35; 7/12. 3. 26/27; 21/25. 4. 1/5; 5/13; 17/25; 1/2; 21/25. 5. 0.030; 

0.146; 0, 0.712. 6. 0.196; 0.288. 7. 0.75; 0.28. 8. 10 an 15. 9. 0.27; 0.35; 0.3375. 10. 0.75; 

0.8. 11. 8/15; 7/15; 3/5; 2/5; 9/16. 12. 0.24; 0.42; 0.706. 14. 2/5; 4/15; 1/8. 15. 0.18; 0.58; 0.6. 

16. 1/2; 5/11. 17. 1/5; 1/10. 18. 5p ; 4p . 20. 0.25; 0.0577; 0.1057; 0.6676. 21. 10/69; 3/253; 

43/138; 11/138; 11/69. 22. 3/7. 24. 0.32; 0.56. 25. 2/11. 28. 8/23. 29. 2/3. 30. 1/5; 1/3. 31. 

34/45; 9/34. 32. 183/250; 44/183. 33. 1/9. 34. 1/6. 35. 5/72. 36. 5/324. 37. 1/2. 38. 1/8; 3/8; 

8/9. 39. 1/3; 2/3; 3/4. 40. 0.8; 0.56. 41. 0.72. 42. 0.5. 43. 9/10. 44. 0.52. 45. 5/32. 

32

Tavi III 

SemTxveviT sidideTa maxasiaTeblebi 

 

x 

a, 

b 

elementarul xdomilebaTa sivrceze gansazRrul ricxviT funqcias 

SemTxveviTi sidide ewodeba. SemTxveviT sidides ewodeba diskretuli tipis 

Tu is Rebulobs calkeul, izolirebul SesaZlo mniSvnelobebs. Sem- 

TxveviT sidides ewodeba uwyveti tipis Tu misi SesaZlo mniSvnelobebis 

simravle mTlianad avsebs raime ricxviT Sualeds. 

cxrils, romelSic CamoTvlilia diskretuli tipis SemTxveviTi sididis 

SesaZlo mniSvnelobebi da maTi Sesabamisi albaTobebi, ganawilebis 

kanoni (an mwkrivi) ewodeba: 

x i x 1 x 2 x n 

p i p 1 p 2 p n 

aq p i 

1. albaTobis statistikuri ganmartebidan gamomdinare: 

i 

Teoriuli sixSire erToblivi sixSire albaToba. 

P( a, b) P( x) 

. Tu mocemulia diskretuli tipis SemTxveviTi 

sidide da raime ricxviTi g funqcia, maSin g( ) isev iqneba diskretuli 

tipis SemTxveviTi sidide ganawilebis kanoniT: 

gx ( i 

) gx ( 

1) 

gx ( 

2) 

gx ( 

n) 

 

p 

i 

p 

1 

p 

2 

p 

n 

 

hipergeometriuli ganawileba. davuSvaT, rom yuTSi N burTia da 

maT Soris M TeTria. SemTxveviT, dabrunebis gareSe yuTidan viRebT n 

burTs. albaToba imisa, rom amoRebul n burTs Soris zustad k cali 

iqneba TeTri gamoiTvleba formuliT: 

m nm 

CM 

CN M 

P( N; M; n; m): P( n 

m) 

, m 0,1,...,min( M, n) 

. 

n 

CN 

ricxvTa am mimdevrobas hipergeometriuli ganawileba ewodeba. 

k1 

geometriuli ganawileba: P( k) 

pq ( q1 p), k 1,2,... . 

k 

 

puasonis ganawileba parametriT : P{ k} e , k 0,1,2,... . 

k! 

SemTxveviTi sididis ganawilebis funqcia: 

F( x): P( x) 

( Sesabamisad, F( x): P( x) 

) 

P( a 

b) F( b) F( a) 

(Sesabamisad, P( a 

b) F( b0) F( a 0) ). 

ganawlebis funqciis Tvisebebi: 

1). nebismieri x -saTvis 0 Fx 

( ) 1; 

2). ganawilebis funqcia araklebadia; 

3). ganawilebis funqcia uwyvetia marcxnidan (Tu ganawilebis funqcias 

ganvmartavT rogorc: F( x): P( x) 

, maSin is iqneba marjvnidan uwyveti); 

4). F( x) P{ xk} 

(Sesabamisad, F( x) P{ xk} 

); 

x x 

k 

x x 

k 

33

5). P{ xk} F( xk 0) F( xk 

) (Sesabamisad, 

P{ xk} F( xk ) F( xk 

0) , roca F( x): P( x) 

). 

SemTxveviT sidides, romelsac aqvs uwyveti ganawilebis funqcia, 

uwodeben uwyvet SemTxveviT sidides. Tu uwyveti SemTxveviTi sididis ganawilebis 

funqcia Fx () warmoebadia, maSin mis warmoebuls SemTxveviTi 

' 

sididis ganawilebis simkvrive ewodeba da aRiniSneba f() 

x -iT: f ( x) F ( x) 

. 

 

b 

x 

f( x) 0; f ( x) dx 1; P( a, b) f ( x) 

dx 

 

; F( x) f ( y) 

dy 

a 

 

uwyveti ganawilebis funqciis p rigis kvantili ewodeba iseT 

ricxvs, romlisTvisac F( xp) 

p . diskretuli ganawilebis SemTxvevaSi, Tu 

p1 p2 pi x p1 p2 pi pi 

1, maSin xp 

xi 

1. p 1/ 2 rigis kvantils 

SemTxveviTi sididis an misi ganawilebis funqciis mediana ewodeba da aRiniSneba 

x . e. i. x x1/2 

. moda ewodeba SemTxveviTi sididis im mniSvnelobas 

(an mniSvnelobebs), romelic Seesabameba ganawilebis simkvrivis lokalur 

maqsimums uwyveti SemTxveviTi sididis SemTxvevaSi an albaTobis 

lokalur maqsimums diskretuli SemTxveviTi sididis SemTxvevaSi. 

organzomilebiani SemTxveviTi sidide. 

diskretuli organzomilebiani ( , ) SemTxveviTi sididis ganawilebis 

kanons (anu da SemTxveviTi sidideebis erTobliv ganawilebis kanons) 

aqvs organzomilebiani cxrilis saxe, romelic gvaZlevs SesaZlo 

mniSvnelobebis calkeuli komponentebis CamonaTvals da im p(x i , y j ) albaTobebs, 

ra albaTobebiTac miiReba mniSvneloba (x i , y j ): 

 

 

x 1 x 2 … x i … x n 

y 1 p(x 1 , y 1 ) p(x 2 , y 1 ) … p(x i , y 1 ) … p(x n , y 1 ) 

… … … … … … … 

y j p(x 1 , y j ) p(x 2 , y j ) … p(x i , y j ) … p(x n , y j ) 

… … … … … … … 

y m p(x 1 , y m ) p(x 2 , y m ) … p(x i , y m ) … p(x n , y m ) 

p ( x , y ) 1 

ij , i j 

; P( xi ) p( xi , y 

j) 

; P( y ) ( , ) 

j 

j 

p x 

i i 

yj 

. 

organzomilebiani ( , ) SemTxveviTi sididis ganawilebis funqcia 

(an da SemTxveviTi sidideebis erToblivi ganawilebis funqcia) ewodeba: 

F( х, у ) = p ( x, y ). 

1). 0 ≤ F(x, y) ≤ 1; 

2). F(x, y) aris TiToeuli argumentis mimarT araklebadi, marjvnidan 

uwyveti funqcia; 

3). adgili aqvs zRvrul Tanafardobebs: F(-∞, y) = 0; F(x, - ∞) = 0; 

F(- ∞, -∞) = 0; F( ∞, ∞) = 1; 

4). F(x, ∞) = F 1 (x); F( ∞, y) = F 2 (y). 

SemTxveviT sidideebs ewodeba damoukidebeli, Tu 

x p 

34

F(x, y) = F 1 (x) F 2 (y). 

diskretuli SemTxveviTi sididebi damoukidebelia maSin da mxolod 

maSin, roca 

p(x i , y j )= p(x i ) p( y j ), i, 

j. 

uwyveti organzomilebiani SemTxveviTi sididis erToblivi ganawilebis 

simkvrive (anu organzomilebiani simkvrive) ewodeba erToblivi ganawilebis 

funqciis Sereul meore rigis kerZo warmoebuls: 

2 

F( 

x, 

y) 

f ( x, 

y) 

. 

xy 

y x 

1). f(x, y) ≥ 0; 2). x, 

y) 

 

 

3). 

 

F ( f ( x, 

y) 

dxdy; 

 

f ( x, 

y) 

dxdy 1 ; 4). p (( X, 

Y) 

D) 

f ( x, 

y) 

dxdy. 

x 

 

d f ( x, y) 

 

dF1 

() x dF( x, ) 

 

5). f1( x) 

 

f ( x, y) 

dy 

dx dx dx , 

analogiurad, 

 

 

f 2( 

y) 

f ( x, 

y) 

dx. 

6). uwyveti SemTxveviTi sidideebi damoukidebelia maSin da mxolod 

maSin, roca f(x, y)= f 1 (x) f 2 ( y). 

diskretuli SemTxveviTi sididis maTematikuri lodini aRiniSneba 

E simboloTi da ewodeba ricxvs: 

E ( ) P( ) 

D 

 

, anu E { } 

 

x P i i 

x x p 

i 

i i i 

. 

uwyveti tipis SemTxveviTi sididis maTematikuri lodini ewodeba 

ricxvs 

 

E 

xf () x dx , (Tu | x | f ( x) 

dx ). 

 

a). Ec c const ; b). E( ) 

E E 

; g). E( c) 

cE; d). E( E) 0 ; 

2 2 2 

2 2 

e). E( c) E( E ) ( c E 

) ; v). min E( c) E( E 

) ; 

 

 

c( , ) 

z). Tu da damoukidebeli SemTxveviT sididebia, maSin 

E( ) 

E E 

(Sebrunebuli debuleba mcdaria). 

SemTxveviTi sididis funqciis maTematikuri lodini: 

Eg ( ) g ( x ) P { x } g ( x ) p 

i 

i i i i i 

. 

Tu SemTxveviTi sididis SesaZlo mniSvnelobebia x 1 

, x 2 

, ..., x 

n 

, xolo 

B raime xdomilebaa ( PB ( ) 0), maSin SemTxveviTi sididis pirobiTi 

maTematikuri lodini B xdomilebis mimarT aRiniSneba E( | B) 

simbolo- 

Ti da ganimarteba Semdegnairad: 

35

E( | B) xiP( xi 

| B) 

. 

1 

cxadia, rom E( | ) E 

da E( | B) E( I ) 

PB ( ) 

B 

. 

SemTxveviTi sididis SemTxveviT sidideze regresiis mrudi 

(funqcia) ewodeba funqcias R( y) E( | y) 

. 

SemTxveviTi sidides R( ) (regresiis funqciaSi Casmulia SemTxveviTi 

sidide) SemTxveviTi sididis pirobiT maTematikur lodins uwodeben 

pirobiT da E( | ) simboloTi aRniSnaven. 

SemTxveviTi sididis dispersia. 

SemTxveviTi sididis dispersia D ganimarteba Semdegnairad: 

2 2 2 

DX E( E ) E ( E 

) . 

E -s ewodeba SemTxveviTi sididis meore rigis momenti (TviTon 

dispersias uwodeben agreTve – meore rigis centralur moments). 

Tu diskretuli tipis SemTxveviTi sididis ganawilebis kanonia 

x i x 1 x 2 x n 

p i p 1 p 2 p n 

maSin 

n n 

2 

( 

i 

 

j j) 

i 

i1 j1 

, anu 

DX x x p p 

36 

n 

i1 

n 

n 

2 2 

i i 

( j j) 

i1 j1 

. 

DX x p x p 

I. mudmivis dispersia nulis tolia -- Dc 0; 

2 

II. D( ab) 

a D 

. 

III. Tu da damoukidebeli SemTxveviTi sidideebia, maSin 

D( ) 

D D 

. 

SemTxveviTi sididis pirobiTi dispersia B xdomilebis mimarT: 

2 2 2 

D( | B): E{[ E( | B)] | B} E( | B) [ E( | B)] 

. 

bernulis ganawilebas (anu bernulis SemTxveviT sidides) aqvs saxe: 

1 0 

P p 1 

p 

pda D p(1 p) 

. 

np da D np(1 p) 

. 

aq E 

binomialuri ganawilebis SemTxvevaSi: E 

hipergeometriuli ganawilebis SemTxvevaSi: 

nM 

nM ( N M) ( N n) 

E 

da D 

 

. 

2 

N 

N ( N1) 

puasonis ganawilebis SemTxvevaSi: E D . 

2 

geometriuli ganawilebis SemTxvevaSi: E 1/ p da D (1 p)/ 

p . 

eqsponencialuri ganawileba. SemTxveviT sidides ewodeba eqsponencialurad 

ganawilebuli parametriT ( 0), Tu: 

0, x 0, 

f ( x) 

x 

e 

, x 0. 

2 

am SemTxvevaSi: E 1/ da D 1/ .

standartuli gadaxra. momentebi. 

SemTxveviTi sididis saSualo kvadratuli gadaxra (an standartuli 

gadaxra): D 

. 

 

E 

SemTxveviTi sididis standartizacia: ( E 0 , D 1). 

 

momentebi, asimetria da eqscesi. 

SemTxveviTi sididis n rigis sawyisi momenti ewodeba sidides 

n 

n 

: E 

( a: 

1 

E 

). 

SemTxveviTi sididis n rigis centraluri momenti ewodeba sidides 

: ( ) n 2 

n E a ( : 2 

D 

). 

eqscesis koeficienti ewodeba sidides: 

4 

4 

E( 

E) 

e 3 3. 

4 

 

2 4 

[ E( E) ] 

asimetriis koeficienti ewodeba sidides: 

3 

3 

E( 

E) 

. 

 

3 2 3 

[ E( E) ] 

kovariacia. korelaciis koeficienti. 

kovariaciis koeficienti an ubralod kovariacia ewodeba sidides: 

cov( , ) E[( E )( E )] E( ) 

E E 

. 

1). Tu da damoukidebelia, maSin cov( , ) 0 (piriqiT, ara); 

2). cov( , ) D 

; 

3). cov( , ) cov( , ) 

; 

4). cov( c, ) c cov( , ) 

; 

5). cov( , ) cov( , ) cov( , ) 

; 

6). D( ) D 2cov( , ) 

D 

; 

7). | cov( , )| D D 

. 

korelaciis koeficienti ewodeba sidides: 

cov ; E E 

 

( ; ): 

E 

 

D 

D 

 

 

. 

a). –1(;)1. 

b). Tu (;)=1, maSin =k+b, sadac k da b— mudmivebia, k>0. 

g). Tu (;)= –1, maSin =k+b, sadac k da b— mudmivebia, k0; (;)= – 1 roca k i

{ ggg,ggs,gsg,sgg,gss,sgs,ssg, sss} 

da, Sesabamisad, saZiebeli SemTxveviTi sidide iqneba -ze gansazRruli 

Semdegi ricxviTi funqcia: 

( 

ggg) 3 ; (ggs) (gsg) (sgg) 2; 

(gss) (sgs) (ssg) 1 da (sss) 0 . 

cxadia es SemTxveviTi sidide diskretuli tipisaa, is Rebulobs 

izolirebul mniSvnelobebs, magaliTad, 1-sa da 2-s Soris is ar 

Rebulobs arcerT mniSvnelobas. 

magaliTi 2. SemTxveviTi sidide iyos ori saTamaSo kamaTlis gagorebisas 

mosul qulaTa jami. am SemTxvevaSi elementarul xdomilebaTa sivrce 

Sedgeba 36 elementaruli xdomilebisagan: 

{( i , j) : i, 

j 1,2,..., 

6} , 

xolo SemTxveviTi sidide calkeul elementarul xdomilebas ( i , j) 

(sadac 

i -- pirvel kamaTelze mosuli qulaa, xolo j -- meore kamaTelze mosuli 

qula) Seusabamebs: ( i, j) 

i j (pirvel da meore kamaTelze mosuli 

qulebis jami). magaliTad, (1,3) (2,2) (3,1) 4. aRniSnuli SemTxveviTi 

sididis SesaZlo mniSvnelobebia: 2, 3, . . . , 12. is dikretuli tipisaa. 

magaliTi 3. ori msroleli TiTojer esvris samiznes. maT mier samiznis 

dazianebis (mizanSi moxvedris) albaTobebia Sesabmisad 0.6 da 0.7. SemTxveviTi 

sidide iyos dazianebul samizneTa raodenoba. SevadginoT 

misi ganawilebis mwkrivi. 

amoxsna. cxadia, rom SemTxveviTma sididem SeiZleba miiRos Semdegi 

mniSvnelobebi: 0 (verc erTma msrolelma ver daaziana samizne), 1 

(mxolod erTma msrolelma daaziana samizne) da 2 (orive msrolelma 

daaziana samizne). vipovoT Sesabamisi albaTobebi. 

bunebrivia SegvZlia vigulisxmoT rom pirveli da meore msrolelis 

srolis Sedegebi erTmaneTisagan damoukidebelia. SemoviRoT xdomilebebi: 

A -- pirvelma msrolelma daaziana samizne da B -- meore msrolelma 

daaziana samizne. mocemulia, rom PA ( ) 0.6 da PB ( ) 0.7 . Sesabamisad, 

PA ( ) 0.4 da PB ( ) 0.3 . garda amisa, A da B damoukidebeli xdomilebebia. 

damoukidebeli xdomilebebia agreTve: A da B , A da B , A da B . 

advili dasanaxia, rom xdomileba – verc erTma msrolelma ver daaziana 

samizne iqneba A B, xdomileba -- mxolod erTma msrolelma daaziana 

samizne iqneba ( A B) 

( 

AB) 

da xdomileba -- orive msrolelma daaziana 

samizne iqneba 

A B. gasagebia, rom ( A B) 

da ( A B) 

uTavsebadi 

xdomilebebia ( A B) 

( 

AB) 

Ø. 

amitom, damoukidebel xdomilebaTa namravlis albaTobisa da uTavsebad 

xdomilebaTa jamis albaTobis formulebis Tanaxmad gveqneba: 

P ( X 0) P( 

AB) 

P( 

A) 

P( 

B) 

0,4 0,3 

0,12; 

P( X 1) P{( AB) ( AB)} P( AB) P( AB) 

 

P( A) P( B) P( A) P( B) 0,6 0,3 0,4 0,7 0,46 ; 

38

P( AB) P( A) P( B) 0,6 0,7 0,42 

Sesabamisad, SemTxveviTi sididis ganawilebis mwkrivi iqneba: 

x 0 1 2 

i 

p 0.12 0.46 0.42 

i 

magaliTi 4. auditoriaSi myofi 15 studentidan 5 vaJia. vipovoT albaToba 

imisa, rom SemTxveviT SerCeul 6 students Soris 3 vaJia? 

amoxsna. Tu mivusadagebT hipergeometriul ganawilebas, gasagebia, 

rom: N 15, M 5, n 6 da k 3. amitom saZiebeli albaToba iqneba: 

3 63 3 3 

C10C 

1510 C10C 

5 

120 10 

P(15;5;6;3) 0.239 . 

6 6 

C15 C15 

5005 

magaliTi 5. saqonlis partiaSi defeqtur nawarmTa ricxvi Rebulobs 

mniSvneloba 0-s albaTobiT – 0.3; mniSvneloba 1-s albaTobiT – 0.4; 

mniSvneloba 2-s albaTobiT – 0.2 da mniSvneloba 3-s albaTobiT – 0.1 anu 

-s ganawilebis mwkrivs aqvs saxe: 

x 0 1 2 3 

i 

p 0.3 0.4 0.2 0.1 

i 

gamovTvaloT -s ganawilebis funqcia da avagoT misi grafiki. 

Tu x 0 , maSin F( x) P( x) P( ) 0 ; 

Tu 0x 

1, maSin F( x) P( x) P( 0) 0.3 ; 

Tu 1x 

2, maSin F( x) P( x) P{( 0) ( 1)} 

 

P( 0) P( 1) 0.3 0.4 0.7 ; 

Tu 2 x 3, maSin F( x) P( x) P{( 0) ( 1) ( 2)} 

P( 0) P( 1) P( 2) 0.3 0.4 0.2 0.9 ; 

da bolos, Tu x 3, maSin F( x) P( x) P( ) 1. 

Sesabamisad, ganawilebis funqciis grafiks eqneba Semdegi saxe: 

F(x) 

1.0 

0.9 

0.7 

0.3 

0 1 2 3 х 

39

magaliTi 6. mocemulia SemTxveviTi sididis ganawilebis funqcia: 

0, x a, 

 

x 

a 

F ( x) 

, a x b, 

b 

a 

1, x b. 

 

(sadac a da b ( a 

b) nebismieri namdvili ricxvebia). vipovoT Sesabamisi 

ganawilebis simkvrive. cxadia, rom ganmartebis Tanaxmad: 

0, x a, 

 

1 

f ( x) 

, a x b, 

b 

a 

0, x b 

 

rac Seexeba x a da x b wertilebs, aq Fx () funqcias warmoebuli 

ara aqvs da iq SegviZlia f() 

x ganvmartoT nebismierad, vTqvaT, 

f ( a) f ( b) 0. SemTxveviT sidides, romelsac aqvs aRniSnuli ganawilebis 

simkvrive, ewodeba Tanabarad ganawilebuli [ ab , ] monakveTze. 

magaliTi 7. davuSvaT, rom naTuris muSaobis dro aris SemTxveviTi 

sidide ganawilebis funqciiT Ft (), xolo naTuris gamocda grZeldeba na- 

Turis mwyobridan gamosvlamde (gadawvamde), Tu es moxdeba gamocdis dawyebidan 

araumetes 100 saaTis ganmavlobaSi, anu t 0 

100 

sT momentamde. 

vTqvaT, Gt () -- aris am gamocdis dros naTuris normalurad muSaobis 

drois ganawilebis funqcia. maSin cxadia, rom: 

 

F( 

t), 

t 100 

G( 

t) 

 

 

1, t 100. 

Gt () funqcias aqvs naxtomi t 0 

wertilSi, vinaidan Sesabamisi SemTxveviTi 

sidide Rebulobs t 0 

mniSvnelobas albaTobiT 1 Ft 

( 

0) 0. 

magaliTi 8. vipovoT p rigis x p 

kvantili Tanabari ganawilebis 

funqciisaTvis. 

p ( 0 p 1) rigis x p 

kvantili unda veZeboT rogorc F() 

x p gantolebis 

amonaxsni. Tanabari ganawilebis funqciis SemTxvevaSi es gantoleba 

miiRebs saxes: 

x a 

p, 

b a 

saidanac cxadia, rom: 

xp 

a p( ba) a(1 p) 

bp . 

roca p 0, maSin nebismieri x a warmoadgens p 0 rigis kvantils, 

xolo p 1 rigis kvantili iqneba nebismieri x b ricxvi. 

magaliTi 9. mocemulia organzomilebiani SemTxveviTi sididis ganawilebis 

kanoni: 

40

-2 3 6 

-0.8 0.1 0.3 0.1 

-0.5 0.15 0.25 0.1 

vipovoT caklkeuli SemTxveviTi sididis ganawilebis kanoni. 

amoxsna. cxrilSi moyvanili albaTobebis svetebis mixedviT SekrebiT 

miviRebT –s ganawilebis mwkrivs: 

-2 3 6 

р 0.25 0.55 0.2 

albaTobebis SekrebiT striqonebis mixedviT, miviRebT –s ganawilebis 

mwkrivs: 

-0.8 -0.5 

p 0.5 0.5 

magaliTi 10. mocemulia da damoukidebeli SemTxveviTi sidideebis 

ganawilebis kanoni: 

-2 3 6 -0.8 -0.5 

р 0.2 0.5 0.3 p 0.4 0.6 

vipovoT Z max{ , } SemTxveviTi sididis ganawilebis kanoni. 

amoxsna. Z SemTxveviTi sididis SesaZlo nmniSvnelobebia: -0.8; -0.5; 3 

da 6. gamovTvaloT Sesabamisi albaTobebi. gvaqvs: 

P( Z 0.8) P{( 2) ( 0.8)} P( 2) P( 0.8) 0.2 0.4 0.08 ; 

P( Z 0.5) P{( 2) ( 0.5)} P( 2) P( 0.5) 0.2 0.6 0.12 ; 

P( Z 3) P{[( 3) ( 0.8)] [( 3) ( 0.5)]} 

 

P[( 3) ( 0.8)] P[( 3) ( 0.5)] 0.5 0.4 0.5 0.6 0.5 ; 

P( Z 6) P{[( 6) ( 0.8)] [( 6) ( 0.5)]} 

 

P[( 6) ( 0.8)] P[( 6) ( 0.5)] 0.3 0.4 0.3 0.6 0.3 . 

magaliTi 11. gamovTvaloT saTamaSo kamaTelze mosuli qulaTa 

ricxvis maTematikuri lodini. gvaqvs: 

1 1 1 1 1 1 

E 1 2 3 4 5 6 3.5 . 

6 6 6 6 6 6 

3 

magaliTi 12. davuSvaT, g( x) x 4x 

da mocemulia SemTxveviTi sididis 

ganawilebis kanoni: 

-2 -1 0 2 

0.1 0.3 0.4 0.2 

davadginoT g( ) 

SemTxveviTi sididis ganawilebis kanoni da gamovTvaloT 

misi maTematikuri lodini. 

cxadia, rom g( 2) g(0) g(2) 0 da g( 1) 3 . amitom SemTxveviTi 

sididis SesaZlo mniSvnelobebia 0 da 3. davadginoT misi ganawilebis kanoni. 

amisaTvis gamovTvaloT albaTobebi: P{ 0} da P{ 3} . radgan xdomilebebi 

{ 2} , { 0} da { 2} uTavsebadia, amitom albaTobaTa Sekrebis 

wesis Tanaxmad gveqneba: 

P{ 0} P{{ 2} { 0} { 2}} 

41

P{ 2} P{ 0} P{ 2} 0.1 0.4 0.2 0.7 . 

garda amisa, P { 3} P{ 

1} 

0. 3. amitom g( ) 

SemTxveviTi sididis 

ganawilebis kanons aqvs saxe: 

0 3 

0.7 0.3 

garda amisa, E 00.7 30.3 0.9 . 

axla gamovTvaloT g( ) 

SemTxveviTi sididis maTematikuri lodini. 

gveqneba: 

E g( 2) 0.1 g( 1) 0.3 g(0) 0.4 g(2) 0.2 

 

00.1 30.3 00.4 00.2 0.9 . 

magaliTi 13. gamovTvaloT puasonis ganawilebis maTematikuri lodini. 

m 

x x 

Tu visargeblebT warmodgeniT e , maSin gveqneba: 

m0 m! 

k k k1 

 

 

EX k e e k e e e . 

k0 k! k1 k! k1 

( k 1)! 

e. i. puasonis ganawilebis parametri warmoadgens am ganawilebis 

maTematikur lodins (saSualo mniSvnelobas). 

magaliTi 14. davuSvaT, rom elementarul xdomilebaTa sivrce Sedgeba 

sami tolalbaTuri elementaruli xdomilebisagan { 1 , 2, 3} 

, 

P( 1 ) P( 2 ) P( 3 

) 1/3. ganvmartoT da SemTxveviTi sidideebi Semdegnairad: 

( 1 ) 1, ( 2 ) 0, ( 3) 1; 

( 1 ) 1, ( 2) 0, ( 3) 1. 

1 1 1 

maSin gasagebia, rom , E( ) E 

1 0 ( 1) 0. Sesabamisad, 

E( ) 

E E 

. meores mxriv, 

3 3 3 

P{ 0} P{ 0} P{ 0, 0} P( 2) 1/3, 

maSin rodesac da SemTxveviTi sidideebi rom iyvnen damoukideblebi 

{ 0, 0} xdomilebis albaToba unda yofiliyo 1 1 1 . 

3 3 9 

magaliTi 15. SemTxveviTi sididis ganawilebis kanonia 

x 

i 

-1 0 1 2 3 

p 0.1 0.15 0.3 0.25 0.2 

i 

gamovTvaloT misi dispersia. 

vinaidan dispersiis gamosaTvlelad gvaqvs ori gansxvavebuli formula, 

Sesabamisad, gveqneba dispersiis gamoTvlis ori xerxi. moxerxebulia 

es gamoTvlebi Caiweros cxrilebis saxiT. 

42

dispersiis gamoTvlis pirveli xerxi: 

i 

2 

2 

x 

i 

p 

i 

xp 

i i ( xi 

E ) ( xi 

E 

) pi 

1 -1 0.10 -0.1 5.29 0.5290 

2 0 0.15 0 1.69 0.2535 

3 1 0.30 0.3 0.09 0.0270 

4 2 0.25 0.5 0.49 0.1225 

5 3 0.20 0.6 2.89 0.5780 

E 

 

5 

 

i1 

x p 1.3 

i 

i 

5 

2 

( 

i 

) i 

1.51 

i1 

D 

x E p 

dispersiis gamoTvlis meore xerxi: 

i 

2 

2 

x 

i 

p 

i 

xp 

i i 

x 

i 

xi 

p 

i 

1 -1 0.10 -0.1 1 0.1 

2 0 0.15 0 0 0 

3 1 0.30 0.3 1 0.3 

4 2 0.25 0.5 4 1 

5 3 0.20 0.6 9 1.8 

E 

 

5 

 

x p 1.3 

i i 

i1 

 

2 

 

2 

 

D E ( E 

) 1.51 

E 

5 

2 2 

 

i 

i1 

x p 

i 

3.2 

magaliTi 16. davuSvaT, rom SemTxveviTi sidide simetriuladaa 

ganawilebuli nulis irgvliv, e. i. E=0. vTqvaT, = 2 . maSin 

E()=E( 3 )=0, vinaidan 3 agreTve, siametriuladaa ganawilebuli nulis 

irgvliv. meores mxriv, EE=0, vinaidan E=0. amitom: 

E 

 

E E 

( , ) 0. 

 

e. i. korelacia (da maSasadame, kovariacia) SeiZleba iyos nuli, maSinac 

ki roca SemTxveviTi sidideebi damokidebulia 

magaliTi 17. davuSvaT, rom da SemTxveviTi sidideebis erToblivi 

ganawilebis kanoni mocemulia Semdegi cxriliT: 

 

 

1 2 

1 1/5 0 1/5 

2 0 3/5 3/5 

3 1/5 0 1/5 

2/5 3/5 

cxadia, rom: 

E 11/5 23/531/5 2; E 12/5 23/5 8/5; 

E 111/5 223/5311/5 16/5 ; E 

E E 

0 . 

43

aqedan gamomdinare, korelaciis koeficienti nulia, maSin rodesac 

(iseve rogorc wina magaliTSi) naTelia, rom adgili aqvs SemTxveviTi 

sididis funqcionalur damokidebulebas SemTxveviT sidideze. 

magaliTi 18. da SemTxveviTi sidideebis qvemoT moyvanili erToblivi 

ganawilebis kanonis mixedviT gamovTvaloT korelaciis koeficienti 

(;). 

1 2 3 

 

10 1/36 0 0 1/36 

20 2/36 1/36 0 3/36 

30 2/36 2/36 2/36 6/36 

40 1/36 9/36 16/36 26/36 

6/36 12/36 18/36 

E 101/36 203/36 306/36 4026/36 35.83 ; 

E 16/36 212/36 318/36 2.3; 

 

40 35,83 2 

26/36 57.64 ; 7.6 ; 

2 2 2 

 

2 2 2 

D 10 35,83 1/36 20 35.83 3/36 30 35.83 6/36 

 

D 12,3 6/36 22.3 12/36 32.3 1 18/36 0.556 ; 

0.746 ; 

E 1011/36 201 2/36 20 21/36 301 2/36 30 22/36 

 

30 32/36 4011/36 40 29/36 40316/36 86.94 ; 

( , ) 6.94 2.3 35.83 / 7.6 0.746 0.8 . 

amocanebi 

1. SemTxveviTi sidide iyos ori kamaTlis agdebisas mosul qulaTa; 

a). jami; b). namravli; g). ganayofi; d). sxvaoba; e). sxvaobis moduli. aageT 

ganawilebis kanoni. 

2. kamaTels agdeben erTjer. 6 qulis mosvlis SemTxvevaSi kamaTels 

agdeben meorejer. SemTxveviTi sidide iyos qulaTa jami (Tu I agdebisas 

ar movida 6 qula, maSin II Sesakrebad CavTvaloT 0). aageT ganawilebis 

kanoni. 

3. CanTaSi devs 2 wiTeli da 3 lurji fanqari. CanTidan SemTxveviT 

iReben or fanqars dabrunebis gareSe. SemTxveviTi sidide iyos maTSi lurji 

fanqrebis ricxvi. aageT ganawilebis kanoni. 

4. SemTxveviTi sididis ganawilebis kanonia: 

1 2 3 4 

P 0.1 0.2 0.3 0.4 

44

gakeTda ori dakvirveba SemTxveviT sidideze. SemTxveviTi sidide iyos: 

a). miRebuli qulebidan udidessa da umciress Soris sxvaoba (roca 

dakvirvebebi tolia -- 0); b). qulebis jami; g). qulebis namravli. aageT 

-s ganawilebis kanoni. -s romeli mniSvnelobaa yvelaze ufro mosalodneli. 

5. wesier monetas agdeben orjer. aageT mosul gerbTa ricxvis ganawilebis 

kanoni. 

6. or wesier kamaTels agdeben erTdroulad. SemTxveviTi sidide iyos 

miRebuli qulebidan udidessa da umciress Soris sxvaoba (roca dakvirvebebi 

tolia – sxvaoba aris 0). aageT ganawilebis kanoni. 

7. wesier kamaTels agdeben erTjer. SemTxveviTi sidide iyos mosuli 

qulis naxevari, roca qula luwia, xolo winaaRmdeg SemTxvevaSi – 

misi gaormagebuli. aageT ganawilebis kanoni. 


qulis mesamedi, roca qula 3-is jeradia, xolo winaaRmdeg SemTxveva- 

Si – misi gaormagebuli. aageT ganawilebis kanoni. 


qulis naxevari, roca qula luwia, xolo winaaRmdeg SemTxvevaSi – 

misi gasammagebuli. aageT ganawilebis kanoni. 

10. or wesier kamaTels agdeben erTdroulad. SemTxveviTi sidide 

iyos miRebuli qulebidan; a). udidesi; b). umciresi. aageT ganawilebis 

kanoni. 

11. erTdroulad agdeben or wesier oTxwaxnagas, tetraedrs (waxnagebi 

samkuTxedebia da gadanomrilia cifrebiT 1, 2, 3, 4). aageT qveda waxnagze 

mosul qulaTa: a). namravlis; b). jamis; g). sxvaobis; d). ganayofis; e). 

gaorkecebuli namravlis ganawilebis kanonebi. 

12. CanTaSi devs 6 wiTeli da 3 mwvane kalkulatori. CanTidan dabrunebis 

gareSe iReben 2 kalkulators. aageT maTSi mwvane kalkulatorTa 

ricxvis ganawilebis kanoni. 

13. CanTaSi devs 6 wiTeli da 3 mwvane kalkulatori. CanTidan dabrunebis 

gareSe iReben 3 kalkulators. aageT maTSi wiTel kalkulatorTa 

ricxvis ganawilebis kanoni. 

14. SemTxveviT iReben erT karts 52 kartidan. Tu amoRebuli karti 

tuzia, maSin Cerdebian. winaaRmdeg SemTxvevaSi agrZeleben kartis amoRebas 

dabrunebis gareSe manam sanam an tuzi ar amova an 4 karti ar iqneba 

amoRebuli. aageT amoRebul kartTa raodenobis ganawilebis kanoni. 

15. SemTxveviT iReben erT karts 36 kartidan. Tu amoRebuli karti 

agurisaa, maSin Cerdebian. winaaRmdeg SemTxvevaSi agrZeleben kartis amo- 

Rebas dabrunebis gareSe manam sanam an aguri ar amova an 4 karti ar iqneba 

amoRebuli. aageT amoRebul kartTa raodenobis ganawilebis kanoni. 

16. qvemoT mocemulia SemTxveviTi sididis ganawilebis kanoni 

1 2 3 4 5 

P q 2q 3q 4q 5q 

ipoveT: a). q , b). P{ 3} , g). P{ 2} . 

45

17. kompiuteri daprogramebulia 0-dan 9-is CaTvliT erTniSna ricxvebis 

misaRebad ( SemTxveviTi sidide) ise, rom kenti cifrebis (1, 3, 5, 7, 

9) miRebis albaToba aris luwi cifrebis (0, 2, 4, 6, 8) miRebis albaTobis 

naxevari. ipoveT ganawilebis kanoni. 

18. kompiuteri daprogramebulia 0-dan 9-is CaTvliT erTniSna ricxvebis 

misaRebad ( SemTxveviTi sidide) ise, rom 3-is jeradi cifrebis (0, 

3, 6, 9) miRebis albaToba aris 3-is ara jeradi cifrebis (1, 2, 4, 5, 7, 8) mi- 

Rebis albaTobis gaormagebuli. ipoveT ganawilebis kanoni. 

19. ipoveT p Tu SemTxveviTi sididis ganawilebis kanonia: 

-1 1 2 3 5 

P p 0.2 0.15 0.2 2p 

20. P{ k} 

p , sadac k 1,2,3,4,5,6,7,8 . ipoveT p . 

21. kubis formis saTamaSo kamaTeli mowyobilia ise, rom masze kenti 

qulis mosvlis albaToba 3-jer metia luwi qulis mosvlis albaTobaze. 

aageT qulaTa ganawilebis kanoni. 

22. kubis formis saTamaSo kamaTeli mowyobilia ise, rom masze 

nebismieri qulis mosvlis albaToba am qulis proporciulia. aageT 

qulaTa ganawilebis kanoni. 

23. kubis formis saTamaSo kamaTeli mowyobilia ise, rom masze nebismieri 

qulis mosvlis albaToba am qulis ukuproporciulia. aageT qulaTa 

ganawilebis kanoni. 

24. ipoveT p Tu SemTxveviTi sididis ganawilebis kanonia: 

-2 -1 0 3 4 

P p 0.3 0.12 0.34 2 

p 

25. 52 kartidan SemTxveviT iReben erT karts dabrunebiT 520-jer. 

gamoTvaleT mosalodneli ricxvi (sixSire) imisa, rom mova: a). aguri; b). 

tuzi; g). suraTiani karti ( K, Q, 

J ); d). an tuzi, an aguri, an orive erTad; 

e). arc tuzi da arc aguri. 

26. kubis formis saTamaSo kamaTeli mowyobilia ise, rom masze luwi 

qulis mosvlis albaToba 3-jer metia kenti qulis mosvlis albaTobaze. 

am kamaTels agdeben 420-jer. gamoTvaleT mosalodneli ricxvi (sixSire) 

imisa, rom mova: a). 1 qula; b). luwi qula; g). martivi ricxvi. 

27. qvemoT moyvanilia SemTxveviTi sididis dagrovili albaTuri 

ganawilebis kanoni (anu P{ k} 

nacvlad P{ k} 

-si): 

0 1 2 3 4 5 

P{ k} 

0.116 0.428 0.765 0.946 0.995 1.000 

gakeTda 100 dakvirveba SemTxveviTi sidideze. gamoTvaleT uaxloes 

mTel ricxvamde damrgvalebuli yvela Sedegis mosalodneli sixSire. 

28. SemTxveviTi sididis ganawilebis kanons aqvs saxe: 

k 

0.3 (0.7) , Tu k 1,2,3,4; 

 

P{ k} 

p, Tu k 5; 

 

0, yvela danarCeni k Tvis 

. 

46

ipoveT p da { 3} xdomilebis mosalodneli sixSire, roca SemTxveviTi 

sidideze tardeba 1000 dakvirveba. 

29. yuTSi Zevs 49 erTnairi burTi, romlebic gadanomrilia 1-dan 49- 

mde. SemTxveviT irCeven 6 burTs dabrunebis gareSe. ipoveT albaToba imisa, 

rom aqedan oTxze iqneba luwi qula. 

30. monetas agdeben gerbis pirvelad gamoCenamde. vipovoT albaToba 

imisa, rom gerbi pirvelad mova meoTxe agdebisas. 

31. gamoTvaleT zemoT moyvanili SemTxveviTi sidideebis maTematikuri 

lodinebi. 


1 2 3 4 

P 0.1 a 0.3 b 

cnobilia, rom E 3. ipoveT a da b . 

33. cnobilia, rom arcerTi soko ar cocxlobs momaval wlamde. nebismieri 

soko momdevno wels iZleva raodenobis axal sokos. davuSvaT, 

rom mimdinare wels xarobs ori soko. vipovoT momaval wels sokoTa 

raodenobis ganawilebis kanoni, gamovTvaloT -s maTematikuri lodini 

da dispersia, Tu SemTxveviTi sididis ganawilebis kanonia: 

0 1 2 

P 0.2 0.6 0.2 

m i T i T e b a: SeadgineT damoukidebeli , wyvilis erToblivi ganawilebis 

kanoni, . 

34. gamoTvaleT qvemoT moyvanili ganawilebis kanonebis mixedviT 

Sesabamisi SemTxveviTi sidideebis maTematikuri lodini da dispersia: 

0 1 2 3 4 

P 1/8 3/8 1/8 1/4 1/8 

-2 -1 0 1 2 3 

P 0.15 0.25 0.3 0.05 0.2 0.05 

1 2 3 4 5 6 7 

P 0.1 0.2 0.1 0.2 0.1 0.2 0.1 

3 4 5 6 7 

P 1/18 5/18 7/18 1/18 4/18 

35. wesieri saTamaSo kamaTlis waxnagebze dawerilia cifrebi 1, 2, 2, 

3, 3 da 3. avRniSnoT asoTi kamaTlis erTxel gagorebisas mosuli qula. 

vipovoT -s maTematikuri lodini da standartuli gadaxra. 

36. wesieri saTamaSo kamaTlis waxnagebze dawerilia cifrebi 1, 2, 2, 

3, 3 da 3. avRniSnoT asoTi kamaTlis orjer gagorebisas mosul qula- 

Ta jami. aageT -s ganawilebis kanoni da gamoTvaleT misi lodini da 

dispersia. 

47

37. samSeneblo kompanias sTavazoben or A da B proeqts da finansurma 

direqtorma unda urCios kompanias am proeqtebidan romeli unda 

airCios. misi SefasebiT A proeqti iZleva 150000 larian mogebas albaTobiT 

0.5, 250000 larian mogebas albaTobiT 0..2 da 100000 larian wagebas 

albaTobiT 0.3. B proeqti iZleva 100000 larian mogebas albaTobiT 0.6, 

200000 larian mogebas albaTobiT 0..3 da 50000 larian wagebas albaTobiT 

0.1. daadgineT romel proeqts unda dauWiros mxari finansurma direqtorma. 

m i T i T e b a: SeadareT erTmaneTs EA ( ) da EB. ( ) 

38. supermarketSi kvercxebi iyideba yuTebiT, romelSic devs 6 – 6 

kvercxi. gatexili kvercxebis raodenobis ganawilebis kanonia: 

0 1 2 3 4 5 6 

P 0.8 0.14 0.03 0.02 0.01 0 0 

ipoveT: a). -s maTematikuri lodini da dispersia; b). gautexavi kvercxebis 

raodenobis maTematikuri lodini da dispersia. 


1 2 3 4 5 

P a 0.3 0.2 0.1 0.2 

ipoveT: a da SemTxveviTi sididis maTematikuri lodini da standartuli 

gadaxra. 


2 3 4 5 6 7 

P 0.05 0.25 a b 0.1 0.3 

cnobilia, rom E 4.9 . vaCvenoT, rom a b da gamovTvaloT SemTxveviTi 

sididis standartuli gadaxra. 

41. wesier saTamaSo kamaTels agdeben manam sanam ar gamoCndeba 6 qula 

an ar Catardeba 4 agdeba. iyos Catarebul agdebaTa raodenoba, xolo 

-- ki mosuli 6 qulebis raodenoba am TamaSSi. ipoveT: a). -s ganawilebis 

kanoni; b). -s standartuli gadaxra; g). E . 

42. wina amocanis pirobebSi, Tu moTamaSes TamaSis ganmavlobaSi mouvida 

6 qula, maSin is igebs 100 lars, xolo Tu TamaSis ganmavlobaSi 

ar mova 6 qula, maSin moTamaSe agebs 150 lars. vipovoT moTamaSis mier 

mogebuli Tanxis maTematikuri lodini. 

43. komiteti, romlis SemadgenlobaSi Sedis 6 mamakaci da 4 qali, 

irCevs Tavis 2 warmomadgenels. vigulisxmoT, rom komitetis nebismieri 

wevris arCeva Tanabarad SesaZlebelia da avagoT arCeuli qalebis raodenobis 

ganawilebis kanoni. vipovoT arCeuli qalebis mosalodneli ricxvi. 

44. agdeben wesier oTxwaxnaga piramidas, romelzec aRniSnulia 

cifrebi 1, 2, 3 da 4. Semdeg agdeben wesier monetas imdenjer ra cifric 

mova piramidis fuZeze. iyos piramidis fuZeze mosuli qula, xolo -- 

48

ki mosuli gerbebis raodenoba. a). aCveneT, rom P { 2} 1/ 

4 ; b). aageT -s 

1 

ganawilebis kanoni; g). aCveneT, rom E E; d). gamoTvaleT D . 

2 

45. monetas agdeben 5-jer. SemTxveviTi sidide iyos mosul gerbTa 

ricxvi, xolo SemTxveviTi sidide ki bolo or agdebaSi mosul gerbTa 

ricxvi. avagoT am SemTxveviTi sidideebis erToblivi ganawilebis kanoni 

da vipovoT kovariacia. 

46 32 kartidan SemTxveviT iReben 2-s. SemTxveviTi sidide iyos amoRebuli 

tuzebis ricxvi, xolo SemTxveviTi sidide ki amoRebuli 

mefeebis ricxvi. avagoT avagoT am SemTxveviTi sidideebis erToblivi 

ganawilebis kanoni da vipovoT korelaciis koeficienti. 

pasuxebi: 

1. a). 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 – 1/36, 1/18, 1/12, 1/9, 5/36, 1/6, 5/36, 1/9, 1/12, 

1/18, 1/36. 2. 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12 – 1/6, 1/6, 1/6, 1/6, 1/6, 1/36, 1/36, 1,36, 

1,36, 1,36, 1/36. 3. 0, 1, 2 – 1/10, 3/5, 3/10. 16. 1/15, 2/5/, 4/5. 24. 0.2. 25. 130; 40; 120; 

160; 360. 26. 105; 105; 245. 27. 12; 31; 34; 18; 5; 0. 28. 0.468; 103. 32. 0.2; 0.4. 33. 0,1, 2, 

3, 4 – 0.04, 0.24, 0.44, 0.24, 0.04; 2; 0.8. 34. 15/8, . ;0.05, . ; 4, 3.6; 46/9, 116/81. 35. 7/3; 

0.745. 36. 2, 3, 4, 5, 6 – 1/36, 1/9, 5/18, 1/3; 1/4; 14/3; 10/9. 37. EA ( ) 95000 , 

EB ( ) 115000 ; B . 38. 0.3, 0.51; 5.7, 0.51. 39. 0.2; 2.8, 1.4. 40. ab 0.15 ; 1.7. 41. 1, 2, 

3, 4 – 1/6, 5/36, 25/216, 125/216; 1.172; 0.5177. 42. -20.56. 43. 0, 1, 2 – 1/3, 8/15, 2/15; 

0.8. 44. 0, 1, 2, 3 – 11/48, 7/16, 13/48, 1/16; E 7/ 6, E 7/3; 13/18. 

49

Tavi IV 

diskretul ganawilebaTa gamoyenebebi 

binomialuri ganawileba 

•PTu yovel konkretul cdas gaaCnia ori SesaZlo Sedegi (warmateba 

da marcxi) da isini urTierTgamomricxavia; 

• tardeba cdaTa sasruli raodenoba -- n ; 

• yoveli cdis Sedegi damoukidebelia yvela danarCeni cdis Sedegisagan; 

• calkeul cdaSi warmatebis p albaToba mudmivia; 

maSin SemTxveviT sidides, romelic warmoadgens warmatebaTa raodenobas 

n cdaSi uwodeben binomialur SemTxveviT sidides, aRniSnaven 

Bi( n, p ) simboloTi da mas aqvs Semdegi ganawilebis kanoni: 

k k n k 

P{ Bi( n, p) k} Pn( k) Cnp (1 p) 

, k 0,1,..., n. 

amasTanave, E[ Bi( n, p)] 

np ; D[ Bi( n, p)] np(1 p) 

. 

puasonis ganawileba 

k 

 

P{ Po( ) k} 

e , k 0,1, 2,... ; E[ Po( )] 

; D[ Po( )] 

. 

k! 

puasonis ganawileba adeqvaturi modelia im xdomilebebisaTvis, romlebic: 

• xdebian SemTxveviT sivrceSi an droSi; 

• xdebian cal-calke (erTdroulad moxdena ar SeiZleba); 

• xdebian damoukideblad, da 

• xdebian mudmivi intensivobiT (xdomilebaTa raodenoba mocemul 

drois intervalSi am intervalis sigrZis proporciulia). 


1 

magaliTi 1. mocemulia Bi(8, ) . ipoveT: a). P{ 6} ; b). P{ 2} ; g). 

4 

P{ 0} . 

amoxsna. a). visargebloT bernulis formuliT, sadac n 8 da 

6 1 6 3 2 1 6 3 2 

gvaqvs: P{ 6} C8 

( ) ( ) 28 ( ) ( ) 0.00385 0.004 ; 

4 4 4 4 

0 1 0 3 8 1 1 1 3 7 

b). P{ 2} P{ 0} P{ 1} P{ 2} C8 ( ) ( ) C8 ( ) ( ) 

4 4 4 4 

2 1 2 3 6 

C8 

( ) ( ) 0.1001 0.2669 0.3114 0.6785 0.679 ; 

4 4 

g). gadavideT sawinaaRmdego xdomilebaze: 

0 1 0 3 8 

P{ 0} 1 P{ 0} 1 C8 

( ) ( ) 10.1001 0.8998 0.9 . 

4 4 

50 

1 

p . 

4

1 

magaliTi 2. mocemulia Bi(3, ) . ipoveT E da D . 

4 

1 3 

1 3 9 

amoxsna. E np 3 . D np(1 p) 3 . 

4 4 

4 4 16 

magaliTi 3. mocemulia Bi(10, 0.3) . ipoveT: a). E da D ; b). 

P{ E E } 

. 

amoxsna. a). E np 100.3 3, D np(1 p) 100.3 0.7 2.1 ; 

b). P{ E E } P{3 2.1 3 2.1} P{3 1.44 

31.44} 

 

P{1.55 4.44} P{ 2} P{ 3} P{ 4} P{ 4} P{ 1} 

 

0.8497 0.1493 0.7004 0.7 . 

magaliTi 4. mocemulia Bi(10, 0.3) . ipoveT: a). P{ 2} ; b). P{ 2} ; 

g). P{ 2} ; d). { 2} xdomilebis albaToba, Tu cnobilia, rom { 2} . 

2 2 8 

amoxsna. a). P{ 2} C10 

(0.2) (0.8) 0.302 ; 

b). P{ 2} P{ 0} P{ 1} P{ 2} 0.6778 ; 

g). P{ 2} 1 P{ 2} 10.6778 0.3222 ; 

d). unda moiZebnos P{ 2| 2} . pirobiTi albaTobis ganmartebis 

Tanaxmad gvaqvs: 

P{( 2) ( 2} 

P{ 2| 2} 

P{ 2} 

P{( 2) [( 0) ( 1) ( 2)]} P{ 2} 0.302 

0.4456 . 

P{ 2} P{ 2} 0.6778 

magaliTi 5. cnobilia, rom Bi( n, p) 

, E 24 da D 8. ipoveT n 

da p . 

amoxsna. vinaidan E[ Bi( n, p)] 

np da D[ Bi( n, p)] np(1 p) 

, amitom 24 np 

da 8 np(1 p) 24(1 p) 

. saidanac 1p 

1/3 anu p 2/3. Sesabamisad, 

n 24/ p 243/ 2 36 . 

magaliTi 6. cnobilia, rom Bi( n,0.6) 

. vipovoT cdaTa is minimaluri 

raodenoba, romlisTvisac P{ 1} 0.9 . 

0 0 n 

n 

amoxsna. P{ 1} 1 P{ 0} 1 C n 

(0.6) (0.4) 1 (0.4) 0.9 . amitom 

gvaqvs: (0.4) n 

0.1, anu nln(0.4) ln(0.1) . aqedan, imis gaTvaliswinebiT, rom 

ln(0.4) 

ln(0.4) uaryofiTia, davaskvniT, rom n 2.51 . Sesabamisad, cdaTa 

ln(0.1) 

saZiebeli minimaluri ricxvi iqneba n 3. 

magaliTi 7. qvemoT moyvanil situaciebSi miuTiTeT rodisaa puasonis 

ganawileba adeqvaturi modeli: 

a). im avtomobilebis ricxvi, romlebic gadakveTaven erTzolian 

xids Tavisufali moZraobis dros 10 saaTidan 11 saaTamde; 

am SemTxvevaSi puasonis ganawileba iqneba kargi modeli, radganac 

SeiZleba CavTvaloT, rom manqanebi xids gadakveTaven erTmaneTisagan damoukideblad 

da drois SemTxveviT momentebSi (moZraoba Tavisufalia), 

ori manqana erTdroulad ver gadakveTs xids (xidi erTzoliania) da saS- 

51

ualod drois mocemul intervalSi xidze gavlili manqanebis raodenoba 

drois intervalis proporciuli iqneba; 

b). im avtomobilebis ricxvi, romlebic 9 saaTidan 10 saaTamde Sedian 

qalaqis centris avtosadgomze; 

gasagebia, rom am SemTxvevaSi puasonis ganawileba ver iqneba kargi 

modeli, gansakuTrebiT Tu es samuSao dRea -- avtosadgomze iqneba rigebi 

da manqanebi ver imoZraveben Tavisuflad da erTmaneTisagan damoukideblad; 

g). radiaoqtiuri wyaros mier wamSi gamosxivebuli nawilakebis raodenoba; 

puasonis ganawileba iqneba kargi modeli mxolod im SemTxvevaSi, 

Tu gamosxiveba xdeba damoukideblad da SemTxveviT momentebSi da yoveli 

konkretuli gamosxiveba erTnawilakiania. 

magaliTi 8. radioaqtiuri wyaros mier wamSi gamosxivebuli nawilakebis 

raodenoba emorCileba puasonis ganawilebas saSualoTi 5. gamoTvaleT 

albaToba imisa, rom wamSi gamosxivebuli iqneba: a). 0; b). 1; g). 2; d). 

3 an meti nawilaki. 

amoxsna. iyos SemTxveviTi sidide: “radiaqtiuri wyaros mier wam- 

Si gamosxivebuli nawilakebis raodenoba”, maSin pirobis Tanaxmad 

Po(5) . visargebloT puasonis ganawilebis kanoniTa da Sesabamisi cxrilebiT. 

maSin: 

0 

5 5 

a). P{ 0} e 0.0067 ; 

0! 

1 

5 5 

b). P{ 1} e 0.0337 ; 

1! 

2 

5 5 

g). P{ 2} e 0.0842 ; 

2! 

d). gadavideT sawinaaRmdego xdomilebaze: 

P{ 3} 1 P{ 3} 1 P{ 0} P{ 1} P{ 2} 

10.0067 0.0337 0.0842 0.875 . 

magaliTi 9. taqsebis firmaSi taqsebze Semosuli gamoZaxebebis raodenoba 

puasonis kanoniTaa ganawilebuli, saSualod 4 gamoZaxeba yovel 

30 wuTSi. ipoveT albaToba imisa, rom: a). arcerTi gamoZaxeba ar Semova 30 

wuTSi; b). erTi gamoZaxeba Semova 1 saaTSi; g). orze naklebi gamoZaxeba 

Semova 15 wuTSi. 

amoxsna. a). iyos SemTxveviTi sidide: “taqsebis firmaSi 30 wuTis 

ganmavlobaSi Semosuli gamoZaxebebis raodenoba”. maSin cxadia, rom 

Po(4) . Sesabamisad, gvaqvs: 

0 

4 4 

P{ 0} e 0.0183 ; 

0! 

b). iyos SemTxveviTi sidide: “taqsebis firmaSi 1 saaTis ganmavlobaSi 

Semosuli gamoZaxebebis raodenoba”. vinaidan, pirobidan gamomdinare, 

1 saaTSi saSualod 8 gamoZaxeba Semova, amitom Po(8) 

. Sesabamisad, 

52

1 

8 8 

P{ 1} e 0.0027 ; 

1! 

g). analogiurad, Tu aris SemTxveviTi sidide: “taqsebis firmaSi 

15 wuTis ganmavlobaSi Semosuli gamoZaxebebis raodenoba”, maSin 

Po(2) . amitom 

0 1 

2 2 2 2 

P{ 2} P{ 0} P{ 1} e e 0.406 . 

0! 1! 

magaliTi 10. V ml tbis wyalSi organuli nawilakebis raodenoba 

emorCileba puasonis ganawilebis kanons saSualoTi 0.2V . gamoTvaleT 

albaToba imisa, rom: a). 50 ml tbis wyali Seicavs 8-ze nakleb organul 

nawilaks; b). 30 ml tbis wyali Seicavs 2-ze met organul nawilaks; g). 10 

ml tbis wyali Seicavs zustad 3 organul nawilaks. 

amoxsna. a). V 50, 0.2 50 10 , e. i. Po(10) 

. visargebloT puasonis 

dagrovili (kumulatiuri) albaTobebis cxriliT: 

P{ 8} P{ 7} 0.2202 ; 

b). V 30, 0.2 30 6, e. i. Po(6) 

. gadavideT sawinaaRmdego xdomilebaze: 

P{ 2} 1 P{ 2} 10.0619 0.938 ; 

g). V 10, 0.210 2, anu Po(2) 

. amitom 

P{ 3} 0.1804 . 

amocanebi 

1. wesier saTamaSo kamaTels agdeben 10-jer. ipoveT albaToba imisa, 

rom: a). 3 qula mova 4-jer; b). 4 qula mova 5-jer; g). 3 qula mova 6-jer. 

2. yuTSi Zevs 49 erTnairi burTi, romlebic gadanomrilia 1-dan 49- 

mde. SemTxveviT irCeven 6 burTs dabrunebiT. ipoveT albaToba imisa, rom 

aqedan oTxze iqneba luwi qula. 

3. SemTxveviT sidides aqvs binomuri ganawileba parametrebiT 

n 6 da p 0.2. gamoTvaleT: a). P{ 3} ; b). P{ 4} ; g). P{ 6} ; d). E . 

4. cnobilia, rom Bi(7,2/3) 

. gamoTvaleT: a). P{ 4} ; b). P{ 6} ; g). 

P{ 0} ; d). E . 

5. cnobilia, rom Bi(9,0.45) 

. opoveT: a). P{ 3} ; b). P{ 4an 

5} ; g). 

P{ 7} ; d). E . 


. gamoTvaleT: a). P{ 4} ; b). umciresi 

x romlisTvisac P{ x} 0.9 . 


. gamoTvaleT: a). P{2 

5} ; b). 

P{ 1an 5} 

. 

8. cnobilia, rom Bi(7,1/6) 

. ipoveT albaToba imisa, rom aris: 

a). zustad 3; b). sul cota 4. 

53

9. universitetis studentTa 30%-is asaki meryeobs 16-wlidan 19 wlamde. 

ipoveT albaToba imisa, rom SemTxveviT SerCeuli 10 studentidan 4- 

ze naklebis asaki iqneba 16-wlidan 19 wlamde. 

10. sakonditro fabrika amzadebs didi raodenobiT ferad tkbileuls. 

cnobilia, rom saSualod tkbileulis 20% mwvane ferisaa. SekvraSi 

aris 20 tkbileuli. igulisxmeT, rom SekvraSi tkbileuli xvdeba SemTxveviT 

fabrikis mier damzadebuli mTeli produqciidan da gamoTvaleT 

albaToba imisa, rom SekvraSi zustad 7 tkbileuli iqneba mwvane. 

11. mefrinveleobis fabrikaSi warmoebuli 6 – 6 kvercxi Calagebulia 

yuTebSi. TiTeuli kvercxis gatexvis albaToba sxva kvercxebisagan 

damoukideblad aris 0.1. yuTs davarqvaT cudi, Tu masSi devs sul cota 2 

gatexili kvercxi. ipoveT albaToba imisa, rom SemTxveviT SerCeuli yuTi 

cudia. 

12. garkveul tropikul kunZulze romelime konkretul TveSi qari- 

Sxalis moxdenis albaTobaa 0.08. gamoiyeneT binomialuri ganawileba da 

gamoTvaleT albaToba imisa, rom wlis ganmavlobaSi qariSxali iqneba 

orze met TveSi. 

13. cnobilia, rom dRis garkveul monakveTSi qveynis zrdasruli 

mosaxleobis 35% atarebs jinsebs. a). dRis am monakveTSi SeirCa 12 zrdasruli 

adamiani. gamoiyeneT binomialuri ganawileba da gamoTvaleT alba- 

Toba imisa, rom maTgan zustad xuTs acvia jinsi; b). Tu albaToba imisa, 

rom erT adamians mainc acvia jinsi aris 0.95, maSin ramdeni adamiania SerCeuli? 

14. mefrinveleobis fabrikis mier warmoebul A da B tipis kvercxebs 

cal-calke awyoben 6 – 6 calad yuTebSi. A tipis kvercxis gatexvis 

albaTobaa 1/6, xolo B tipis kvercxis – 1/10. SemTxveviT SerCeul iqna 

erTi yuTi A tipis kvercxebiT da erTi yuTi yuTi B tipis kvercxebiT. 

ipoveT albaToba imisa, rom: a). TiTeul yuTSi aris zustad erTi gatexili 

kvercxi; b). orive yuTSi erTad aris 2 gatexili kvercxi. 

15. bankma gasca krediti 10 adamianze, romelTagan TiToeulis mier 

kreditis daubruneblobis albaToba aris 0.15. ipoveT valis ardambrunebel 

kreditorTa: a). ganawilebis kanoni; b). maTematikuri lodini; g). ualbaTesi 

ricxvi. 

16. maRaziaSi yovelTviurad gayiduli sarecxi manqanebis ricxvi ganawilebulia 

Bi (100,0.25) kanoniT. maRaziis yoveldRiuri xarjebi Seadgens 

200 lars. sarecxi manqanis fasia 1000 lari. gamoTvaleT maRaziis yovelTviuri 

mogeba. 

17. SemTxveviTi sidide ganawilebulia puasonis kanoniT saSualo- 

Ti 3. ipoveT: a). P{ 2} ; b). P{ 1} ; g). P{ 3} . 

18. mocemulia, rom Po(3.25) 

. gamoTvaleT: a). P{ 3} ; b). P{ 2} ; 

g). P( 2) . 

19. SemTxveviTi sidide ganawilebulia puasonis kanoniT saSualo- 

Ti 2.4. ipoveT: a). P{ 3} ; b). P{ 2} ; g). P{ 3} . 

54

20. avtomagistralze kviraSi saSualod 2 avaria xdeba. CavTvaloT, 

rom kviraSi avariaTa ricxvi emorCileba puasonis ganawilebis kanons da 

gamovTvaloT albaToba imisa, rom: a). konkretul kviras ar moxdeba avaria; 

b). konkretul kviras moxdeba 2 avaria; g). or kviraSi moxdeba 3-ze 

naklebi avaria. 

21. wuTSi saSualod 15 momxmarebeli gaivlis supermarketis salaro-aparatTan 

Semowmebas. vigulisxmoT, rom gvaqvs miaxloebiT puasonis 

ganawileba da gamovTvaloT albaToba imisa, rom: a). 10 wuTian interval- 

Si arcerTi momxmarebeli ar gaivlis salaro-aparatTan Semowmebas; b). 15 

wuTian intervalSi 3-ze meti momxmarebeli gaivlis salaro-aparatTan Semowmebas. 

22. mimdinare wlis ianvris ganmavlobaSi Tqven 15-jer dagpatiJes 

wveulebaze. CaTvaleT, rom momavali wlis ianvarSi Tqvens mier miRebul 

mopatiJebaTa raodenoba ganawilebulia puasonis kanoniT imave saSualo- 

Ti da gamoTvaleT albaToba imisa, rom: a). momavali wlis ianvris konkretul 

dRes Tqven ar miiRebT arcerT mipatiJebas; b). momavali wlis ianvris 

Tvis konkretul 7 dRian kviraSi Tqven miiRebT 3-ze met mipatiJebas. 

23. teqnikuri momsaxurebis sadgurSi 1 saaTSi Semodis 100 avtomobili. 

Semosul avtomobilTa raodenoba ganawilebulia puasonis kanoniT. 

ipoveT: a). albaToba imisa, rom teqnikuri momsaxurebis sadgurSi 3 wuTis 

ganmavlobaSi ar Semova avtomobili; b). drois intervali, romelSic sul 

cota 0.25-is toli albaTobiT teqnikuri momsaxurebis sadgurSi ar Semova 

avtomobili. 

24. radioaqtiuri wyaro saSualod wamSi asxivebs 1 nawilaks. CaTvaleT, 

rom gamosxivebul nawilakTa raodenoba emorCileba puasonis kanons. 

a). gamoTvaleT albaToba imisa, rom 4 wamSi gamosxivebuli iqneba 0 

an 1 nawilaki; b). gamosxivebis reJimis Secvlis Semdeg albaToba imisa, 

rom 4 wamSi gamosxivebuli iqneba 0 an 1 nawilaki aris 0.8. amJamad wamSi 

saSualod ramden nawilaks asxivebs radioaqtiuri wyaro? 

25. qvemoT moyvanil magaliTebSi miuTiTeT rodisaa puasonis ganawileba 

adeqvaturi modeli: a). wvimis wveTebis raodenoba, romelic Cvardeba 

limonaTis biTlSi 1 wuTis ganmavlobaSi; b). satvirTo avtomobilebis 

raodenoba, romlebic gadakveTen konkretul adgils gadatvirTul avtomagistralze; 

g). Tvis ganmavlobaSi sadazRvevo kompaniaSi Semosul pretenziaTa 

raodenoba. 

26. qvemoT moyvanilia raodenoba gzaze manqanagadavlili zRarbebis 

13 kviris ganmavlobaSi: 10 8 9 7 11 5 4 12 7 6 11 3 11. a). gamoTvaleT 

am monacemebis saSualo da dispersia da axseniT ra SemTxvevaSi iqneba 

puasonis ganawileba adeqvaturi modeli; b). gamoiyeneT puasonis ganawileba 

da a) punqtSi gamoTvlili saSualoTi gamoTvaleT albaToba imisa, 

rom kviraSi manqanagadavlili zRarbebis raodenoba metia an toli 7- 

is; g). Tu albaToba imisa, rom kviraSi manqanagadavlili zRarbebis raodenoba 

metia an toli 7-is iqneboda 0.0244, maSin ramdeni unda iyos 

puasonis ganawilebis saSualo. 

55

27. martis TveSi Semosuli satelefono zarebis raodenobis ganawilebis 

kanonia 

dReSi Semosuli zarebis 0 1 2 3 4 

raodenoba ( x ) 

dReebis raodenoba 9 12 5 4 1 

a). gamoTvaleT fardobiTi sixSireebi; b). gamoTvaleT ganawilebis saSualo 

da dispersia. axseniT esadageba Tu ara puasonis ganawileba; g). gamoiyeneT 

puasonis ganawileba da wina punqtSi gamoTvlili saSualos mixedviT 

gamoTvaleT P{ x} 

, x 0,1, 2, 3, 4; d). SeadareT Teoriuli 

albaTobebi da fardobiTi sixSireebi. eTanxmebiT Tu ara b) punqtis 

daskvnas? 

28. mocemulia, rom Po( m) 

da P{ 1} 4 P{ 2} . ipoveT: a). m; b). 

P{ 3} ; g). P{ 4| 3} . 

29. avtomagistralis me-100 kilometrze SuadRis 10-wamian interval- 

Si gavlili manqanebis raodenoba emorCileba puasonis ganawilebas 

Po(0.8) . a). ipoveT albaToba imisa, rom aRniSnul adgils gadakveTs 3, 

4 an 5 avtomobili; b). cnobilia, rom aRniSnul adgili gadakveTa 3, 4 an 

5 avtomobilma. ipoveT albaToba imisa, rom avtomobilebis raodenoba iyo 

zustad 4. 

30. satelefono sadgurSi 8.00 – 8.05 intervalSi Semosul gamoZaxebaTa 

raodenoba emorCileba puasonis kanons saSualoTi 2. ipoveT albaToba 

imisa, rom: a). SemTxveviT SerCeul samSabaTs satelefono sadgurSi 

8.00 – 8.05 intervalSi ar Semova arcerTi gamoZaxeba; b). samSabaTsa da 

oTxSabaTs erTad Semova zustad 1 gamoZaxeba imave intervalSi; g). gamoZaxeba 

Semovida samSabaTs, Tu cnobilia, rom samSabaTsa da oTxSabaTs er- 

Tad Semovida zustad 1 gamoZaxeba. 

31. avtoqarxnis yovelkvireuli mocdenebis raodenoba ganawilebulia 

Semdegnairad: 

x 0 1 2 3 4 5 6 

P{ x} 

0.04 0.24 0.28 0.16 0.16 0.08 0.04 

a). ipoveT -s saSualo; b). ra iqneba mocdenebis erToblivi mosalodneli 

raodenoba momavali 48 kviris ganmavlobaSi? 

32. maTematikoss daaviwyda paroli da is cdilobs Sevides kompiuterSi. 

kompiuteri mxolod 4 mcdelobis saSualebas iZleva. yoveli mcdelobisas, 

damoukideblad sxva mcdelobebisagan, warmatebis albaTobaa 0.4. 

a). ipoveT kompiuterSi warmatebuli SeRwevis albaToba; b). SeadgineT 

mcdelobaTa raodenobis albaTuri ganawilebis kanoni. 

33. 52 kartidan SemTxveviT iReben sam karts dabrunebis gareSe. SeadgineT 

amoRebuli tuzebis ricxvis ganawilebis kanoni. 

34. yoveli CarTvisas eleqtronuli mowyobiloba gamoyofs 0, 1 an 3 

voltis tol Zabvas Sesabamisad 1/2, 1/3 da 1/6 albaTobebiT. iyos am mowyobilobis 

ori damoukidebeli CarTvisas gamoyofili Zabvis jami. a). Se- 

56

adgineT -s albaTuri ganawilebis kanoni; b). am mowyobilobis 360-jer 

damoukideblad CarTvisas ramdenjer unda velodoT Sedegs 1 volti? 

35. damwyebi meisre mizanSi axvedrebs albaTobiT 0.3. ipoveT albaToba 

imisa, rom mizanSi 6-jer srolisas meisre mizans moartyams 2-jer mainc. 

36. nikasa da mis oTx amxanags erTmaneTisagan damoukideblad aqvs 

prizis mogebis 0.45-is toli albaToba. ipoveT albaToba imisa, rom: a). 

zustad ori xuTi amxanagidan moigebs prizs; b). nika da mxolod erTi misi 

amxanagi moigebs prizs. 

37. agdeben 10 wesier saTamaSo kamaTels. Tqven giTxres, rom erT-er- 

Ti agdebis Sedegi aris 6 qula. ras udris albaToba imisa, rom sul cota 

orjer movida 6 qula. 

38. 8.00 saaTidan 22.00 saaTamde garkveul benzingasamarT sadgurze 

Semodis saSualod 0.8 avtomobili wuTSi. igulisxmeT, rom Semosvlis 

droebi SemTxveviTia da gamoTvaleT albaToba imisa, rom sul cota 2 

avtomobili Semova benzingasamarT sadgurze konkretuli erTi wuTis 

ganmavlobaSi 8.00 saaTidan 22.00 saaTamde. 

39. patara qalaqis saxanZro sadgurSi Ramis gamoZaxebaTa ricxvi modelirdeba 

puasonis ganawilebiT saSualoTi 4.2 Ramis ganmavlobaSi. a). 

ipoveT albaToba imisa, rom konkretul Rames saxanZro sadgurSi Semova 

3 an meti gamoZaxeba. b). ras unda akmayofilebdes saxanZro sadgurSi Semosuli 

gamoZaxebebi, rom is iyos puasonis modelis adeqvaturi? 

40. maRazia yidis ori firmis televizors. kviraSi gayiduli sonis 

firmis televizorebis saSualo ricxvia 1.8. gayidvebs adgili aqvs erTmaneTisagan 

damoukideblad SemTxveviT drois momentebSi. ipoveT albaToba 

imisa, rom: a). mocemul kviras gaiyideba zustad 2 sonis firmis televizori; 

b). mocemul orkvirian periodSi gaiyideba zustad 4 sonis firmis 

televizori. 

41. saxlis mepatrones surs baRSi daTesos balaxi. Teslis mobneva 

xdeba SemTxveviT da baRis konkretul 1 kv. santimetri farTobis mqone 

nawilSi davardnili Teslis raodenoba warmoadgens puasonis kanoniT 

ganawilebul SemTxveviT sidides, romlis saSualo baRis nawilis farTobis 

proporciulia. baRis farTobia 50 kv. metri da iTeseba 10 balaxis 

6 

Tesli. gamoTvaleT: a). -s saSualo; b). albaToba imisa, rom 1 kv. sm. far- 

Tobis mqone nawilSi ar daecema arcerTi Tesli; g). P{ 0an 4} 

. 

42. firmis TanamSromlebis mier samuSaos Sesrulebisas kviris ganmavlobaSi 

miRebuli tramvebis raodenobis ganawilebaa: 

tramvebis raodenoba kviraSi 0 1 2 3 4 da meti 

kvireebis raodenoba 31 17 3 1 0 

a). axseniT, rodis SeiZleba am ganawilebis modelad CaiTvalos puasonis 

ganawileba; b). gamoTvaleT am monacemebis saSualo da dispersia da axseniT 

ratom SeiZleba modelad gamodges puasonis ganawileba; g). gamoTvlili 

saSualos mixedviT gamoTvaleT puasonis ganawilebis Teoriuli sixSireebi 

imisa, rom kviraSi moxdeba 0, 1, 2, 3, 4 an meti sawarmoo tramva. 

57

43. umaRlesi ligis fexburTis matCebSi gatanili burTebis raodenobis 

analizma aCvena, rom SemTxveviT SerCeul matCSi gatanili golebis 

raodenobis modelad SeiZleba ganvixiloT puasonis ganawileba 

parametriT 2.7. sxvadasxva matCebSi gatanili golebis raodenoba 

erTmaneTisagan damoukidebelia. ipoveT albaToba imisa, rom: a). matCi 

dasruldeba golis gareSe; b). matCSi gatanili iqneba 4 an meti goli. 

44. wina amocanis pirobebSi cnobilia, rom SabaT dRes Catarda 11 

Sexvedra. a). ipoveT im Sexvedrebis mosalodneli ricxvi, romlebic dasruldeba 

golis gareSe; b). ipoveT albaToba imisa, rom golebi gatanili 

iqneba yvela 11 SexvedraSi; g). rogori ganawileba eqneba 11 SexvedraSi gatanili 

golebis jamur raodenobas; d). gamoTvaleT albaToba imisa, rom 

11 SexvedraSi gatanili iqneba 30 golze meti. 

amocanebi gamocdisaTvis 

45. eka esvris samiznes 10-jer. yoveli srolisas eka samiznes azianebs 

albaTobiT 0.4 damoukideblad sxva srolebis Sedegebisagan. iyos 

samiznis dazianebaTa ricxvi. ipoveT: a). -s saSualo da standartuli gadaxra; 

b). P{ 2} . 

46. agdeben wesier saTamaSo kamaTels, romelzec aRniSnulia cifrebi 

1, 1, 2, 3, 4, 5. ipoveT mosuli qulis saSualo da dispersia. 

47. nikas SeuZlia miznSi moartyas qva albaTobiT 0.4. is isvris qvas 

6-jer. ipoveT albaToba imisa, rom: a). qva mizans moxvdeba zustad 4-jer; 

b). qva samiznes pirvelad moxvdeba mesame gasrolisas. 

48. yoveli TamaSisas biWi igebs prizs albaTobiT 0.25. is TamaSobs 

10-jer. vigulisxmoT, rom TamaSebi urTierTdamoukidebelia. iyos mogebul 

prizTa raodenoba. ipoveT: a). E ; b). P{ 2} . 

49. dazga awarmoebs qsovils garkveuli dazianebebiT. dazianebebis 

raodenoba 1 metr qsovilSi aris SemTxveviTi sidide, romelic emorCileba 

puasonis ganawilebas saSualoTi 3. ipoveT albaToba imisa, rom qsovilis 

1 metri Seicavs 5 an met dazianebas. 

50. nika da rezo agdeben or-or wesier saTamaSo kamaTels erTdroulad. 

TiTeul kamaTelze mosuli qulebi ikribeba. ipoveT albaToba imisa, 

rom: a). nika moagrovebs 9 qulas; b). nika da rezo orive moagrovebs 9 

-- 9 qulas; g). nika da rezo moagroveben erTi da igive qulas; d). nikas 

mier mogrovili qula aRemateba rezos mier mogrovil qulas. 

51. wina amocanis pirobebSi davuSvaT, rom aris udidesi oTxive 

1 4 

kamaTelze mosuli qulidan. a). aCveneT, rom P{ x} ( x) 

, x 1, 2, , 6 ; b). 

6 

gadawereT da daasruleT Semdegi ganawilebis kanoni: 

x 1 2 3 4 5 6 

P{ x} 

1/1296 15/1296 671/1296 

58

g). gamoTvaleT E . 

52. ori telegrafisti gadascems garkveul teqsts. juanSeri teqstis 

gadacemisas saSualod uSvebs 2.7 Secdomas, xolo RvTisavari 2.5 Secdomas. 

vigulisxmoT, rom TiToeuli telegrafistis mier daSvebuli Secdomebis 

raodenoba emorCileba puasonis ganawilebas. gamoTvaleT albaToba 

imisa, rom teqstis gadacemisas: a). juanSeri dauSvebs 2 Secdomas; b). 

RvTisavari dauSvebs 3 Secdomas; g). juanSeri dauSvebs 2 Secdomas da 

RvTisavari dauSvebs 3 Secdomas. 

53. wina amocanis pirobebSi cnobilia, rom garkveuli teqstis gadacemisas 

juanSerma da RvTisavarma erTad dauSves 5 Secdoma. gamoTvaleT 

albaToba imisa, rom juanSerma meti Secdoma dauSva, vidre RvTisavarma. 

54. warsuli dakvirvebebidan iagom icis, rom fostalionis mier 

dRis ganmavlobaSi mis saxlSi motanili werilebis raodenoba ganawilebulia 

puasonis kanoniT saSualoTi 3. a). ipoveT albaToba imisa, rom Sem- 

TxveviT SerCeul dRes fostalioni iagos saxlSi moitans or werils; b). 

erT dRes iagom dainaxa, rom fostalioni uaxlovdeba mis saxls da Sesabamisad, 

man icis, rom fostalions moaqvs werili. gamoTvaleT albaToba 

imisa, rom am dRes fostalioni moitans or werils. 

55. SemTxveviT sidides aqvs puasonis ganawileba parametriT . 

albaToba imisa, rom miiRebs mniSvnelobas 1 an 2 aris p . a). dawereT 

p -s gamosaTvleli gamosaxuleba; b). daxazeT p -s grafiki, roca 0 

4; 

g). ipoveT -s is mniSvneloba, roca p miaRwevs maqsimums. 

56. SemTxveviT sidides aqvs puasonis ganawileba parametriT da 

akmayofilebs Tanafardobas: P{ 3} P{ 0} P{ 1} . a). gamoTvaleT -s 

mniSvneloba; b). gamoTvaleT P{2 

4} . 

57. satelevizo arxze axal ambebs uCveneben dRis erTi da igive 

dros. albaToba imisa, rom goderZi uyurebs axal ambebs konkretul 

dRes aris 0.4. ipoveT albaToba imisa, rom xuTi momdevno dRis ganmavlobaSi 

goderZi axal ambebs uyurebs ara umetes 3-jer. 


x 0 1 2 3 

P{ x} 

0.2 a b 0.25 

cnobilia, rom E 1.55 . a). ipoveT a da b ; b). ipoveT D . 

59. SemTxveviT sidides aqvs puasonis ganawileba parametriT . a). 

mocemulia, rom P{ 4} P{ 2} P{ 3} , ipoveT -s mniSvneloba; b). mocemulia 

3.2 , ipoveT albaTobebi: P{ 2} da P{ 3| 2} . 

60. moTamaSes Seaqvs n lari, irCevs 1, 2, 3, 4, 5 da 6 ricxvebidan 

erT ricxvs da agdebs sam wesier saTamaSo kamaTels. Tu arCeuli ricxvi 

mova samive kamaTelze, maSin is igebs Setanili Tanxis gaoTxmagebuls. 

Tu arCeuli ricxvi mova zustad or kamaTelze, maSin is igebs Setanili 

Tanxis gasammagebuls. Tu arCeuli ricxvi mova zustad erT kamaTelze, 

maSin is igebs Setanili Tanxis gaormagebuls. Tu arCeuli ricxvi ar mo- 

59

va arcerT kamaTelze, maSin is arafers igebs. a). gadawereT da daasruleT 

mogebuli Tanxis raodenobis ganawilebis kanoni: 

mogeba n n 2n 3n 

albaToba 75/216 

17 

b). aCveneT, rom moTamaSis mosalodneli mogeba aris 

216 n lari; g). ra 

Tanxa unda Seitanos moTamaSem, rom misi mosalodneli danakargi iyos 34 

TeTri? 

61. SemTxveviT sidides aqvs puasonis ganawileba da P{ 2} 0.404 . 

ipoveT P{ 2} . 

62. diskretul SemTxveviT sidides aqvs Semdegi ganawilebis kanoni: 

k 

, roca x 1, 2, 3, 4, 

P{ x} 

x 

0, sxvagan. 

gamoTvaleT: a). k mudmivis mniSvneloba; b). E . 

pasuxebi: 

1. Bi (10,1/ 6) : P 10 

(4) ; P 10 

(5) ; P 10 

(6) . 2. Bi (6, 24/ 49) : P 6 

(4) . 3. 0.0819; 0.0154; 0.0001; 

1.2. 4. 0.2561; 0.2048; 0.0005; 14/3. 5. 0.2119; 0.4728; 0.0498; 4.05. 6. 0.0017; 6. 7. 0.6123; 

0.3877. 8. 0.0781; 0.0176. 9. 0.6496. 10. 0.0545. 11. 0.1143. 12. 0.0652. 13. 0.2039; 7. 14. 

0.1424; 0.2821. 15. Bi (10,0.15) ; 1.5; 1. 16. 19000. 17. 0.224; 0.1991; 0.5768. 18. 0.2218; 

0.3696; 0.8352. 19. 0.7787; 0.6916; 0..209. 20. 0.1353; 0.2707; 0.2381. 21. 0.0821; 0.5162. 22. 

0.6065; 0.4634. 23. 0.0067; 49.9wm. 24. 0.0916; 0.206 nawilaki wamSi. 25. ki; ara; ki. 

26. 8, 8 ki, vinaidan saSualo = dispersia; 0.6866; 2.8. 27. 0.29, 0.39, 0.16, 0.13, 

0.03; 1.23, 1.21, puasoni gamosadegia; 0.29, 0.36, 0.22, 0.09, 0.03; puasoni 

misaRebia. 28. 0.5; 0.0144; 0.1217. 29. 0.0472; 0.162. 30. 0.135; 0.0733; 0.5. 31. 2.56; 122.9. 

32. 0.8704; 1, 2, 3, 4 – 0.4, 0.24, 0.144, 0.216. 33. 0, 1, 2, 3 – 248/1105, 496/1105, 

304/1105, 57/1105. 34. 0, 1, 2, 3, 4, 5, 6 – 1/4, 1/3, 1/9, 1/6, 1/9, 1/36; 120. 35. 0.58. 36. 

0.337; 0.135. 37. 0.615. 0.615. 38. 0.191. 39. 0.79; gamoZaxebebi unda Semodiodes Sem- 

TxveviT, damoukideblad, TiTo-TiTod da mudmivi intensivobiT. 40. 0.268; 

0.191. 41. 2; 0.135; 0.188. 42. tramvebi unda xdebodes SemTxveviT, erTmaneTisagan 

damoukideblad, TiToTiTod da mudmivi intensivobiT; 0.5, 0.481, SeiZleba 

radganac saSualo da dispersia daaxloebiT tolia; 32, 16, 4, 1, 0. 43. 

0.067; 0.286. 44. 0.739; 0.465; Po (29.7) ; 0.430. 

60

Tavi V 

uwyveti tipis ganawilebebi 

SemTxveviT sidides ewodeba uwyveti tipis, Tu misi ganawilebis 

funqcia -- F ( x): P{ x} 

uwyvetia. Tu ganawilebis funqcia warmoidgineba 

 

x 

F( x) f( u) 

du saxiT, maSin f () x funqcias ewodeba uwyveti SemTxvevi- 

 

Ti sididis ganawilebis simkvrive. simkvrives aqvs Semdegi Tvisebebi: 

a). f( x) 0 yoveli x -saTvis; 

 

b). f 

( x) dx 1; 

 

b 

g). P{ a, b} F ( b) F ( a) f ( x) 

dx , sadac ab 

, aris nebismieri 

 

a 

( ab , ), ( ab , ] , [ ab , ) , [ ab , ] intervalebidan. 

uwyveti SemTxveviTi sididis mediana M aris is mniSvneloba, romelic 

simkvrivis grafikis qveS moTavsebul farTobs yofs or tol nawilad. 

maTematikurad is ase ganimarteba: 

M 

1 

P{ M} F( M) f( x) 

dx . 

 

2 

uwyveti SemTxveviTi sididis p -kvantili ewodeba iseT x p 

mniSvnelobas, 

romel mniSvnelobamdec simkvrivis grafikis qveS moTavsebuli far- 

T-obi p -s tolia: 

gasagebia, rom x 0.5 

M . 

P{ xp} F( xp) f( x) 

dx p . 

uwyveti SemTxveviTi sididis qveda kvartili, Q 1 

(Sesabamisad, zeda 

kvartili, Q 3 

) ewodeba 1 4 -kvantils (Sesabamisad, 3 4 -kvantils). 

sxvaobas Q 3 

Q 1 

kvartilTSorisi gabnevis diapazoni ewodeba. 

ganawilebis (1 ) 

-kvantils -zeda kritikuli wertili ewodeba. 

uwyveti SemTxveviTi sididis moda m ewodeba argumentis im mniSvnelobas, 

sadac simkvrive aRwevs maqsimums: f ( m) max f ( x) 

. 

uwyveti SemTxveviTi sididis maTematikuri lodini (anu saSualo) 

ganimarteba Semdegnairad: 

 

x p 

 

E 

xf () x dx 

 

. 

uwyveti SemTxveviTi sididis dispersia moicema formuliT: 

 

 

x 

 

61

2 2 2 

() 

 

D x f x dx . 

asimetriis koeficienti a gamoiTvleba formuliT: 

 

1 

3 

a ( x ) f ( x) 

dx 

3 

 

. 

 

eqscesis koeficienti e tolia: 

 

1 

4 

e ( x ) f ( x) dx 3 

4 

 

. 

 


magaliTi 1. uwyveti SemTxveviTi sididis ganawilebis simkvrivea: 

 

 

2 x , Tu 1 x 

2; 

f 

() x 3 

0, sxvagan. 

a). SeamowmeT, rom f () x akmayofilebs simkvrivis a) da b) Tvisebas; 

b). gamoTvaleT P{1.5 

2} . 

x 0 ; garda amisa, 

amoxsna. a). f( x) 0 yoveli x -saTvis, vinaidan 2 0 

2 2 x 1 

 

2 2 

2 2 2 

f 

( x) dx xdx | 

1 

(2 1 ) 1. 

 

1 

3 3 2 3 

2 2 

2 x 2 1 2 2 1 

 

1.5 

1.5 

3 2 3 3 

b). N P{1.5 

2} f ( x) dx | 

(2 1.5 ) 1.75 0.583 . 

3 x , roca 


2 

 

k(1 x ), Tu 1 x 1; 

f 

() x 0, 

sxvagan, 

sadac k mudmivia. ipoveT: a). k ; b). P{0.3 

0.6} ; g). P{| | 0.2} . 

amoxsna. a). visargebloT simkvrivis b) TvisebiT: 

 

1 

2 1 3 1 

1 f 

( x) dx k(1 x ) dx k( x x ) | 

1 

 

1 

3 

1 3 1 3 4 4 8k 

k(1 1 ) k[( 1) ( 1) ] k k 

( ) , 

3 3 3 3 3 

e. i. 8 k 1 . saidanac k 3/8 ; 

3 

0.6 

3 2 3 1 3 0.6 

b). 

P{0.3 

0.6} (1 x ) dx ( x x )| 

0.3 

8 8 3 

0.3 

62

3 1 3 3 1 3 

(0.6 0.6 ) (0.3 0.3 ) 0.136 ; 

8 3 8 3 

0.2 

3 2 3 1 3 0.2 

g). P{| | 0.2} P{ 0.2 0.2} (1 x ) dx ( x x )| 

0.2 

 

0.2 

8 8 3 

3 1 3 3 1 3 

(0.2 0.2 ) [ 0.2 ( 0.3) ] 0.152 . 

8 3 8 3 

magaliTi 3. savaWro centris gamyidvelTa wliuri xelfasi , gazomili 

1000 larebSi, modelirdeba albaTuri ganawilebis simkvriviT: 

7/2 

cx , Tu x 16; 

f 

() x 0, 

sxvagan. 

ipoveT: a). c -s mniSvneloba; b). albaToba imisa, rom SemTxveviT SerCeuli 

gamyidvelis wliuri xelfasi moTavsebulia 20 000 larsa da 30 000 lars 

Soris. 

 

 

7/2 2 5/2 2 5/2 

c 

amoxsna. a). 1 f 

( x) dx cx dx cx | 

16 ( 0) ( c16 ) , 

 

16 

5 5 2560 

saidanac c 2560 ; 

30 

7/2 2 5/2 30 

b). 

P{20 30} 2560 x dx 2560 ( ) x | 

20 

20 

5 

2 5/2 2 5/2 

2560 ( ) 30 2560 ( ) 20 0.365 . 

5 5 

magaliTi 4. 100 000 litrebSi gazomili benzinis yovelkvireuli gayidvebi 

aRiwereba ori A da B modeliT. A modelis mixedviT: 

2 x, Tu 0 x1; 

f 

() x 

0, sxvagan, 

xolo B modelis Tanaxmad: 

3 2 

 

12 x (1 x ), Tu 0 x 1; 

f 

() x 0, 

sxvagan. 

a). ipoveT pirveli modelis mediana M 

A 

; b). aCveneT, rom meore 

modelsac igive mediana aqvs, MB MA. 

amoxsna. a). ganmartebis Tanaxmad: 

MA 

MA 

2 

1 

x MA 

2 

f 

( x) dx 2xdx 2 | 

0 

( M 

A) 

2 . 

2 

 

amitom MA 

1/ 2 ; 

b). analogiurad: 

MB 

MB 

1 

3 2 4 6 MB 

f 

( x) dx 12 x (1 x ) dx (3x 2 x ) | 

0 

 

2 

 

0 

0 

4 6 4 2 

B B B B 

3 ( M ) 2 ( M ) ( M ) [3 2 ( M ) ] . 

63

gavixsenoT, rom ( MA) 

. meore mxriv, radganac 

2 

1 2 1 1 1 

( ) [3 2 ] 2 

, amitom cxadia, rom MB 

MA 

1/ 2 . 

2 2 4 2 

magaliTi 5. amocana 3-Si gamoTvaleT wliuri xelfasis: a). mediana; 

b). qveda da zeda kvartilebi; g). moda; d). maTematikuri lodini. 

amoxsna. a). gvaqvs: 

M 

M 

1 7/2 2 5/2 M 

5/2 

f 

( x) dx 2560 x dx 2560 x | 

16 

1024 M 1 

2 . 

5 

 

16 

2 1 

5/2 

amitom medianisaTvis vRebulobT gantolebas: 1024 M 1 1/ 2. 

5/2 

saidanac advilad davaskvniT, rom M 2048 , anu M 21.1 . Sesabamisad, 

wliuri xelfasis mediana iqneba 21.11000 21100 lari; 

b). ganmartebis Tanaxmad: 

Q1 Q1 

1 7/2 2 5/2 Q1 

5/2 

f 

( x) dx 2560 x dx 2560 x | 

16 

1024Q 

1 

1 

4 . 

 

16 

5 

Sesabamisad, qveda kvartilisaTvis gvaqvs gantoleba: 

5/2 

4 

1024 Q 

2/5 

 

1 

1 1/4, saidanac gvaqvs: Q1 

(1024 ) 18 . analogiurad 

3 

davrwmundebiT, rom zeda kvartili Q 3 

27.9 ; 

g). vinaidan simkvrive klebadia funqciaa intervalze [16, ) , amitom 

moda iqneba m 16 . Sesabamisad, wliuri xelfasis modaa 161000 16000 

lari. 

d). ganmartebis Tanaxmad: 

 

7/2 5/2 

E 

xf 

( x) dx x 2560 x dx 2560 x dx 

 

16 16 

 

2 3/2 2 3/2 2 

2560 ( ) x | 16 02560 ( ) 16 | 

16 26 . 

3 3 3 

e. i. gamyidvelTa xelfasis saSualoa 

2 

26 1000 26700 

3 lari. 

SevniSnavT, rom saSualo metia medianaze, vinaidan ganawileba dadebiTad 

asimetriulia. 

magaliTi 6. gamoTvaleT moda amocana 4-is B modelSi. 

amoxsna. gavawarmooT simkvrive da gavutoloT nuls: 

3 2 ' 2 4 2 2 

[12 x (1 x )] 36x 60x 12 x (35 x ) 0 . 

am gantolebis fesvebia: 0, 3/5 da 3/5 , romelTagan 0x 

1 

3 2 

SualedSi vardeba mxolod 3/5 da aq 12 x (1 x ) funqcia aRwevs 

maqsimums. Sesabamisad, moda aris 3/5 0.775 . amitom gayiduli benzinis 

modaa: 

0.775 100 000 77 500 litri. 

64

magaliTi 7. benzingasamarTi sadguris yovelkvireuli moTxovna benzinze 

gazomili 1000 litrebSi modelirdeba simkvrivis funqciiT: 

 

 

f 

() x 0, 

sxvagan. 

gamoTvaleT saSualokvireuli moTxovna benzinze. 

amoxsna. ganmartebis Tanaxmad: 

2 

120 x (1 x), Tu 0 x 1; 

 

1 1 

2 

 

3 4 

 

0 0 

4 5 1 

(30x 

24 x )| 

030 24 6 . 

E 

xf ( x) dx x120 x (1 x) dx 120 ( x x ) dx 

Sesabamisad, benzingasamarTi sadguris saSualo moTxovna benzinze 

kviraSi Seadgens: 61000 6000 litrs. 


 

 

3 x (2 x ), Tu 0 x 2; 

f 

() x 4 

0, sxvagan. 

ipoveT: a). saSualo; b). dispersia; g). P{ } 

. 

amoxsna. a). 

b). 

 

3 3 3 

E 

xf x dx x x x dx x x dx 

 

2 2 

2 3 

( ) (2 ) ( ) 

0 

4 

0 

2 4 

 

3 4 

x 3x 

2 

( )| 

0 43 1; 

2 16 

 

2 2 

2 2 2 2 3 2 3 3 3 4 

D x f 

( x) dx 

x x(2 x) dx 1 ( x x ) dx 1 

4 2 4 

 

0 0 

4 5 

3x 

3x 

2 24 1 

( )| 

0 

1 (6 ) 0 1 0.2 

; 

8 20 5 5 

g). P{ } P{1 0.2 

1 0.2} 

3 3 3 3x 

x 

 

1 

0.2 1 

0.2 2 3 

2 1 

0.2 

x(2 x) dx ( ) ( ) | 

1 0.2 

4 x x dx 

 

2 4 4 4 

1 

0.2 1 

0.2 

2 3 2 3 

3 (1 0.2) (1 0.2) 3 (1 0.2) (1 0.2) 

( ) ( ) 

4 4 4 4 

2 3 2 3 

3 (1.447) (1.447) 3 (0.552) (0.552) 

( ) ( ) 0.626 . 

4 4 4 4 

magaliTi 9. mocemulia ganawilebis simkvrive: 

0, x 0, 

 

f ( x) cos x, 0 x / 2, 

 

0, x / 2. 

ipoveT ganawilebis funqcia. 

65

x 

amoxsna. visargebloT formuliT: F( x) f ( u) 

du . 

Tu u 0 , maSin fu ( ) 0 . Sesabamisad, F( x) 0du 

0; 

Tu 0 u 

/ 2, maSin 

Tu x /2, maSin 

sabolood gvaqvs: 

0 

66 

 

x 

 

F( x) 0du cos udu sin x 

x 

; 

0 

0 /2 

x 

/2 

0 

. 

0 /2 

F( x) 0du cos udu 0du sin x | 1 

0, x 0, 

 

F( x) sin x, 0 x 

/ 2, 

 

0, x 

/ 2. 

amocanebi 

1. SemTxveviTi sididis ganawilebis simkvrivea: 

1 

c(1 x), Tu 0 x 8; 

f 

() x 8 

0, sxvagan. 

ipoveT: a). c mudmivis mniSvneloba; b). P{ 6} ; g). P{4 

6} . 


2 

cx 

, Tu 0 x 

3; 

f 

() x 0, 

sxvagan. 

ipoveT: a). c mudmivis mniSvneloba; b). P{ 2} ; g). P{1.5 

2.5} ; d). x -is 

mniSvneloba, Tu cnobilia, rom P{ x} 0.2 . 


2 

cx 

, Tu 0 x 

3; 

f 

() x 0, 

sxvagan. 

ipoveT: a). c mudmivis mniSvneloba; b). P{ 1.5} . 


2 

 

c(4 x ), Tu 2 x 2; 

f 

() x 0, 

sxvagan. 

ipoveT: a). c mudmivis mniSvneloba; b). P{ 0} ; g). P{ 1} ; d). P{| | 1} ; e). 

P{ 0.5 

0.5} . 

5. kompiuteris “kartrijis” muSaobis xangrZlivobaa saaTi. Sem- 

TxveviTi sididis ganawilebis simkvrivea: 

2 

cx 

, Tu x 400; 

f ( x) 

 

0, 

sxvagan.

gamoTvaleT c mudmivis mniSvneloba. ipoveT albaToba imisa, rom: a). “kartriji” 

imuSavebs sul cota 500 saaTi; b). “kartriji” Sesacvleli gaxdeba 

manam sanam is imuSavebs 600 saaTi; g). ori “kartriji” Sesacvleli 

iqneba manam sanam TiToeuli imuSavebs 600-600 saaTi; d). oTxi “kartrijidan” 

ori imuSavebs 600 saaTze mets, xolo ori 600 saaTze naklebs. 


2 

 

c( x a) , Tu 0 x a; 

f 

() x 0, 

sxvagan. 

3 

ipoveT: a). a da c mudmivis mniSvnelobebi, Tu a c 3; b). P{ a/ 2} . 


1 

2 

x , Tu 0 x3; 

f 

() x 9 

0, sxvagan. 

ipoveT SemTxveviTi sididis: a). mediana; b). qveda da zeda kvartilebi. 

8. kompiuteris “kartrijis” saRebavis muSaobis xangrZlivoba , gazomili 

saaTebSi, modelirdeba albaTuri ganawilebis simkvriviT: 

2 

 

400 x , Tu x 

400; 

f 

() x 0, 

sxvagan. 

ipoveT “kartrijis” muSaobis xangrZlivobis mediana. 


2x 4, Tu 2 x 

3; 

f 

() x 

0, sxvagan. 

a). daxazeT simkvrivis grafiki; b). ipoveT SemTxveviTi sididis mediana 

da moda; g). ipoveT kvartilTSorisi gabnevis diapazoni. 


2 1 

(1 x), Tu 0 x 

5; 

f 

() x 5 5 

0, sxvagan. 

a). daxazeT simkvrivis grafiki; b). ipoveT SemTxveviTi sididis mediana 

da modaluri mniSvneloba. 


 

 

3 x (2 x ), Tu 0 x 2; 

f 

() x 4 

0, sxvagan. 

ipoveT SemTxveviTi sididis modaluri mniSvneloba. 


2 

cx (1 x), Tu 0 x 1; 

f 

() x 0, 

sxvagan. 

67

a). ipoveT c ; b). gamoTvaleT moda; g). dawereT gantoleba, romelsac akmayofilebs 

mediana M . 


1 

2 

x , Tu 0 x3; 

f 

() x 9 

0, sxvagan. 

ipoveT saSualo da dispersia. 


2x 4, Tu 2 x 

3; 

f 

() x 

0, sxvagan. 

ipoveT saSualo da dispersia. 


1 1 

(1 x), Tu 0 x 

8; 

f 

() x 4 8 

0, sxvagan. 

a). daxazeT simkvrivis grafiki; b). ipoveT saSualo da dispersia. 

16. sawarmos mier gamoSvebuli silikonis masa , gazomili kilogramebSi, 

modelirdeba ganawilebis simkvriviT: 

3 (4 2 

x x ), Tu 0 x 4; 

f 

() x 32 

0, sxvagan. 

a). daxazeT simkvrivis grafiki; b). ipoveT silikonis masis saSualo da 

dispersia. 

17. elementebis muSaobis xangrZlivoba , gazomili saaTebSi, modelirdeba 

ganawilebis simkvriviT: 

4 

 

3000 x , Tu x10; 

f 

() x 0, 

sxvagan. 

a). daxazeT simkvrivis grafiki; b). ipoveT elementebis muSaobis 

xangrZlivobis saSualo da dispersia. 

18. kompiuteris “kartrijis” muSaobis xangrZlivoba , gazomili 

saaTebSi, modelirdeba albaTuri ganawilebis simkvriviT: 

 

2 

cx 

, Tu 400 x 

900; 

f 

() x 0, 

sxvagan. 

a). gamoTvaleT c -s mniSvneloba; b). daxazeT simkvrivis grafiki; g). 

ipoveT “kartrijis” muSaobis xangrZlivobis saSualo da dispersia. 

19. SemTxveviTi sididis ganawilebis funqciaa: 

0, x 2, 

 

F( x) 0.5 x, 2 x 4, 

 

1, 

x 4. 

68

ipoveT albaToba imisa, rom SemTxveviTi sidide miiRebs mniSvnelobas, 

romelic: a). naklebia 0.2-ze; b). naklebia 3-ze; g). ar aris naklebi 3-ze; d). 

ar aris naklebi 5-ze. 

20. SemTxveviTi sididis ganawilebis funqciaa: 

0, x 0, 

2 

F( x) x , 0 x 1, 

 

1, x 1. 

ipoveT albaToba imisa, rom 4 damokidebeli dakvirvebis Sedegad SemTxveviTi 

sidide 3-jer miiRebs mniSvnelobas (0.25, 0.75) Sualedidan. 

21. ipoveT ganawilebis simkvrive, Tu ganawilebis funqciaa: 

0, x 0, 

 

F( x) sin x, 0 x 

/ 2, 

 

1, x 

/ 2. 

22. ipoveT ganawilebis simkvrive, Tu ganawilebis funqciaa: 

0, x 0, 

 

F( x) sin 2 x, 0 x / 4, 

 

1, x / 4. 

23. ipoveT ganawilebis funqcia, Tu ganawilebis simkvrivea: 

sin x, 0 x 

/ 2, 

f() 

x 0, sxvagan. 

24. ipoveT ganawilebis funqcia, Tu ganawilebis simkvrivea: 

x1/ 2,1 x 

2, 

f() 

x 0, sxvagan. 

25. ipoveT [2, 8] monakveTze Tanabrad ganawilebuli SemTxveviTi sididis 

ganawilebis simkvrive da standartuli gadaxra. 

pasuxebi: 

1. 1/4; 1/16; 3/16. 2. 1/9; 8/27; 147/324; 1.75. 3. 1/15; 11/40. 4. 3/32; 1/2; 5/32; 5/16; 47/128. 

1 

5. 400; 4/5; 1/3; 1/9; 8/27. 6. 3; 1/9; 1/8. 7. 2.38; 2.73; 1.89. 8. 800. 9. --; 2 2, 2; 

2 

1 5 

( 3 1) 

4 3 

. 10. --; 5 2, 0 . 11. 1. 12. 12; 2/3; 6M 

8M 

1 0. 13. 9/4, 27/80. 14. 

2 

2 

8/3, 1/18. 15. --; 8/3, 32/9. 16. --; 2, 4/5. 17. --; 15, 75. 18. 720; --; 584, 19100. 19. 0; 0.5; 

0.5; 0. 20. 0.25 ( Bi(4,0.5): P4 

(3) 0.25 ). 21. cos x (0, /2]( x) 

. 22. 2cos2 x (0, /4]( x) 

. 23. 

2 

F( x) 0, x 0; 1cos x, 0 x / 2; 1, x / 2 . 24. F( x) 0, x 1; 

1/ 2( x x) 

, 

1 

1x 

2; 1, x 2. 25. 

[2,8]() 

x ; 3 . 

6 

69

Tavi VI 

normaluri ganawileba 

SemTxveviT sidides ewodeba normaluri da aRiniSneba simboloTi 

2 

Na ( , ), Tu mis ganawilebis simkvrives (Sesabamisad, ganawilebis funqcias) 

aqvs saxe: 

2 

( xa) 

 

2 

2 

x 

1 

 

1 

2 

f 2 () x e (Sesabamisad, 

2 

2 () 

N( a, ) 

( , ) 

2 

F x Na 

2 

e 

 

dt 

 

 

). 

 

standartuli normaluri ganawilebis ( N (0,1) ) simkvrive (Sesabamisad, 

ganawilebis funqcia) aRiniSneba simboloTi: 

x 

1 

( x): f 

2 

N(0,1) 

( x) 

e (Sesabamisad, 

2 

2 

2 

2 

( ta) 

2 

x t 

1 

( x): F 

2 

N(0,1) 

( x) 

e dt 

2 

; garda 

x t 

1 

amisa, ixmareba aRniSvna: 

2 

0( x): 

e dt 

2 

). 

0 

cxadia, rom: ( x) ( x) 

; ( x) 1 ( x) 

; ( x) 0.5 

0( x) 

, x 0 ; 

2 

N( a, ) a 

N(0,1) 

; P{ N(0,1) c, d} ( d) ( c) 

; 

 

2 

a, 

 

2 

d a c a 

2 

x 

a, 

0.1 0.1 p 

a 

P{ N( a, ) c, d} ( ) ( ) ; xp 

 

xp 

a ; xp 

, sadac 

 

 

2 

x 

2 

aris Na ( , )-is p -kvantili. 

a, 

p 

Tu 0. 5, maSin 0( 

z ) 0. 5 

; xolo Tu 0. 5, maSin 

0( 

z 

) 0.5 , sadac z standartuli normaluri ganawilebis zeda - 

kritikuli wertilia. 

 


magaliTi 1. vipovoT albaToba imisa, rom standartuli normaluri 

SemTxveviTi sidide naklebia 1.18-ze, PN ( (0,1) 1.18) ? 

amoxsna. davxazoT standartuli normaluri ganawilebis simkvrivis 

wiri da abscisTa RerZze avRniSnoT 1.18-is Sesabamisi wertili (am SemTxvevaSi 

z 1.18 ). 

70

normaluri ganawilebis cxrilis pirvel svetSi movZebnoT ricxvi 

1.1, xolo pirvel striqonSi ki – ricxvi .08. 

1.1-is Sesabamisi striqonisa da .08-is Sesabamisi svetis gadakveTaze 

vpoulobT ricxvs – 0.8810. amitom saZiebeli albaToba iqneba 

PN ( (0,1) 1.18) 0.8810 anu 88.10%. 


SemTxveviTi sidide naklebia -0.63-ze, PN ( (0,1) 0.63) ? 

amoxsna. am SemTxvevaSi z 0.63 da saZiebeli albaToba iqneba 

P( N(0,1) 0.63) P( N(0,1) 0.63) 1 (0.63) 10.7357 0.2643 anu 26.43%. 


SemTxveviTi sidide metia -1.48-ze, PN ( (0,1) 1.48) ? 

amoxsna. am SemTxvevaSi z 1.48 da gamosaTvlelia normaluri ganawilebis 

wiris qveS -1.48-is marjvniv moTavsebuli aris farTobi. 

Tu visargeblebT normaluri ganawilebis simetriulobiTa da () z 

funqciis cxrilebiT, miviRebT: 

P( N(0,1) 1.48) P( N(0,1) 1.48) (1.48) 0.9306 . 

SeniSvna. SegviZlia visargebloT sawinaaRmdego xdomilebis albaTobis 

gamosaTvleli formuliT da maSin magaliTi daiyvneba wina magali- 

Tze: 

P( N(0,1) 1.48) 1 P( N(0,1) 1.48) 1 P( N(0,1) 1.48) 10.0694 0.9306 . 

71


SemTxveviTi sidide moTavsebulia SualedSi -1.65-sa da 1.65-s Soris, 

P( 1.65 N(0,1) 1.65) ? 

amoxsna. am SemTxvevaSi saZiebeli albaToba iqneba; 

P( 1.65 N(0,1) 1.65) (1.65) ( 1.65) 

 

(1.65) [1 (1.65)] 2 (1.65) 1 20.9505 1 0.9010 . 

SevniSnavT, rom amoxsnis procedurebi sqematurad gamosaxulia qvemoTmoyvanil 

sam naxazze: 

magaliTi 5. vipovoT z -is iseTi mniSvneloba, romlis marjvniv standartuli 

normaluri ganawilebis wiris qveS moTavsebuli aris farTobi 

tolia 0.2000-is? 

amoxsna. cxadia, rom es amocana tolfasia z -is iseTi mniSvnelobis 

moZebnis, romlis marcxniv normaluri ganawilebis wiris qveS moTavsebuli 

aris farTobi tolia 1-0.2000=0.8000-is. normaluri ganawilebis cxril- 

Si vpoulobT 0.8000-sTan yvelaze axlos mdgom ricxvs -- 0.7995-s. es ricxvi 

dgas 0.8-is Sesabamisi striqonisa da .04-is Sesabamisi svetis gadakveTaze. 

amitom gasagebia, rom z -is saZiebeli mniSvnelobaa z 0.84 . 

72

magaliTi 6. davuSvaT, rom avtomobilis dazianebis SemTxvevaSi avariul 

gamoZaxebaze reagirebis saSualo dro aris 25 wuTi. CavTvaloT, 

rom es sidide ganawilebulia daaxloebiT normalurad da misi standartuli 

gadaxra tolia 4.5 wuTis. SemTxveviT SerCeul iqna 80 gamoZaxeba. 

daaxloebiT ramden maTganze moxdeba regireba 15 wuTze nakleb droSi? 

amoxsna. am amocanis amosaxsnelad vipovoT normaluri wiris qveS 

15-is marcxniv mdebare aris farTobi. 

nabiji 1. davxazoT grafiki da movniSnoT saZiebeli are. 

nabiji 2. vipovoT 15-is z mniSvneloba: 

X 

15 25 10 

z 2.22 . 

4.5 4.5 

nabiji 3. vipovoT z 2.22 

-sa da z 0-s Soris mdebare aris farTobi. 

is tolia 0.4868-is. 

nabiji 4. gamovakloT 0.5000-s 0.4868. miviRebT 0.0132-s. 

nabiji 5. imisaTvis, rom gavigoT randen gamoZaxebaze moxda reagireba 

15 wuTze nakleb droSi, gavamravloT SerCevis moculoba (80) miRebul 

farTobze (0.0132). maSin miviRebT 1.056-s. maSasadame, 1.056 anu daaxloebiT 

erT gamoZaxebaze moxdeba reagireba 15 wuTze nakleb droSi. 

magaliTi 7. cnobilia, rom abiturientebis mxolod 10% SeiZleba 

gaxdes studenti. CavTvaloT, rom abiturientebis mier mogrovili qulebis 

mniSvnelobebi normaluradaa ganawilebuli saSualoTi 200 da standartuli 

gadaxriT 20. vipovoT is minimaluri qula, romelic saWiroa raTa 

abiturienti gaxdes studenti. 

amoxsna. vinaidan abiturientebis mier mogrovili qulebis mniSvnelobebi 

normaluradaa ganawilebuli, amitom im qulis mniSvneloba ( X ), 

romlis zeviTac abiturienti gaxdeba studenti, aris iseTi ricxvi, romlis 

marjvniv normaluri wiris qveS moTavsebuli aris farTobi tolia 

10%-is anu 0.1000-is: 

nabiji 1. imisaTvis, rom vipovoT normaluri wiris qveS 200-sa da 

73

X -s Soris mdebare aris farTobi 0.5000-s gamovakloT 0.1000, miviRebT 

0.4000-s. 

nabiji 2. vipovoT z mniSvneloba, romelic Seesabameba normaluri 

ganawilebis cxrilSi 0.4000-s. im SemTxvevaSi, roca cxrilSi ar iZebneba 

zustad es mniSvneloba viRebT masTan yvelaze axlos myofs, am SemTxveva- 

Si 0.3997-s. Sesabamisi z 1.28 . 

nabiji 3. SevitanoT 1.28 z mniSvnelobis gamosaTvlel formulaSi 

z( X )/ 

da amovxsnaT X . 

1.28 X 

 

 

 

1.28 20 200 X 

X 25.60 200 225.60 

X 226 . 

e.i. imisaTvis, rom abiturienti gaxdes studenti, man unda moagrovos 

226 qula. 

SeniSvna. im SemTxvevaSi, roca farTobis mniSvneloba vardeba cxrilis 

ori mniSvnelobis zustad SuaSi, maSin viRebT maTi Sesabamisi z mni- 

Svnelobebidan ufro dids. magaliTad, Tu farTobis mniSvnelobaa 0.4500, 

is imyofeba 0.4495-isa da 0.4505-is SuaSi da z mniSvnelobad viRebT 1.65-s 

da ara 1.64-s. 

sawyisi mniSvnelobis gamoTvla z mniSvnelobis mixedviT: 

X z . 

magaliTi 8. samedicino gamoklvelis mizniT mkvlevars surs SearCios 

arteriuli wnevis safuZvelze Sedgenili populaciis SuaSi mdgomi 

adamianebis 60%. vigulisxmoT, rom arteriuli wneva ganawilebulia normalurad 

saSualoTi 120 da standartuli gadaxriT 8. davadginoT SesarCevi 

adamianebis arteriuli wnevis zeda da qveda sazRvari. 

amoxsna. gasagebia, rom SesarCevi adamianebis arteriuli wnevis zeda 

da qveda sazRvari iqneba is ori mopirdapire ricxvi, romelTa gareT normaluri 

wiris qveS moqceuli TiToeuli kudis farTobia 20%. 

X 

2 

120 X 

1 

cxadia, rom z mniSvneloba, romelic Seesabameba normaluri ganawilebis 

cxrilSi 0.5000-0.2000=0.3000 farTobs tolia 0.84-is. SevitanoT igi 

X z formulaSi. miviRebT, rom 

X1 z 0.84 8120 126.72 . 

74

meores mxriv, standartuli normaluri ganawilebis saSualos mimarT 

simetriulobis gamo, X 

2 

-is gamosaTvlelead unda aviRoT z 0.84 . 

Sesabamisad gveqneba: 

X2 z 0.84 8120 113.28 . 

e. i. mkvlevarma unda SearCios iseTi adamianebi, romelTa arteriuli 

wneva moTavsebulia 113.8-sa da 126.72-s Soris, 

113.28 X 

126.72 . 

2 

magaliTi 9. mocemulia N(23, ) da P{ 27} 0.83 . ipoveT . 

23 

amoxsna. cxadia, rom Z N(0,1) . garda amisa, xdomilebebi 

 

23 27 23 

{ 27} da { } eqvivalenturia. amitom 

 

23 27 23 4 

P{ 27} P{ } P{ Z } 0.83 , 

 

4 

anu ( ) 0.83 . saidanac, magaliTi 5-is analogiurad: 4 0.9542 da, Sesabamisad, 

4.19 . 

 

 

4 

0.9542 

magaliTi 10. mas Semdeg rac nika Camosvams ekas manqanidan qalaqis 

centralur fostasTan, is moZraobs qalaqSi da garkveuli drois Semdeg 

brundeba ekas wasayvanad. nikas varaudiT ekas mier fostaSi gatarebuli 

dro daaxloebiT normaluradaa ganawilebuli saSualoTi 6 wuTi da 

standartuli gadaxriT 1.3 wuTi. ramdeni wuTis Semdeg unda dabrundes 

nika fostasTan, rom ar mouwios lodini sul cota 95%-iani garantiiT. 

amoxsna. T iyos ekas mier fostaSi gatarebuli dro. maSin 

2 

T N(6,1.3 ). t iyos fostasTan ekas Camosmidan nikas ukan dabrunebamde 

gasuli dro. mosaZebnia iseTi t , romlisTvisac: P{ T t} 0.95 . es Tanafardoba 

tolfasia: P{ 

t 6 

} 0.95 , anu ( ) 0.95 (1.645) , saidanac 

T 6 t 6 

1.3 1.3 

1.3 

t 6 gvaqvs: 1.645 . aqedan advilad miviRebT, rom t 8.1385 8.14 . maSasadame, 

1.3 

nika ar unda dabrundes 8.14 wuTze adre. 

magaliTi 11. biologi agrovebs monacemebs konkretuli saxeobis kaqtusis 

simaRlis Sesaxeb. misi dakvirvebiT kaqtusebis 34.25%-is sigrZe 12 

sm-ze naklebia, xolo 18.4%-is sigrZe 16 sm-ze metia. biologma dauSva, 

rom simaRle ganawilebulia normalurad. vipovoT am ganawilebis saSualo 

da standartuli gadaxra. 

2 

amoxsna. kaqtusis simaRle avRniSnoT H , H N( a, ) . mosaZebnia a 

da . biologis dakvirvebis Tanaxmad: PH { 12} 0.342 da PH { 16} 0.184 . 

Sesabamisad, 

12a 

16a 

( ) 0.342 da ( ) 10.184 0.816 . 

 

 

75

amitom 12 a 

16a 

0.407 da 0.900 . saidanac vaskvniT, rom: a 13.2 

 

 

da 3.06 . 

magaliTi 12. supermarketSi dRis ganmavlobaSi gayiduli citrusebis 

raodenoba modelirdeba normaluri ganawilebiT. aRmoCnda, rom xangr- 

Zlivi periodis manZilze dReSi saSualod iyideboda 35 kg. citrusi, xolo 

15 kg-ze naklebi gayidul iqna saSualod yoveli 20 dRidan erT dRe- 

Si. a). gamoTvaleT gayidvebis standartuli gadaxra ; b). cnobilia, rom 

konkretul dRes gaiyida 53 kg-ze meti citrusi. ipoveT albaToba imisa, 

rom am dRes gaiyida 56 kg-ze meti citrusi. 

amoxsna. dReSi gayiduli citrusebis raodenoba aRvniSnoT -Ti. ma- 

2 

1 

Sin: N(35, ) da P{ 15} 0.05 . 

20 

1535 

a). 

P{ 15} 0.05 ( ) 0.05 

 

15 35 

1.6448 

12.2 ; 

 

P{ 56 53} P{ 56} 

b). 

P{ 56| 53} 

P{ 53} P{ 53} 

56 35 

1 ( ) 

12.2 10.9574 

0.608 . 

5335 1 ( ) 

10.9299 

12.2 

amocanebi 

1. cnobilia, rom Z N(0,1) 

. normaluri ganawilebis funqciis cxrilis 

gamoyenebiT ipoveT albaTobebi: a). PZ { 1.23} ; b). PZ { 2.47} ; g). 

PZ { 0.16} ; d). PZ { 1.24} ; e). PZ { 2.38} ; v). PZ { 0.59} ; z). PZ { 1.83} ; T). 

PZ { 2.06} ; i). PZ { 0.07} ; k). PZ { 1.83} ; l). PZ { 2.76} ; m). PZ { 0.21} . 


. ipoveT albaTobebi: a). P{1.15 Z 1.35} ; b). 

P{1.11 Z 2.22} ; g). P{0.39 Z 2.42} ; d). P{0 Z 1.55} ; e). P{ 1.82 Z 2.33} ; 

v). P{ 0.85 Z 2.03} ; z). P{ 2.51 Z 1.09} ; T). P{ 0.55 Z 0} ; i). 

P{ 2.82 Z 1.82} ; k). P{ 1.75 Z 1.00} ; l). P{ 2.57 Z 0.12} ; m). 

P{ 1.96 Z 1.96} . 


. ipoveT Sesabamisad s , t , u an v , Tu: a). 

P{ Z s} 0.67 ; b). P{ Z t} 0.88 ; g). P{ Z u} 0.98 ; d). P{ Z v} 0.85 ; e). 

P{ Z s} 0.41 ; v). P{ Z t} 0.12 ; z). P{ Z u} 0.01 ; T). P{ Z v} 0.22 ; i). 

P{ Z s} 0.99 ; k). P{ Z t} 0.97 ; l). P{ Z u} 0.85 ; m). P{ Z v} 0.5 . 

4. mocemulia N(20,16) 

. ipoveT Semdegi albaTobebi: a). P{ 26} ; b). 

P{ 30} ; g). P{ 17} ; d). P{ 13} . 

76


. ipoveT Semdegi albaTobebi: a). P{ 29} ; b). 

P{ 31} ; g). P{ 22} ; d). P{ 16} . 


. ipoveT Semdegi albaTobebi: a). 

P{54 

58} ; b). P{40 

44} ; g). P{47 

57} ; d). P{39 

53} ; e). 

P{44 

56} . 

7. SemTxveviTi sidide ganawilebulia normalurad saSualoTi 3 

da dispersiiT 4. ipoveT albaToba imisa, rom miiRebs uaryofiT mniSvnelobas. 

8. SemTxveviTi sidide ganawilebulia normalurad saSualoTi a 

( a 0) da dispersiiT a 2 /4. a). ipoveT P{ 1.5 a} 

; b). ipoveT albaToba imisa, 

rom uaryofiTia. 



P{ s} 0.98 ; b). P{ t} 0.77 ; g). P{ u} 0.05 ; d). P{ v} 0.33 . 


. ipoveT Sesabamisad s , t , u , v an w, Tu: a). 

P{ s} 0.91 ; b). P{ t} 0.57 ; g). P{ u} 0.10 ; d). P{ v} 0.39 ; e). 

P{15 w 

15 w} 0.9 . 

11. mocemulia N(35.4,12.5) 


P{ s} 0.96 ; b). P{ t} 0.94 ; g). P{ u} 0.29 ; d). P{ v} 0.15 . 

12. SemTxveviTi sidide ganawilebulia normalurad saSualoTi 32 

da dispersiiT 2 . albaToba imisa, rom naklebia 33-ze aris 0.64. ipoveT 

2 

. 

13. SemTxveviTi sidide ganawilebulia normalurad dispersiiT 18 

da P{ 73} 0.03 . ipoveT saSualo. 

14. SemTxveviTi sidide ganawilebulia normalurad, P{ 59} 0.02 

da P{ 29} 0.93 . ipoveT SemTxveviTi sididis saSualo da standartuli 

gadaxra. 

2 

15. mocemulia Na ( , ), P{ 9.81} 0.16 da P{ 8.82} 0.01 . ipoveT 

a da . 

16. afTiaqSi wamlis damzadebaze daxarjuli dro ganawilebulia 

normalurad saSualoTi 15 wuTi da standartuli gadaxriT 2.8 wuTi. ipoveT 

albaToba imisa, rom wamlis damzadebaze daxarjuli dro: a). 20 wuTze 

metia; b). 8 wuTze naklebia; g). moTavsebulia 10 wuTsa da 18 wuTs Soris. 

17. jgufSi 16 wlis gogonebis simaRle ganawilebulia normalurad 

saSualoTi 161.2 sm da standartuli gadaxriT 4.7 sm. ipoveT albaToba imisa, 

rom am jgufidan erTi gogonas simaRle: a). 165 sm-ze metia; b). 150 smze 

naklebia; g). moTavsebulia 165 sm-sa da 170 sm-s Soris; d). moTavsebulia 

150 sm-sa da 163 sm-s Soris. 16 wlis 500 SerCeuli gogonasaTvis SeafaseT 

raodenoba im gogonebis, romelTa simaRle gava zemoT moyvanili 4 

diapazonidan. 

18. avtomobilis SuSis sawmendi rezinis sigrZe ganawilebulia normalurad 

saSualoT 25 sm da standartuli gadaxriT 0.2 sm. SeafaseT 200 

77

cali SuSis sawmendidan ramdenis sigrZes unda movelodeT, rom iqneba: a). 

25.3 sm an meti; b). moTavsebuli 24.89 sm-sa da 25.11 sm-s Soris; g). moTavsebuli 

24.89 sm-sa da 25.25 sm-s Soris. 

19. avtomobilis gacveTili samuxruWe xundis Secvlis dro ganawilebulia 

normalurad saSualoTi 90 wuTi da standartuli gadaxriT 5.8 

wuTi. ipoveT albaToba imisa, rom xundis Secvlas dasWirdeba: a). 105 wu- 

Tze meti; b). 85 wuTze naklebi. daadgineT saSualos mimarT simetriuli 

( ab , ) intervali, romelSic moxvdeba xundis Secvlis dro 90%-iani saimedoobiT. 

20. kompaniis mier warmoebuli dRis ganaTebis muSaobis xangrZlivoba 

ganawilebulia normalurad saSualoTi 2010 saaTi da standartuli 

gadaxriT 20 saaTi. kompaniam gadawyvita gazardos gayidvebis ricxvi, ris- 

Tvisac man valdebuleba aiRo sagarantio drois gasvlamde gafuWebuli 

nebismieri dRis ganaTeba ufasod Secvalos axliT. vipovoT is sagarantio 

dro, romlis dawesebis SemTxvevaSi kompanias mouwevs ufasod Secvalos 

dRis ganaTebis mxolod 3%. 

21. yvavilis foTolis sigrZe ganawilebulia normalurad saSualo- 

Ti 18.2 sm da standartuli gadaxriT 2.3 sm. a). ipoveT albaToba imisa, 

rom yvavilis foTlis sigrZe moTavsebulia 16 sm-sa da 20 sm-s Soris; b). 

yvavilis foTlebis 12% h sm-ze grZelia, xolo 20% l sm-ze moklea. ipoveT 

h da l ; g). SeafaseT yvavilis 500 foTlidan ramdeni iqneba 14 sm-ze 

mokle. 

22. boTls, romelSic asxia ualkoholo sasmeli, aweria 330 ml. sinamdvileSi 

Camosxmuli ualkoholo sasmelis moculoba ganawilebulia 

normalurad standartuli gadaxriT 2.5 ml. ramdeni unda iyos Camosxmuli 

ualkoholo sasmelis moculobis saSualo, raTa darwmunebuli viyoT, 

rom boTlebis sul cota 99%-Si sasmeli meti iqneba 330 ml-ze. 

23. fuTas, romelSic iyideba Saqari, aqvs warwera – 1 kg. Saqari. faqtiurad, 

fuTaSi moTavsebuli Saqris wona ganawilebulia normalurad 

saSualoTi 1.08 kg. fuTebis Semowmebam aCvena, rom fuTebis 2.5% naklulia 

(Seicavs miTiTebul 1 kg-ze nakleb Saqars). a). ipoveT am ganawilebis standartuli 

gadaxra; b). Tu mocemuli fuTa naklulia, maSin gamoTvaleT 

albaToba imisa, rom misi wona saSualoze 3 standartuli gadaxriT naklebia. 

24. generatoris muSaobis xangrZlivoba ganawilebulia normalurad 

saSualoTi 210 sT. aRmoCnda, rom generatorebis 4% muSaobs 222 saaTze 

mets. ipoveT am ganawilebis dispersia. 

25. albaTobis gamocdaze studentebis 15% Rebulobs 63 qulaze 

mets, xolo 10% -- 32 qulaze naklebs. vigulisxmoT, rom qulebi ganawilebulia 

normalurad da vipovoT qulebis saSualo da standartuli gadaxra. 

26. mziani saaTebis raodenoba TveSi H garkveul kurortze ganawilebulia 

normalurad saSualoTi 130 sT. cnobilia, rom PH { 179} 0.975 . 

a). gamoTvaleT H -is standartuli gadaxra; b). gamoTvaleT 

P{100 H 150} . 

78

amocanebi gameorebisaTvis 

27. uwyveti SemTxveviTi sididis ganawilebis simkvrivea: 

k(4 x), Tu 0 x 4, 

f 

() x 

0, sxvagan . 

gamoTvaleT: a). k ; b). P{ 2.5} ; g). E . 


 

 

1 x , Tu 0 x 

2, 

f 

() x 2 

0, sxvagan . 

ipoveT: a). SemTxveviTi sididis mediana da qveda da zeda kvartili; b). 

E ; g). D . 

29. uwyveti U SemTxveviTi sidide Tanabradaa ganawilebuli [0.5, 2.5] 

segmentze. ipoveT am SemTxveviTi sididis: a). ganawilebis simkvrive; b). ma- 

Tematikuri lodini; g). dispersia. 

30. xis naWrebis sigrZe modelirdeba uwyveti Tanabari ganawilebiT, 

romlis simkvrivea: 

 

 

1 , Tu 0.2 x 

0.8, 

fU 

() x k 

0, sxvagan . 

ipoveT: a). k ; b). EU ; g). DU . d). SemTxveviT SeirCa sami naWeri. risi tolia 

albaToba imisa, rom maTgan mxolod oris sigrZe iqneba 0.6-ze meti. 


3 

kx 

, Tu 0 x 

2, 

f 

() x 0, sxvagan . 

ipoveT: a). k ; b). EU ; g). DU ; d). ganawilebis mediana; e). albaToba imisa, 

rom dakvirveba moTavsebuli iqneba saSualodan erTi standartuli gadaxris 

farglebSi; v). albaToba imisa, rom 4 dakvirvebidan 2 iqneba saSualoze 

meti da 2 – saSualoze naklebi. 


( x a)(2 a x), Tu 0 x 2, 

f 

() x 

0, sxvagan . 

ipoveT: a). a; b). E . 

33. cnobilia, rom N(10,8) 

. ipoveT P{ 6} . 

34. cnobilia, rom N(140,56.25) 

. ipoveT albaToba imisa, rom metia 

128.75-ze. 

35. garkveuli modelis avtomobili 56 mili/sT siCqariT moZraobisas 

erTi galoni benziniT gadis manZils, romlis saSualo mniSvnelobaa 

32.4 mili da standartuli gadaxra 1.4 mili. CavTvaleT, rom saqme gvaqvs 

normalur ganawilebasTan da gamoTvaleT albaToba imisa, rom es avtom- 

79

obili 56 mili/sT siCqariT moZraobisas erTi galoni benziniT gaivlis 

30 milze mets. 

36. normalurad ganawilebuli SemTxveviTi sididis saSualoa 20, 

xolo dispersia 4.15. ipoveT albaToba imisa, rom 18 

21. 

37. farTobi, romlis SeRebvac SeiZleba 1 litri bunebrivi saRebaviT 

ganawilebulia normalurad saSualoTi 13.2 kv. metri da standartuli 

gadaxriT 0.197 kv. metri. farTobi, romlis SeRebvac SeiZleba 1 litri xelovnuri 

saRebaviT ganawilebulia normalurad saSualoTi 13.4 kv. metri 

da standartuli gadaxriT 0.343 kv. metri. SesaRebia 12.9 kv. metri farTobi. 

gansazRvreT, romeli saRebavis 1 litri iqneba sakmarisi ufro meti 

albaTobiT am farTobis SesaRebad. 

38. kvercxis wona , gazomili gramebSi, ganawilebulia normalurad: 

N(85,0.36) 

. kvercxebi iyofa sam kategoriad: didi, saSualo da patara. 

didi kvercxis wona metia an toli 90 gramze. kvercxebis 25% pataraa. 

gamoTvaleT: a). kvercxebis ramdeni procentia didi; b). patara kvercxis 

maqsimaluri wona. 


da dispersiiT a. cnobilia, rom P{ 3} 0.2 . ipoveT a. 

40. ori firma, A da B , awarmoebs gamoZaxebis SeCerebis mowyobilebas 

sakomunikacio organizaciisaTvis. gamoZaxebis SeCerebis dro izomeba 

nanowamebSi (nwm). a). SeCerebis dro A firmis mier warmoebuli SeCerebis 

mowyobilebisaTvis modelirdeba normaluri ganawilebiT saSualoTi 283 

nwm da standartuli gadaxriT 8 nwm. ipoveT albaToba imisa, rom A firmis 

mowyobilebisaTvis SeCerebis dro moTavsebulia 275 nwm-sa da 286 nwms 

Soris; b). B firmis mowyobilebisaTvis SeCerebis dro isev igulisxmeba, 

rom normaluria. aq SeCerebis drois 10% naklebia 274.6 nwm-ze da 7.5% 

metia 288.2 nwm-ze. vipovoT am ganawilebis saSualo da standartuli gadaxra. 


da dispersiiT 2 . mocemulia, rom P{ 81.89} 0.01 da P{ 27.77} 0.1. ipoveT 

a da 2 . 

42. cnobilia, rom Na ( ,4) . ipoveT a, Tu cnobilia, rom Sesabamisi 

ganawilebis simkvrivis wiris wves moqceuli aris farTobi 12-dan -mde 

aris 0.8. 

43. danadgari Wris grZel milebs patar-patara milebad. patara milis 

sigrZe ganawilebulia normalurad saSualoTi a sm da standartuli 

gadaxriT 0.25 sm. a-s mniSvnelobis Secvla SesaZlebelia danadgaris regulirebis 

xarjze. a). ipoveT a, romlisTvisac albaToba imisa, rom SemTxveviT 

SerCeuli patara milis sigrZe naklebia 6.5 sm-ze aris 0.1; b). danadgari 

daregulirda ise, rom a 6.4 , xolo standartuli gadaxra ar Secvlila. 

ipoveT albaToba imisa, rom SemTxveviT SerCeuli patara milis 

sigrZe moTavsebulia 6.3 sm-sa da 6.6 sm-s Soris. 

80


44. SemTxveviTi sidide ganawilebulia normalurad saSualoTi 

250 da dispersiiT 50. ipoveT misi simkvrivis wiris qveS 180-dan 280-mde mo- 

Tavsebuli aris farTobi. 

45. garkveuli saxeobis Citis wona ganawilebulia normalurad saSualoTi 

0.8 kg da standartuli gadaxriT 0.12 kg. ipoveT albaToba imisa, 

rom SemTxveviT SerCeuli am saxeobis Citis wona moTavsebuli iqneba 0.74 

kg-sa da 0.95 kg-s Soris. 

46. SemTxveviTi sidide ganawilebulia normalurad da 

P{ 10} 0.67 , P{ 12} 0.937 . ipoveT E . 


2 

kx 

, Tu 0 x 

2, 

f 


a). aCveneT, rom k 3/8 ; b). gamoTvaleT E ; g). ipoveT SemTxveviTi sididis 

mediana. 

48. saqmiani qali dRis ganmavlobaSi telefonze laparaks andomebs 

T saaTs. T SemTxveviTi sididis ganawilebis simkvrivea: 

1 (8 3 

x x ), Tu 0 x 2, 

fT 

() x 12 

0, sxvagan . 

ipoveT SemTxveviTi sididis: a). maTematikuri lodini; b). mediana; g). 

moda. 


1 (4 2 

x x ), Tu 0 x 2, 

f 

() x 4 

0, sxvagan . 

ipoveT SemTxveviTi sididis mediana. 


2 

 

c(2 x x ), Tu 0 x 2, 

f 


ipoveT: a). c mudmivis mniSvneloba; b). albaToba P{0.25 

0.5} . 

51. uwyveti SemTxveviTi sididis ganawilebis funqciaa: 

0, Tu x 0, 

 

F 

( x) sin x, Tu 0 x / 2, 

 

1, Tu x 

/ 2. 

ipoveT SemTxveviTi sididis: a). ganawilebis simkvrive: b). mediana. 

81


f ( x) sin x , Tu 0 x 

/ 2, =0 sxvagan. ipoveT SemTxveviTi sididis mediana. 


2 

da dispersiiT . cnobilia, rom P{ 6.2} 0.9474 da P{ 9.8} 0.6368 . 

ipoveT a da 2 . 

54. uwyveti SemTxveviTi sididis ganawilebis simkvrive ganmartebulia 

[0, a ] intervalze TanafardobiT: 

x/ 2, Tu 0 x3, 

f 

() x 2 

27/8 x , Tu 3 x a. 

ipoveT a-s mniSvneloba. 

55. birTvis srolis SejibrebaSi monawileobs ori aTleti, ivane da 

erekle. TiToeuli aTletisaTvis birTvis srolis manZili ganawilebulia 

normalurad. ivanesaTvis gasul wels birvTis srolis manZilis saSualo 

iyo 60.33 m, xolo standartuli gadaxra 1.95 m. a). gasul wels ivanesaTvis 

birvTis srolis manZilis 80% meti iyo vidre x metri. ipoveT x ; 

b). gasul wels ereklesaTvis birvTis srolis manZilis 80% meti iyo vidre 

59.5 metri. cnobilia, rom gasul wels ereklesaTvis birvTis srolis 

manZilis saSualo iyo 59.39 m. ipoveT misi standartuli gadaxra; mimdinare 

wels ereklesaTvis birvTis srolis manZilis saSualoa 59.5 m, xolo 

standartuli gadaxra 3 m. ivanesaTvis Sesabamisi parametrebia 60.33 m da 

1.95 m. Sejibrebis dros, Semdeg turSi gasasvlelad aTletma birTvi unda 

isrolos sul cota 65 metrze. pirvel turSi TiToeuli aTleti birTvs 

isvris 3-jer; g). gansazRvreT romeli aTletis gasvlaa ufro mosalodneli 

Semdeg turSi pirveli srolis Sedegis mixedviT; d). ipoveT albaToba 

imisa, rom orive aTleti gava Semdeg turSi. 


da dispersiiT 2 , ise, rom P{ 50.32} 0.119 da P{ 43.56} 0.305 . ipoveT 

a da 2 da gamoTvaleT P{| a| 5} . 

57. sawarmo yidulobs WanWikebis 44%-s A firmidan, xolo danarCens 

B firmidan. TiToeuli firmis mier damzadebuli WanWikebis diametri ganawilebulia 

normalurad standartuli gadaxriT 0.16 mm. A firmis mier 

damzadebuli WanWikebis diametris saSualoa 1.56 mm. B firmis mier damzadebuli 

WanWikebis 24.2%-is diametri naklebia 1.52 mm-ze. a). ipoveT B firmis 

mier damzadebuli WanWikebis diametris saSualo; b). ipoveT albaToba 

imisa, rom sawarmos sawyobidan SemTxveviT SerCeuli WanWikis diametri 

naklebi iqneba 1.52 mm-ze; g). SemTxveviT SerCeuli WanWikis diametri naklebia 

1.52 mm-ze. vipovoT albaToba imisa, rom es WanWiki damzadebulia 

B firmaSi; d). B firma dReSi amzadebs 8000 WanWiks. TiTeuli WanWiki iZleva 

1.5 larian mogebas, Tu misi diametri moTavsebulia 1.52 mm-sa da 1.83 

mm-s Soris. WanWiki, romlis diametri naklebia 1.52 mm-ze unda gadagdon 

82

da iZleva 0.85 larian danakargs. dabolos, WanWiki, romlis diametri metia 

1.83 mm-ze iyideba, magram mogebas amcirebs 0.5 laramde. gamoTvaleT B 

firmis mosalodneli mogeba. 

58. albaToba imisa, rom mSeneblobaze miitanen samSeneblo masalebs 

PA ( ) 0.85 , albaToba imisa, rom mSenebloba dasruldeba droulad 

P ( B) 

11/17 

. albaToba imisa, rom samSeneblo masalebs miitanen da mSenebloba 

droulad dasruldeba aris 0.55. a). aCveneT, rom A da B xdomilebebi 

damoukidebelia; b). cnobilia, rom samSeneblo masalebi droulad 

miitanes mSeneblobaze. ipoveT albaToba imisa, rom mSenebloba ar 

dasruldeba droulad. 

59. mSeneblobaze muSaobs 10 kaciani jgufi, romelTa Soris 3 eleqtrikia 

da 2 santeqnikosi. arqiteqtorma daibara saTaTbirod 5 muSa da 

movida SemTxveviT SerCeuli 5 maTgani. a). ipoveT albaToba imisa, rom am 

xuTSi moxvdeba 2 eleqtriki da 1 santeqnikosi; b). mSeneblobaze kviraSi 

muSaobis saaTebis raodenoba ganawilebulia normalurad saSualoTi 42 

sT. jgufis 10% muSaobs kviraSi 48 an met saaTs. ipoveT albaToba imisa, 

rom orive santeqnikosi konkretul kviraSi imauSavebs 40 saaTze mets. 

pasuxebi: 

1. 0.8907; 0.9932; 0.5636; 0.1075; 0.0087; 0.2776; 0.9664; 0.0197; 0.5279; 0.0336; 0.0029; 

0.4168. 2. 0.0366; 0.1203; 0.3405; 0.4394; 0.9557; 0.7816; 0.8561; 0.2088; 0.0320; 0.1186; 

0.4472; 0.9500. 3. 0.4399; 1.1750; 2.0537; 1.0364; 0.2275; 1.1750; 2.3263; 0.7722; -2. 3263; - 

1.8808; -1.0364; 0. 4. 0.9332; 0.0062; 0.7734; 0.0401. 5. 0.9522; 0.0098; 0.7475; 0.0038. 6. 

0.1359; 0.0606; 0.7333; 0.7704; 0.8664. 7. 0.0668. 8. 0.1587; 0.0228. 9. 54.27; 40.31; 52.22; 

41.80. 10. 17.68; 14.65; 12.44; 14.44; 3.29. 11. 41.6; 29.9; 37.4; 31.7. 12. 7.78. 13. 65.0. 14. 

41.5; 8.50. 15. 9.51; 0.298. 16. 0.0371; 0.0062; 0.8209. 17. 0.2094; 0.0086; 0.1788; 0.6405; 

105; 4; 89; 320. 18. 13; 84; 121. 19. 0.0049; 0.1943; (80, 100). 20. 1972 sT. 21. 0.614; 20.9, 

16.3; 17. 22. 336 ml. 23. 0.041 kg; 0.054. 24. 47. 25. 49.1; 13.4. 26. 25; 0.673. 27. 1/8; 

0.141; 4/3. 28. 2 , 1, 3 ; 4/3; 2/9. 29. fU 

( x) 1/ 2, Tu x [0.5, 2.5] , da =0 sxvagan; 

3/2; 1/3. 30. 0.6; 0.5; 0.03; 2/9. 31. 1/4; 1.6; 0.107; 1.68; 0.697; 0.351. 32. a 3 6 ; 2.73. 33. 

0.921. 34. 0.933. 35. 0.957. 36. 0.525. 37. bunebrivi: 0.936 > 0.928. 38. 20.2%; 81 gr. 39. 

18.94. 40. 0.488; 281, 5. 41. 47; 15. 42. 10.3. 43. 6.82; 0.444. 44. 0.645. 45. 0.5858. 46. 

9.193. 47. 3/2; 2 2/3 . 48. 56/45; 1.294; 1.633. 49. 1.082. 50. 3/4; 0.1133. 51. f ( x) cos x, Tu 

0 x 

/ 2, =0 sxvagan; /6. 52. /3. 53. 9.161; 1.827. 54. 54/11. 55. 58.69; 3.41; 

ereklesaTvis (0.0334>0.0083); 0.119. 56. 38.41; 10.09; 0.3798. 57. 1.632; 0.312; 0.775; 

6587. 58. 6/17. 59. 5/21; 0.44 

83

Tavi VII 

albaTobis Teoriis zRvariTi Teoremebi 

CebiSevis utoloba. 

CebiSevis utoloba afasebs SemTxveviTi sididis gadaxras Tavisi 

maTematikuri lodinidan. Tu raime SemTxveviTi sididea, maSin 

nebismieri 0 ricxvisaTvis samarTliania utoloba: 

2 

P(| E | ) 1 D / . 

did ricxvTa kanoni. 

CebiSevis Teorema. vTqvaT, SemTxveviTi sidideebi 1, 

2,... 

wyvilwyvilad 

damoukidebelia ( Ei 

) da arsebobs iseTi ricxvi C , rom 

D i 

C, i 1,2,... . maSin nebismieri dadebiTi ricxvisaTvis sruldeba 

utoloba: 

1 2 n 

E 1 

E 2 

E 

n 

lim P{| | } 1. 

n 

n 

n 

am mtkicebulebas did ricxvTa kanons uwodeben. 

bernulis Teorema. davuSvaT, m aris n damoukidebel eqsperiment- 

Si A xdomilebis moxdenaTa ricxvi, xolo p aris A xdomilebis 

moxdenis albaToba calkeul eqsperimentSi. maSin nebismieri 0 

ricxvisaTvis samarTliania utoloba 

(1 ) 

{| m | } 

p 

P p 

p ( 0 , roca n). 

2 

n 

n 

erTTan ragind axlos myofi albaTobiT SeiZleba vamtkicoT, rom 

damoukidebel cdaTa sakmaod didi ricxvisaTvis dakvirvebadi xdomilebis 

moxdenaTa sixSire ragind umniSvnelod gansxvavdeba misi moxdenis albaTobisagan 

calkeul cdaSi. 

centraluri zRvariTi Teorema. 

did ricxvTa kanoni ar ikvlevs SemTxveviT sidideTa jamis ganawilebis 

kanonis saxes. es sakiTxi Seiswavleba Teoremebis jgufSi, romlebsac 

centraluri zRvariTi Teorema ewodeba. es Teorema amtkicebs, rom 

SemTxveviT sidideTa jamis ganawilebis kanoni, romelTagan calkeul Sesakrebs 

SeiZleba hqondes gansxvavebuli ganawileba, uaxlovdeba normalurs 

SesakrebTa sakmaod didi ricxvis SemTxvevaSi. amiT aixsneba normaluri 

ganawilebis kanonis uaRresad didi mniSvneloba praqtikul gamoyenebebSi. 

Teorema 1. Tu 1 

, 2 

,... – damoukidebeli SemTxveviTi sidideebis mimdevrobaa, 

erTi da igive ganawilebis kanoniT, maTematikuri lodiniT a da 

dispersiiT σ 2 n 

, maSin п –is usasrulod zrdisas S 

jamis ganawilebis 

kanoni uaxlovdeba standartul normalur ganawilebis kanons: 

Sn 

a 

lim P{ x} ( x) 

. 

n 

n 

n 

k1 

k 

84

Teorema 2 (liapunovis Teorema). Tu SemTxveviTi sidide warmoadgens 

damoukidebel SemTxveviT sidideTa Zalian didi ricxvis jams, romelTaTvisac 

Sesrulebulia piroba: 

lim( n 

n 

 

bk)/[ Dk 

] 0 

n 

k 1 k 1 

, 

sadac b k – mesame rigis absoluturi centraluri momentia 

SemTxveviTi sididis , xolo D k – misi dispersia, maSin SemTxveviT 

sidides gaaCnia ganawileba, romelic axlosaa normalur ganawilebasTan. 

Tu SemTxveviTi sidide warmoadgens urTierTdamoukidebel SemTxveviT 

sidideTa Zalian didi raodenobis jams, romelTagan TiToeulis gavlena 

jamze mizerulad mcirea, maSin SemTxveviT sidides gaaCnia ganawileba, 

romelic axlosaa normalurTan. 

Teorema 3 (muavr-laplasis Teorema). Tu tardeba п damoukidebeli 

cda, romelTagan TiToeulSi А xdomileba xdeba albaTobiT р, np 15 , 

maSin samarTliania Tanafardoba: 

Sn 

np 

 

p 

 

( ) ( ), 

npq 

 

 

sadac S n 

– А xdomilebis moxdenaTa ricxvia п cdaSi, q = 1 – p, xolo 

2 

x t 

1 

() x 

e 

2 

dt 

2 

 

 

Sedegi (muavr-laplasis lokaluri Teorema). muavr-laplasis Teoremis 

pirobebSi р n 

(k) 

-- albaToba imisa, rom А xdomileba п cdaSi moxdeba 

zustad k –jer, cdaTa didi ricxvis SemTxvevaSi, Tu np 15 , SeiZleba gamoiTvalos 

Semdegi formuliT: 

1 

p n 

( k) 

( 

x), 

npq 

x 

k np 

1 

sadac x , xolo 

2 

( x) 

e . 

npq 

2 

magaliTi 1. dadgenilia, rom im adamianebis 94%-s, romelTac gakeTebuli 

aqvT tuberkuliozis sawinaaRmdego acrebi, gamoumuSavdeba Sesabamisi 

imuniteti. rogoria albaToba imisa, rom 100000 tuberkuliozze acrili 

adamianidan 5800 araa daculi tuberkulioziT daavadebisagan? 

amoxsna. am amocanis amoxsna marTalia Teoriulad SesaZlebelia bernulis 

formuliT, sadac n 100000, 

k 5800, p 0.06, q 0. 94, da Sesabamisad, 

5800 5800 94200 

P 

100000 

( 5800) C100000(0.06) 

(0.94) 

, 

magram praqtikulad amis gakeTeba Zalian Znelia. am SemTxvevaSi unda visargebloT 

muavr-laplasis lokaluri formuliT, romelic gvaZlevs: 

5800100000 

0.06 

P 

100000(5800) 

( 

)/ 100000 

0.060.94 

( 

2.7)/75. 

100000 

0.060.94 

85 

2 

3/2 

 

k

Tu axla gavixsenebT, rom ( x) 

( 

x) 

da visargeblebT standartuli normaluri 

ganawilebis simkvrivis cxrilebiT (romlidanac ( 2.7) 0. 0104), 

advilad gamoviTvliT, rom 

P 

100000(5800) 

0.000139 . 

magaliTi 2. vipovoT albaToba imisa, rom monetis 100-jer agdebisas 

gerbTa mosvlis ricxvi aRmoCndeba sazRvrebSi 40-dan 60-mde. 

amoxsna. visargebloT muavr-laplasis integraluri TeoremiT. Cvens 

SemTxvevaSi p 0.5, пр = 100·0.5 = 50. amitom Tu 40 S n 

60, maSin 

Sn 

50 

2 2. 

5 

Sesabamisad, gvaqvs: 

Sn 

50 

 

p40 Sn 

60 

p2 2 (2) ( 2) 0.9772 0.0228 0.9544. 

5 

 

 

 

magaliTi 3. magaliTi 2-is pirobebSi vipovoT albaToba imisa, rom 

gerbi mova 45-jer. 

k np 

45 50 

amoxsna. am SemTxvevaSi x 1, amitom 

npq 5 

1 1 1 

p100(45) ( 1) (1) 0.2420 0.0484. 

5 5 5 

SemoviRoT ganawilebis funqciebisaTvis Semdegi aRniSvnebi: 

hipergeometriuli ganawilebis funqcia – 

k sk 

Ct 

Cn t 

H( x; t, s, n) 

; 

s 

kx 

Cn 

binomuri ganawilebis funqcia – 

k k n k 

Bi( x; p, n) 

C p q 

; 

kx 

puasonis ganawilebis funqcia – 

k 

 

( x; ) 

e ; 

0kxk! 

standartuli normaluri ganawilebis funqcia – 

2 

x t 

1 

() 

x e 

2 

dt 

2 

; 

 

xi kvadrat ganawilebis funqcia -- 2 ( xk ; ) ; 

stiudentis ganawilebis funqcia -- T( x; k ) ; 

fiSeris ganawilebis funqcia -- F( x; k1, k 

2) 

. 

aRsaniSnavia, rom zemoT moyvanili ganawilebebis funqciebs Soris 

adgili aqvs qvemoT moyvanil sqemaze gamosaxul zRvrul Tanafardobebs: 

n 

86

t/s p 

H(x;t,s,n) 

Bi(x;n,p) 

n , p0, np 

(x;) 

n, p=const 

Bi(x npq+np;n,p 

) 

(x) 

 

(x +;) 

T(x;k) 

k 

2 (x 

) 

k 

2 k +k;x 

2 (x; k) 

F(x; k 1 ,k) 

k 1 

qvemoT moyvanil naxazze Sedarebulia normaluri da puasonis 

ganawilebebi 7-is SemTxvevaSi: 

magaliTi 4. studenti quCaSi Tavis leqtors SeiZleba Sexvdes albaTobiT 

0.002. ras udris albaToba imisa, rom 1200 SemTxveviT gamvlels 

Soris studenti Sexvdeba Tavis araumetes 3 leqtors? 

87

amoxsna. SemoviRoT xdomilebebi: A { studentiSexvda i leqtors} 

, 

i 0,1,2,3 . maSin CvenTvis saintereso xdomileba iqneba Ai 

. cxadia, rom 

i 

A 3 i 

0 

A 

i 

xdomilebebi wyvil-wyvilad uTavsebadia da Sesabamisad 

 

P ( A) 

P( 

). 

3 i0 

amasTanave, TiToeuli A i 

xdomilebis albaTobis gamosaTvlelad unda visargebloT 

muavr-laplasis lokaluri formuliT. gvaqvs: 

01200 

0.002 

P ( A0 ) P1200 

(0) ( 

)/ 1200 

0.002 0.998 

0.0775 ; 

1200 

0.002 0.998 

112000.002 

P ( A1 ) P1200(1) 

( 

)/ 12000.0020.998 

0.1719; 

12000.0020.998 

2 12000.002 

P ( A2 ) P1200(2) 

( 

)/ 12000.0020.998 

0.2505; 

12000.0020.998 

31200 

0.002 

P ( A3 ) P1200 

(3) ( 

)/ 1200 

0.002 0.998 

0.2389 . 

1200 

0.002 0.998 

amitom saZiebeli albaToba iqneba 

P ( A) 

0.0775 

0.1719 

0.2505 

0.2389 

0.7388. 

magaliTi 5. rogoria albaToba imisa, rom wesieri monetis 200-jer 

agdebisas gerbi mova aranakleb 95-jer da araumetes 105-jer? 

amoxsna. unda visargebloT muavr-laplasis integraluri formuliT, 

sadac n 200, 

a 95, b 105, 

p q 1/ 

2. gvaqvs: 

105 

2000.5 

95 

2000.5 

P ( 95 S200 

105) 

( 

) ( 

) (0.707) 

( 

0.707) 

 

2000.5 

0.5 

2000.5 

0.5 

2(0.707) 

1 

20.761 

0.52. 

magaliTi 6. qarxnis nawarmis 4% mdare xarisxisaa. ramdeni nawarmi 

unda SeirCes produqciis xarisxis Sesamowmeblad, rom albaTobiT 0.9 SegveZlos 

imis mtkiceba, rom qarxnis nawarmis SemTxveviT SerCeul partia- 

Si mdare xarisxis produqciis wilis gansxvaveba 4%-dan absoluturi sididiT 

ar aRemateba 1%-s? 

amoxsna. amocanis pirobis Tanaxmad , . iyos qarxnis 

nawarmis SemTxveviT SerCeul partiaSi mdare xarisxis produqciis 

wili. liapunovis Teoremis Tanaxmad SegviZlia vigulisxmoT, rom SemTxveviTi 

sididis ganawileba axlosaa normalurTan da amitom 

P {| 0.04| 

0.01} 2(0.01/ 

) 

1 

0.9 . 

aqedan ( 0.01/ ) 

0. 95da normaluri ganawilebis cxrilidan vpoulobT, 

rom 0.01/ 

1. 

65. 

2 

meores mxriv, p (1 p)/ 

n 0.040.96/ 

n anu 0.040.96/ 

n . Sesabamisad, 

SegviZlia davweroT gantoleba 

0.01/ 

0.040.96/ 

n 1.65, 

saidanac advilad vipoviT, rom . 

A i 

88

magaliTi 7. mowyobileba Sedgeba 10 damoukideblad momuSave elementisagan, 

romelTagan TiToeulis mwyobridan gamosvlis albaToba T 

droSi aris 0.05. SeafaseT albaToba imisa, rom T droSi mwyobridan gamosul 

elementTa raodenobisa da elementebis mwyobridan gamosvlaTa sa- 

Sualo ricxvs Soris gansxvavebis moduli aRmoCndeba: a). 2-ze naklebi; b). 

2-ze aranaklebi. 

amoxsna. a). aRvniSnoT S n 

simboloTi T droSi mwyobridan gamosul 

elementTa raodenoba. cxadia, rom am SemTxveviT sidides aqvs binomialuri 

ganawileba da amitom: 

ESn 

np 100.05 0.5 da DSn 

npq 10 0.05 0.95 0.475 . 

visargebloT CebiSevis utolobiT. gvaqvs: 

2 

P(| Sn ESn | ) 1 DS 

n 

/ , 

P(| Sn 

0.5| 2) 10.475/ 4 0.88 . 

b). ramdenadac | Sn 

0.5| 2 da | Sn 

0.5| 2 sawinaaRmdego xdomilebebia, 

amitom SegviZlia davweroT, rom: 

P(| Sn 

0.5| 2) 10.88 0.12 . 

magaliTi 8. A xdomilebis moxdenis albaToba TiToeul eqsperimentSi 

aris 0.5. SevafasoT albaToba imisa, rom 100 damoukidebel eqsperimentSi 

A xdomilebis moxdenaTa S n 

ricxvi moTavsebulia 40-dan 60-mde. 

amoxsna. am SemTxvevaSi: 

ESn 

np 100 0.5 50 da DSn 

npq 100 0.5 0.5 25 . 

gasagebia, rom xdomileba 40 S n 

60 eqvivalenturi xdomilebis: 

40 50 S n 

50 60 50 10 S n 

50 10 | Sn 

50| 10. 

Sesabamisad, CebiSevis utolobis Tanaxmad SegviZlia davweroT, 

rom: 

P(40 Sn 

60) P(| Sn50| 10) 125/100 0.75 . 

magaliTi 9. rogor davadginoT albaToba imisa, rom SemTxveviT SeCeuli 

adamiani agrovebs markebs? SeiZleba gamovkiTxoT garkveuli raodenoba 

SemTxveviT SerCeuli adamianebis. Tu n gamokiTxulSi S 

n 

filatelistia, maSin saZiebeli albaToba p Sn / n. ramdeni adamiani unda 

gamoikiTxos, rom cdomileba ar aRematebodes 0.005-s, Tu Cven gvinda, rom 

swori Sedegi miviRoT albaTobiT 0.95? 

amoxsna. amocanis pirobis Tanaxmad P(| S n 

/ n p | 0.005) 0. 95. amitom 

muavr-laplasis integraluri formula gvaZlevs: 

0.95 

P(| 

S / n p | 0.005) P( 

0.005 

( S np)/ 

n 0.005) 

 

n 

n 

n 

n 

n 

P( 

0.005 

( Sn np)/ 

npq 0.005 ) (0.005 

) ( 

0.005 

). 

pq 

pq 

pq 

pq 

amitom, Tu visargeblebT TanafardobiT ( x) 

1( 

x) 

, miviRebT, 

rom 

n 

89

n 

n 

2(0.005 

) 1 

0.95, anu ( 0.005 ) 0. 975. 

pq 

pq 

saidanac standartuli normaluri ganawilebis funqciis cxrilis gamoyenebiT 

vRebulobT, rom 

n 

0.005 

1.96. 

pq 

Tuki axla gaviTvaliswinebT, rom pq 1/ 4 , advilad davaskvniT, rom 

n 40. 

magaliTi 10. mocemulia, rom P(| E | ) 0.8 da D 0.018 . ipoveT 

. 

2 

amoxsna. CebiSevis utolobis Tanaxmad P(| E | ) 1 D / . maSasadame, 

SegviZlia davweroT, rom 

2 

1 D/ 0.8 . 

SevitanoT am gantolebaSi dispersiis mniSvneloba da amovxsnaT . 

gvaqvs: 

2 

10.018/ 0.8 , 

aqedan, cxadia, rom 0.3 . 

magaliTi 11. damoukidebel SemTxveviT sidideTa mimdevroba 1 

, 2 

,... 

mocemulia ganawilebis kanoniT: 

 

n 

2 

P 

n 0 n 

1/ 2n 

2 

1 

1/ n 

2 

1/ 2n 

akmayofilebs Tu ara es mimdevroba did ricxvTa kanons? 

amoxsna. SevamowmoT CebiSevis Teoremis pirobebi. gasagebia, rom Sem- 

TxveviTi sidideebi wyvil-wyvilad damoukidebelia. gamovTvaloT n 

-is 

maTematikuri lodini da dispersia. gvaqvs: 

2 2 2 

E n (1/2 n ) 0 (11/ n ) n 

(1/2 n ) 0 ; 

n 

2 2 2 2 2 2 2 2 2 2 2 2 

n 

(1/2 ) 0 (1 1/ ) (1/2 ) (1/ ) 

2 2 2 

D n 

E n 

( E n) 

. 

E n n n n n n n ; 

amrigad, maTematikuri lodinebi sasrulia da dispersiebi Tanabrad 

SemosazRrulia. amitom CebiSevis Teoremis Tanaxmad aRniSnuli mimdevroba 

akmayofilebs did ricxvTa kanons. 

amocanebi 

1. vigulisxmoT, rom qalaqSi kacebisa da qalebis raodenoba erTidaigivea. 

risi tolia albaToba imisa, rom 100 SemTxveviTi gamvlelidan 32 qali 

iqneba? 

2. cnobilia, rom detali wundebulia albaTobiT 0.6. ramden wundebul 

detals unda velodoT 1000 detalidan albaTobiT 0.9? 

90

3. cnobilia, rom mkurnalobis axali meTodiT avadmyofis gamojanmrTelebis 

albaTobaa 0.8. ramden gamojanmrTelebuls unda velodoT 100 pacientidan 

albaTobiT 0.75? 

4. kalaTburTeli burTs kalaTSi agdebs albaTobiT 0.8. rogoria albaToba 

imisa, rom igi 20 srolidan ocives Caagdebs? 

5. rogoria albaToba imisa, rom wesieri monetis 10-jer agdebisas arcer- 

Txel mova gerbi? 

6. nika da luka agorebenn or-or wesier saTamaSo kamatels. nika igebs partias, 

Tu 20 gagorebidan 2-jer mova 11-is toli jamuri qula. luka igebs 

partias, Tu 10 gagorebidan 2-jer mova 9-is toli jamuri qula. romlis 

mogebaa ufro albaTuri? 

7. ras udris albaToba imisa, rom wesieri monetis 40-jer agdebisas 25- 

jer mova gerbi? 

8. teleatelies monacemebiT, sagarantio periodSi saSualod kineskopebis 

12% gamodis mwyobridan. rogoria albaToba imisa, rom 46 SemTxveviT Ser- 

Ceuli kineskopidan 36 imuSavebs sagarantio periodis manZilze? 

9. biWis dabadebis albaTobaa 0.515. risi tolia albaToba imisa, rom 80 

axalSobils Soris 42 iqneba biWi? 

10. rogoria albaToba imisa, rom 30 gasrolisas samizne daziandeba 8- 

jer, Tu cnobilia, rom erTi gasrolisas samiznis dazianebis albaTobaa 

0.3. 

11. eqsperimentis Catarebisas A xdomilebis moxdenis albaTobaa 0.5. ramdenjer 

unda velodoT A xdomilebis moxdenas 100 eqsperimentSi albaTobiT 

0.048? 

12. 10 burTi SemTxveviT nawildeba 9 yuTSi. rogoria albaToba imisa, rom 

I yuTSi aris aranakleb 1100 burTi da araumetes 1200 burTi? 

13. rogoria albaToba imisa, rom wesieri saTamaSo kamaTlis 12000-jer gagorebisas 

eqvsiani mova aranakleb 1900-jer da araumetes 2100-jer? 

14. ipoveT iseTi ricxvi b , rom wesieri monetis 1000-jer agdebisas mosul 

gerbTa raodenobam S 1000 daakmayofilos piroba 470 S 1000 

b albaTobiT 

0.727. 

15. I satelefono sadguri emsaxureba 2000 abonents da aerTebs maT II sadgurTan. 

I sadguridan II sadguramde 2000 satelefono xazis gayvana araracionaluria. 

ramdeni satelefono xazi unda gaviyvanoT I sadguridan II 

sadguramde, rom I sadguris 100 abonentidan mxolod erTs, romelic Sem- 

TxveviT irCevs laparakis dros II sadguris abonentTan, yvela xazi daxvdes 

dakavebuli? albaToba imisa, rom SemTxveviTi darekvisas xazi dakavebulia 

aris 1/30. 

16. 10 Tofidan 4 Semowmebulia mizanSi srolaSi. Semowmebuli Tofidan 

mizanSi moxvedris albaTobaa 0.9, xolo Seumowmebeli Tofidan ki _ 0.3. 

SemTxveviT SerCeuli Tofidan moaxdines 200 gasrola. rogoria albaToba 

imisa, rom mizanSi moxvedraTa raodenoba S 200 daakmayofilebs pirobas? 

17. ipoveT iseTi ricxvi a, rom albaTobiT 0.9 SeiZlebodes imis mtkiceba, 

rom 900 axalSobils Soris a-ze meti biWia, Tu cnobilia, rom biWis dabadebis 

albaTobaa 0.515. 

91

18. detalebis 80% I xarisxisaa. ramdeni detali unda aviRoT, rom albaTobiT 

0.996 SegveZlos mtkiceba: albaToba p xdomilebisa _ SemTxveviT SerCeuli 

detali I xarisxisaa akmayifilebdes pirobas 0.75 

p 0. 85? 

19. 1000 yuTidan TiToeulSi 5000 TeTri da 5000 Savi burTia. TiToeuli 

yuTidan SemTxveviT iReben 3 burTs. rogoria albaToba imisa, rom raodenoba 

yuTebisa, saidanac amoRebulia 3 erTnairi feris burTi, aris aranakleb 

200 da araumetes 310 yuTi? 

20. “simensis” mier gamoSvebuli naTurebis 70% umaRlesi xarisxisaa. rogoria 

albaToba imisa, rom “simensis” 1000 naTuridan umaRlesi xarisxis 

iqneba aranakleb 682 da araumetes 760 naTura? 

21. biWis dabadebis albaTobaa 0.515. rogoria albaToba imisa, rom 1000 axalSobilidan 

biWi iqneba aranakleb 480 da araumetes 540? 

22. albaToba imisa, rom muxis nergi gaixarebs aris 0.8. rogoria albaToba 

imisa, rom muxis 400 nergidan gaixarebs aranakleb 250? 

23. ramdeni marcvali unada daiTesos amosvlis albaTobiT 0.99, rom amosvlis 

sixSire (amosulebis raodenoba gayofili daTesilebis raodenobaze) 

gansxvavdebodes 0.95-sagan 0.01-ze naklebiT? 

24. albaToba imisa, rom adamiani daiRupeba 21 wlis asakSi aris 0.006. 

dazRveulia 20 wlis 1000 adamiani. TiToeulis wliuri Sesatania 15 lari. 

dazRveuli adamianis daRupvis SemTxvevaSi ojaxi miiRebs 1200 lars. rogoria 

albaToba imisa, rom wlis bolosaTvis kompaniis gasacemi Tanxa gadaaWarbebs 

sawevro Semonatanebs? 

25. SeafaseT albaToba imisa, rom SemTxveviTi sidide gadaixreba Tavisi 

maTematikuri lodinidan aranakleb vidre; a). gaormagebuli standartuli 

gadaxra; b). gasammagebuli standartuli gadaxra. 

26. xdomilebis moxdenis albaToba TiToeul cdaSi aris 0.25. SeafaseT 

albaToba imisa, rom 800 damoukidebel cdaSi am xdomilebis moxdenaTa 

ricxvi aRmoCndeba 150-dan 250-mde sazRvrebSi. 

27. mocemulia diskretuli SemTxveviTi sididis ganawilebis kanoni: 

0.1 0.4 0.6 

P 0.2 0.3 0.5 

SeafaseT | E| 0.4 xdomilebis albaToba. 

28. damoukidebel SemTxveviT sidideTa mimdevroba 1 

, 2 

,... mocemulia ganawilebis 

kanoniT: 

a a 

n 

P n/(2n 1) ( n1)/(2 n 

1) 



, 2 


kanoniT: 

n 1 n 

n 

P n/(2n 1) ( n1)/(2 n 

1) 

92

akmayofilebs Tu ara es mimdevroba CebiSevis Teoremis pirobebs? 


, 2 


kanoniT: 

 

n 3 0 3 

P 1/3 1/3 1/3 


31. cnobilia, rom wonebis SemTxveviTi cdomilebis saSualo mniSvnelobaa 

0.03, xolo dispersia _ 0.0016. a) rogoria albaToba imisa, rom morigi awonvis 

dros sasworis Cvenebis cdomilebis absoluturi sidide ar aRemateba 

0.04-s? b) ipoveT 95%-iani saimedoobis intervali wonebis cdomilebisaTvis. 

32. qarxnis nawarmis 15% mdare xarisxisaa. rogoria albaToba imisa, rom 

am qarxnis 1000 nawarmSi mdare xarisxis mqone nawarmis raodenoba Seadgens 

-s? 

33. gamokvlevebma aCvena, rom moswavleTa 20%-ma ar icis sagzao moZraobis 

wesebi. SemTxveviT SearCies 1600 moswavle. ramdenma moswavlem icis sagzao 

moZraobis wesebi 95%-iani garaantiiT. 

34. Zabvis gasazomad iyeneben specialur mowyobilobas. gansazRvreT am 

mowyobilobis Secdomis saSualokvadratuli gadaxra, Tu mas ar gaaCnia 

sistematuri Secdomebi, ganawilebulia normalurad da albaTobiT 0.8 ar 

gadis sazRvrebidan . 

35. Tu burTula ar gadis naxvretSi, romlis diametria , magram gadis 

naxvretSi, romlis diametria ( ), maSin misi zoma iTvleba misaRebad. 

winaaRmdeg SemTxvevaSi burTula uvargisia. cnobilia, rom burTulis 

diametri warmoadgens SemTxveviT sidides saSualoTi 

da 

dispersiiT. ipioveT albaToba imisa, rom burTula iqneba uvargisi. 

36. qarxnis nawarmis 5% mdare xarisxisaa. ramdeni nawarmi unda SeirCes 

produqciis xarisxis Sesamowmeblad, rom albaTobiT 0.95 SegveZlos imis 

mtkiceba, rom qarxnis nawarmis SemTxveviT SerCeul partiaSi mdare xarisxis 

produqciis wili Seadgens -s? 

37. vaCvenoT, rom damoukidebel SemTxveviT sidideTa mimdevroba { 

n} 

n1 

akmayofilebs 

centralur zRvariT Teoremas, Tu: a) P{ 

1} 

(1 2 )/ 2 

n 

n 

, 

n n1 

P{ 

 

n 

2 

} 1/ 2 , 1,2 ,... 

n , b) P{ n 

n 

} 1/ 

2 . 

pasuxebi: 

1. 0.00012. 2. 600. 3. 80. 4. 0.0132. 5. 0.0017. 6. TiTqmis erTidaigivea. 7. 0.036. 

8. 0.972. 9. 0.009. 10. 0.147. 11. 55. 12. 0.26. 13. 0.985. 14. 511. 15. 86. 16. 0.96. 17. 583. 

18. 576. 19. 0.131. 20. 0.9991. 21. 0.929. 22. 1. 23. 31618. 24. 0.00217. 25. 1/ 4 ; 1/9. 

26. 0.94 . 27. 0.909 . 28. ki. 29. ara. 30. ki. 31. 0.5586; [-0.0484, 0.1084]. 32. 0.9239. 

33. [1248, 1312]. 34. 1.0156. 35. 0.0456. 36. 456. 

93 

n

s t a t i s t i k a 

Tavi I 

wertilovani da intervaluri Sefasebebi. ndobis 

intervali saSualosaTvis 

Sefasebis amocana: n moculobis X ( X1,..., X n 

) SerCevis safuZvelze 

gavakeToT daskvnebi generaluri erTobliobis ucnobi parametris Sesaxeb. 

SerCevis nebismier Tn ( X) Tn ( X1,..., Xn) 

funqcias statistika (Sefaseba) 

ewodeba. wertilovani Sefasebis amocanaa moiZebnos iseTi statistika 

Tn( X1,..., X 

n) 

, romlis SerCeviTi mniSvneloba Tn( x1,..., x 

n) 

, garkveuli azriT, 

SeiZleba CaiTvalos ucnobi parametris WeSmariti (realuri) mniSvnelobis 

miaxloebad (Sefasebad) da gamoyenebuli iqnas mis nacvlad. aseT 

statistikas (Sefasebas) wertilovani statistika (Sefaseba) ewodeba. 

Tn ( X) Tn ( X1,..., Xn) 

Sefasebas ewodeba Caunacvlebeli (anu gadauadgilebadi), 

Tu E 

T( X) 

. 

T ( X) T ( X ,..., X ) Sefasebas ewodeba Zalmosili (anu Zaldebuli), 

n n 1 n 

P 

Tu Tn 

( T n 

albaTobiT krebadia -sken), roca n. 

Caunacvlebel Sefasebas ewodeba optimaluri (anu efeqturi), Tu 

mas sxva Caunacvlebel Sefasebebs Sori gaaCnia umciresi dispersia. 

Tu n moculobis SerCeva warmodgenilia variaciuli mwkrivis saxiT, 

maSin SerCeviTi saSualo ewodeba sidides: 

x1n 1 

x2n2 ... 

xknk n1 n2 

nk 

x x1 x2 

... 

. 

n n n n 

Tu SerCevidan miRebuli mniSvnelobebi ara dajgufebuli, maSin SerCeviTi 

saSualo 

1 

n 

x x i . 

n i1 

SerCeviTi saSualo warmoadgens ucnobi maTematikuri lodinis Caunacvlebel 

da Zalmosil Sefasebas. 

SerCeviTi dispersia ewodeba sidides: 

k 

n 

2 

2 ni 

1 

2 

s xi 

x xi 

x 

. 

i1 n n i1 

Sesworebuli SerCeviTi dispersia ewodeba sidides: 

n 

' n 

2 2 1 

s s x 

2 

i 

x . 

n1 n1 

i1 

Sesworebuli SerCeviTi dispersia warmoadgens ucnobi dispersiis 

Caunacvlebel da Zalmosil Sefasebas. 

SerCeviTi standartuli gadaxra (Sesabamisad, Sesworebuli SerCevi- 

Ti standartuli gadaxra) ewodeba ariTmetikul kvadratul fesvs SerCev- 

94

iTi dispersiidan 

(Sesabamisad, Sesworebuli SerCeviTi dispersiidan 

' 

s 

2 

' 

s ) 

s 

s 

2 

SerCeviTi asimetriis SerCeviTi eqscesis 

koeficienti 

koeficienti 

n 

n 

1 

3 

1 

4 

( xi 

x) 

( xi 

x) 

n i1 

n i1 

aSer 

 

e 

n 

Ser 

3 

n 

1 

2 3 

1 

2 4 

( ( xi 

x) ) 

( ( xi 

x) ) 

n i1 

n i1 

SerCeviTi korelaciis koeficienti ewodeba sidides: 

xy x y 

1 

n 

rxy 

, sadac xy x ss 

i y i ; 

x y 

n i1 

2 1 n 

x x i 

n i 1 

s s x x 

2 

; 2 

2 1 

s n 

y 

sy yi 

y 

95 

. 

n i 1 

SerCeviTi parametrebis ganawileba normaluri populaciisaTvis. 

davuSvaT, rom X1, X2,..., X 

n 

warmoadgens SerCevas normaluri generaluri 

erTobliobidan, 

Xi 

2 

N( a, ) ( i 1,2,..., 

n), maSin: X da S 2 2 

' 

( S ) damou- 

2 

kidebeli SemTxveviTi sidideebia; X N( a, / n) 

; 

n 

'2 

2 2 2 ( n1) 

S 2 

( i 

a) / 

( n) 

; 

( n 1) ; 

2 

i1 

 

X a 

X a X a 

Z N(0,1) 

; T t( n1) 

. 

' 

/ n 

S / n1 S / n 

maqsimaluri dasajerobis meTodi: vTqvaT, px ( 

i, ) aris albaToba imisa, 

rom eqsperimentis Sedegad diskretuli SemTxveviTi sidide miiRebs 

х i mniSvnelobas. maqsimaluri dasajerobis funqcia ewodeba argumentis 

funqcias: L (х 1 , х 2 , …, х п ; ) = p(x 1 , )p(x 2 , )…p(x n , ) ( ln L funqcias maqsimaluri 

dasajerobis logariTmuli funqcia ewodeba). maqsimaluri dasajerobis 

Sefasebas uwodeben -s im mniSvnelobas, sadac maqsimaluri dasajerobis 

funqcia (an rac igivea ln L) aRwevs Tavis maqsimums. mis mosaZebnad sa- 

Wiroa: 1). vipovoT warmoebuli ln L/ 

; 2). gavutoloT warmoebuli nuls 

(miviRebT e. w. maqsimaluri dasajerobis logariTmul gantolebas) da vipovoT 

kritikuli wertilebi; 3). vipovoT meore warmoebuli 2 ln L/ 

 

2 ; Tu 

is uaryofiTia kritikul wertilSi, maSin es wertili – maqsimumis 

wertilia. 

uwyveti SemTxveviTi sididis SemTxvevaSi, romlis f(x, ) ganawilebis 

simkvrivis saxe cnobilia, magram igi Seicavs ucnob parametrs, maqsimaluri 

dasajerobis funqcias aqvs Semdegi saxe: 

L( x , x ,..., x ; ) f ( x , ) f ( x , ) f ( x , ) 

. 

1 2 n 

1 2 

n

ucnobi parametris maqsimaluri dasajerobis Sefasebis sapovnelad 

unda CavataroT igive procedurebi, rac diskretul SemTxvevaSi. 

momentTa meTodi: momentTa meTodi dafuZnebulia im garemoebaze, 

rom sawyisi da centraluri empiriuli momentebi warmoadgenen Sesabamisi 

sawyisi da centraluri Teoriuli momentebis Zalmosil Sefasebebs. amitom 

ama Tu im Sefasebis misaRebad Teoriuli momentebi unda gavutoloT 

imave rigis Sesabamis empiriul momentebs da amovxsnaT miRebuli gantoleba 

an gantolebaTa sistema. 

ucnobi parametris (an 1 ) saimedoobis mqone anu 100 % -iani 

(an 100(1 )% 

-iani) ndobis intervali ewodeba intervals ( TT 1 

, 2 

) , romlisTvisac:P 

P{ T1 T2} 

(an 1 ), sadac T 1 

da T 2 

parametris garkveuli 

wertilovani Sefasebebia, (0,1) ( (0,1) ); -s mniSvnelovnebis done ewodeba; 

ndobis intervalis sigrZis naxevars Sefasebis sizuste ewodeba. 

mniSvnelovnebis donis mqone ndobis intervali normaluri populaciis 

maTematikuri lodinisaTvis cnobili dispersiis SemTxvevaSi 

2 

aris: 

z/2 z/2 

 

X 

, X 

 

n n , 

sadac z /2 

standartuli normaluri ganawilebis zeda /2-kritikuli wertilia. 

aq Sefasebis sizustea l 

. 

z 

/2 

n 

Sefasebis winaswar dafiqsirebuli l sizustis uzrunvelsayofad 

saWiro SerCevis minimaluri moculobaa: 

* 2 

n [( z 

/2) ] 1. 

l 

mniSvnelovnebis donis mqone ndobis intervals normaluri populaciis 

maTematikuri lodinisaTvis ucnobi dispersiis SemTxvevaSi aqvs 

Semdegi saxe: 

' 

' 

S 

S 

( X tn 

1, /2 , X tn 

1, /2 

), 

n 

n 

sadac tn 1, /2 

aris Tavisuflebis n 1 xarisxis mqone stiudentis ganawilebis 

zeda /2-kritikuli wertili. 

mniSvnelovnebis donis mqone asimptoturi ndobis intervali (aranormaluri) 

populaciis saSualosaTvis SerCevis didi moculobis SemTxvevaSi 

( n 30), roca dispersia cnobilia aris: 

 

 

( X n z/2 , X n z/2 

) , 

n 

n 

xolo ucnobi dispersiis SemTxvevaSi ki: 

96

' 

' 

S 

S 

( X n z/2 , X n z/2 

) . 

n 

n 

mniSvnelovnebis donis mqone ndobis intervali (aranormaluri) 

populaciis saSualosaTvis, roca ucnobia da n 30: 

' 

' 

S 

S 

( X tn 

1, /2 , X tn 

1, /2 

). 

n 

n 

rodis gamoiyeneba Z an T ganawileba 

cnobiliaTuara ? 

diax gamoiyene SerCevis 

ara 

aris Tu ara n 30? 

ara 

gamoiyene t 

formulaSi 

n1, /2 

** 

' 

da s 

 

diax 

 

z /2 

* 

moculobis miuxedavad 

gamoiyene z 

/2 

' 

Secvale s -iT 

daformulaSi 

* sidide unda iyos normalurad ganawilebuli roca n 30. 

** sidide unda iyos miaxloebiT normaluri. 


magaliTi 1. universitetis reqtors surs Seafasos wels Caricxuli 

studentebis saSualo asaki. warsuli gamokvlevebidan cnobilia, rom 

standartuli gadaxra aris 2 weli. aRebulia 50 studentisagan Semdgari 

SerCeva da misTvis gamoTvlili saSualo tolia 23.2 wlis. ipoveT 95%- 

iani ndobis intervali populaciis saSualosaTvis. igulisxmeba, rom asaki 

daaxloebiT normaluradaa ganawilebuli. 

amoxsna. 95%-iani ndobis intervali niSnavs, rom ndobis intervalis 

saimedooba 1 

95% 

100% 

0.95 , anu 0.05 . amitom normaluri ganawilebis 

cxrilidan vpoulobT z /2 

z0.025 1.96 . Sesabamisad, saZiebeli ndobis 

intervali iqneba: 

2 2 

(23.2 1.96 , 23.2 1.96 ) , (22.6,23.8) , 

50 50 

anu 23.2 0.6 . amrigad, 50 studentis asakidan gamomdinare, 95%-iani saimedoobiT, 

universitetis reqtors SeuZlia Tqvas, rom studentebis saSualo 

asaki moTavsebulia 22.6 welsa da 23.8 wels Soris. 

97

magaliTi 2. cnobilia, rom garkveuli medikamentis gamoyenebis Sedegad 

pulsis ricxvi matulobs. cnobilia, rom pulsis ricxvis standartuli 

gadaxra aris 5 dartyma wuTSi. SerCeul iqna am medikamentis 30 momxmarebeli 

da maTTvis pulsis ricxvis saSualo aRmoCnda 104 dartyma wuT- 

Si. vipovoT 99%-iani ndobis intervali pulsis ricxvis WeSmariti saSualosaTvis. 

igulisxmeba, rom Sesabamisi ganawileba daaxloebiT normaluria. 

amoxsna. aq 1 

0.99 , anu 0.01 . amitom z /2 

z0.005 2.58 . Sesabamisad, 

ndobis intervali iqneba: 

5 5 

104 2.58 a 104 2.58 

, anu 102 a 

106 . 

30 30 

maSasadame, 30 momxmareblisagan Sedgenili SerCevis safuZvelze, 

99%-iani saimedoobiT, Cven SegviZlia davaskvnaT, rom am medikamentis yvela 

momxmareblis pulsis ricxvis saSualo moTavsebulia (cvalebadobs) 

102-sa da 106-s Soris. 

magaliTi 3. kolejis prezidentma daavala statistikis maswavlebels 

Seafasos kolejis studentebis saSualo asaki. ra moculobis SerCevaa 

aucilebeli? statistikis maswavlebeli Tvlis, rom saimedooba unda 

iyos 99%, raTa Sefaseba iyos swori erTi wlis sizustiT. wina gamokvlevebidan 

cnobilia, rom asakTa standartuli gadaxra aris 3 weli. igulisxmeba, 

rom asaki daaxloebiT normalurdaa ganawilebuli 

amoxsna. am SemTxvevaSi l 1; 3; 10.99 0.01 . amitom 

z /2 

z0.005 2.58 . Sesabamisad, SerCevis minimaluri moculoba iqneba: 

* 2.58 2 

n [( 3) ] 1 [59.9] 1 60. 

1 

amrigad, imisaTvis, rom maswavlebelma Seafasos studentTa asakis 

saSualos WeSmariti mniSvneloba erTi wlis sizustiT 99%-iani saimedoobiT, 

mas sWirdeba sul cota 60 studentisagan Semdgari SerCeva. 

magaliTi 4. SemTxveviT SearCies 10 avtomobili da gazomes marjvena 

wina saburavis proteqtoris siRrme. saSualo aRmoCnda 0.32 diumi (1 diumi 

aris 2.54 sm.), xolo Sesworebuli standartuli gadaxra -- 0.08 diumi. 

ipoveT 95%-iani saimedoobis ndobis intervali proteqtoris siRrmis saSualosaTvis. 

igulisxmeT, rom sidide miaxloebiT normaluradaa ganawilebuli. 

' 

amoxsna. am SemTxvevaSi n 10, X 0.32, s 0.08 . vinaidan populaciiis 

is standartuli gadaxra ucnobia, is unda SevcvaloT SerCevis Sesworebuli 

standartuli gadaxriT s 0.08 . Tavisuflebis xarisxia 

' 

n110 1 9 . 10.95 0.05 , / 2 0.05/ 2 0.025 . Sesabamisad t ganawilebis 

zeda kritikuli wertilebis cxrilSi me-9 striqonisa da 0.025-is Sesabamisi 

svetis gadakveTaze vpoulobT tn 1, /2 

-is mniSvnelobas: 

tn 1, /2 

t9,0.025 2.262 . amitom saZiebeli ndobis intervali iqneba: 

0.08 0.08 

(0.32 2.262 ,0.32 2.262 ), anu (0.26,0.38) . 

10 10 

98

amrigad, 10 moculobis mqone SerCevis safuZvelze SeiZleba davaskvnaT, 

rom avtomobilebis wina marjvena saburavebis proteqtorebis siRrmis 

populaciis saSualo 95%-iani saimedobiT moTavsebulia 0.26 diumsa da 

0.38 diums Soris. (1 diumi = 2.54 sm) 

magaliTi 5. mocemulia generaluri erToblioba garkveuli maxasia- 

TebliT, romelic ganawilebulia normalurad 6.25-is toli dispersiiT. 

Catarebulia n 27 moculobis SerCeva da miRebulia maxasiaTeblis saSualo 

SerCeviTi mniSvneloba x 12 . vipovoT ndobis intervali, romelic 

faravs generaluri erTobliobis gamosakvlevi maxasiaTeblis ucnob maTematikur 

lodins saimedoobiT 0.99 . 

amoxsna. gvaqvs: 6.25 2.5; n 27 ; x 12 ; 10.99 0.01 ; 

z /2 

z0.005 2.57 . aqedan vRebulobT saZebn ndobis intervals: (10.76, 13.24). 

magaliTi 6. 20 eleqtronaTuris sakontrolo Semowmebisas maTi muSaobis 

saSualo xangrZlivoba aRmoCnda 2000 saaTis toli, xolo Sesworebuli 

standartuli gadaxra ki 11 saaTis toli. cnobilia, rom naTuris 

muSaobis xangrZlivoba warmoadgens normaluri kanoniT ganawilebul SemTxveviT 

sidides. ganvsazRvroT am SemTxveviTi sididis maTematikuri 

lodinis ndobis intervali saimedoobiT 0.95. 

amoxsna. am SemTxvevaSi 10.95 0.05 . stiudentis ganawilebis 

cxrilidan (Tavisuflebis 19-is toli xarisxiT) vpoulobT, rom 

tn 1, /2 

t19,0.025 2.093 . amitom saZiebeli ndobis intervali iqneba (1994.8, 

2005.2). 

amocanebi ( cnobilia an n 30) 

1. ipoveT z /2 

(standartuli normaluri ganawilebis /2-zeda 

kritikuli wertili): a). 99%-iani; b). 98%-iani; g). 95%-iani; d). 90%-iani da 

e). 94%-iani ndobis intervalebisaTvis. 

2. gamoTvaleT SerCeviTi saSualo, Sesworebuli standartuli gadaxra, 

aageT 95%-iani ndobis intervali populaciis saSualosaTvis da ipoveT 

Sefasebis sizuste. populaciis standartuli gadaxraa 11. isargebleT 

Semdegi SerCeviT: 

43 52 18 20 25 45 43 21 42 32 

24 32 19 25 26 44 42 41 41 53 

22 25 23 21 27 33 36 47 19 20 

3. qvemoT moyvanili SerCevis monacemebis mixedviT gamoTvaleT SerCeviTi 

saSualo, Sesworebuli standartuli gadaxra da aageT 95%-iani 

ndobis intervali populaciis saSualosaTvis: 

47.596 68.751 5.838 69.831 28.843 53.107 31.391 48.829 50.706 32.785 

62.892 55.105 63.974 56.674 38.362 51.549 31.938 31.851 56.088 48.321 

34.906 38.359 72.086 34.009 50.850 43.801 46.127 49.926 54.960 49.671 

4. 35 mexuTe klaselis kiTxvis qulebis saSualo aris 82, xolo Sesworebuli 

standartuli gadaxra 15. a). ipoveT 95%-iani ndobis intervali 

mexuTe klaselebis kiTxvis qulebis saSualosaTvis; b). ipoveT 99%-iani 

99

ndobis intervali mexuTe klaselebis kiTxvis qulebis saSualosaTvis; g). 

romeli intervalia ufro ganieri? axseniT ratom. 

5. gamoTvaleT SerCeviTi saSualo, Sesworebuli standartuli gadaxra 

da aageT 90%-iani ndobis intervali evropeli moqalaqis wliuri Semosavlis 

saSualosaTvis, Tu qvemoT moyvanilia SemTxveviT SerCeuli 50 

evropeli moqalaqis Semosavlebi gazomili aTasobiT evroebSi: 

84 14 31 72 26 49 252 104 31 8 

3 18 72 23 55 133 16 29 225 138 

85 24 391 72 158 4340 346 19 5 846 

461 254 125 61 123 60 29 10 366 47 

28 254 6 77 21 97 6 17 8 82 

6. 40 pedagogze dakvirvebam aCvena, rom isini saSualod 12.6 wuTs andomeben 

erTi naweris gasworebas. a). aageT 90%-iani ndobis intervali gasworebis 

drois saSualosaTvis, Tu cnobilia, rom standartuli gadaxra 

tolia 2.5 wuTis; b). ramdenad mosalodnelia, rom romelime pedagogma 

Tqvas, rom is saSualod 30 wuTs andomebs naweris gasworebas? 

7. SemTxveviT SerCeuli 40 Tavdamsxmelis saSualo qula aris 186, 

xolo TavdamsxmelTa populaciis standartuli gadaxraa 6. a). aageT 95%- 

iani ndobis intervali TavdamsxmelTa populaciis saSualo qulisaTvis; 

b). aageT 95%-iani ndobis intervali TavdamsxmelTa populaciis saSualo 

qulisaTvis im SemTxvevaSi, roca 40 Tavdamsxmelis nacvlad ganixileba 

100 Tamdamsxmelisagan Sedgenili SerCeva imave saSualo quliT; g). romeli 

intervalia ufro patara (zusti) da ratom? 

8. SemTxveviT SerCeul 49 bavSvze dakvirvebam (romelTa asaki meryeobda 

8-dan 12 wlamde) aCvena, rom isini savaWro centrSi saSualod xarjaven 

18.5 lars, xolo Sesworebuli standartuli gadaxraa 1.56 lari. aageT 

90%-iani ndobis intervali 8-dan 12 wlamde asakis bavSvebis mier sava- 

Wro centrSi daxarjuli Tanxis saSualosaTvis. 

9. gamokvlevebma aCvena, rom aSS-s moqalaqes saSualod Wirdeba 5.9 

Tve axali samuSaos mosaZebnad. aageT 95%-iani ndobis intervali samuSaos 

moZebnis saSualosaTvis, Tu gamokiTxuli 36 samuSaos mZebnelisaTvis 

Sesworebuli standartuli gadaxra aRmoCnda 0.8 Tve. 

10. qvemoT moyvanilia aSS-s SemTxveviT SerCeul 40 kompaniaSi droebiTi 

samuSao adgilebis raodenoba: 

7685 3100 725 850 11778 7300 3472 540 11370 5400 

1570 160 9953 3114 2600 2821 6200 3483 8954 8 

1000 1650 1200 390 1999 400 3473 600 1270 873 

400 713 11960 1195 2290 175 887 1703 4236 1400 

gamoTvaleT SerCeviTi saSualo, Sesworebuli standartuli gadaxra 

da aageT 90%-iani ndobis intervali droebiTi samuSao adgilebis 

saSualosaTvis. 

11. 48 SemTxveviT SerCeuli dRis monacemebis mixedviT did hospitalSi 

dRis ganmavlobaSi Semosuli pacientebis saSualod 38% saWiroebs 

gadaudebel saswrafo daxmarebas. populaciis standartuli gadaxraa 4. 

a). aageT 99%-iani ndobis intervali im pacientebis raodenobis saSualos- 

100

aTvis, vinc saWiroebs gadaudebel saswrafo daxmarebas; b). aageT 99%-iani 

ndobis intervali im pacientebis raodenobis saSualosaTvis, vinc saWiroebs 

gadaudebel saswrafo daxmarebas, Tu populaciis standartuli gadaxraa 

8 nacvlad 4-sa; g). ratomaa meore SemTxvevaSi ndobis intervali 

ufro ganieri? 

12. centraluri respublikuri saavadmyofos 84 sxvadasxva adgilas 

gazomili xmauris donis saSualo iyo 61.2 decibali, xolo Sesworebuli 

standartuli gadaxra 7.9 decibali. aageT 95%-iani ndobis intervali saavadmyofoSi 

xmauris donis realuri saSualosaTvis. 

13. mkvlevars surs Seafasos dedaqalaqiis policiis oficris saSualo 

xelfasi. misi mizania, rom Sefasebis saimedooba iyos 95%. cnobilia, 

rom policiis oficris xelfasis standartuli gadaxraa 1050 lari. 

ra moculobis SerCeva dasWirdeba mkvlevars imisaTvis, rom Sefasebis sizuste 

iyos 200 lari? 

14. fakultetis dekanats surs Seafasos mowveuli pedagogebis saSualo 

kvireuli datvirTva. wina gamokvlevis mixedviT standartuli gadaxra 

iyo 2.6 sT. ramdenad didi moculobis SerCevis aReba iqneba saWiro, 

Tu dekanats surs 99%-iani saimedoobiT WeSmarit saSualosa da SerCevis 

saSualos Soris gansxvaveba ar aRematebodes 1 saaTs. 

15. centraluri samxedro hospitalis 117 sxvadasxva oTaxSi xmauris 

saSualo done iyo 58 decibali, xolo Sesworebuli standartuli gadaxra 

4.8 decibali. aageT 90%-iani ndobis intervali hospitalSi xmauris 

donis WeSmariti saSualosaTvis. 

16. sadazRvevo kompania cdilobs Seafasos avadmyofobis dReebis 

saSualo raodenoba makdonaldsis qselSi momuSave personalisaTvis. sapilote 

SemowmebiT dadginda, rom Sesabamisi standartuli gadaxraa 2.5 

dRe. ra moculobis SerCevaa saWiro, rom kompaniam 95%-iani saimedoobiT 

daadginos intervali, romelic Seicavs avadmyofobis dReebis realur sa- 

Sualos maqsimaluri cdomilebiT 1 dRe? 

17. restornis mepatrones surs daadginos 99%-iani ndobis intervali 

Sampaniuris fasis realuri saSualosaTvis. ra moculobis unda iyos 

SerCeva, rom Sefasebis sizuste iyos 0.1 lari, Tu wina gamokvlevis Tanaxmad 

fasis standartuli gadaxra iyo 0.12 lari. 

18. samSobiaro saxlis personals surs Seafasos axalSobilTa wona. 

ra moculobis SerCeva iqneba saWiro, rom 90%-iani saimedoobiT wonis 

realuri saSualo moTavsebuli iyos SerCevis saSualosagan 6 unciis (1 

uncia = 28.3 gr) farglebSi, Tu cnobilia, rom axalSobilTa wonis standartuli 

gadaxraa 8 uncia. 

amocanebi ( ucnobia da n 30) 

19. ipoveT tn 1, /2 

(Tavisuflebis ( n 1) xarisxis mqone stiudentis ganawilebis 

/2-zeda kritikuli wertili): a). 99%-iani ndobis intervalisaTvis, 

roca n 18; b). 95%-iani ndobis intervalisaTvis, roca n 23 ; 

g). 98%-iani ndobis intervalisaTvis, roca n 15; d). 90%-iani ndobis in- 

101

tervalisaTvis, roca n 10 ; e). 95%-iani ndobis intervalisaTvis, roca 

n 20 . 

qvemoT moyvanil amocanebSi igulisxmeT, rom yvela sidide ganawilebulia 

daaxloebiT normalurad: 

20. qvemoT moyvanili monacemebisaTvis gamoTvaleT SerCeviTi saSualo, 

Sesworebuli standartuli gadaxra, aageT 95%-iani ndobis intervali 

populaciis saSualosaTvis da ipoveT Sefasebis sizuste: 

625 675 535 406 512 680 483 522 619 575. 

21. qvemoT moyvanili monacemebis mixedviT gamoTvaleT SerCeviTi sa- 

Sualo, Sesworebuli standartuli gadaxra da aageT 95%-iani ndobis intervali 

populaciis saSualosaTvis: 

62 81 86 79 73 88 90 98 78 93 87 82 78 59 63 97 93 84. 

22. SemTxveviT SerCeuli 20 pacientisis 100 mililitr sisxlSi hemoglobinis 

saSualo Semcvleloba aRmoCnda 16 grami, xolo Sesworebuli 

standartuli gadaxra ki 2 gr. aageT 99%-iani ndobis intervali WeSmariti 


23. meteorologma 15 civi haeris masivze dakvirvebis Sedegad daadgina, 

rom maTi gavrcelebis saSualo siCqarea 18 mili/sT, xolo standartuli 

gadaxra ki 2 mili/sT. aageT 95%-iani ndobis intervali realuri 

saSualo siCqarisaTvis. 

24. Statis warmomadgenels surs Seafasos Statis sakanonmdeblo 

organoSi qalebis saSualo ricxvi. man SemTxveviT SearCia 17 Stati da 

miiRo Semdegi monacemebi: 5 33 35 37 24 31 16 45 19 13 18 29 15 39 

18 58 132. gamoTvaleT saSualos wertilovani Sefaseba, Sesworebuli 

standartuli gadaxra da aageT 90%-iani ndobis intervali populaciis 

saSualosaTvis. ratomaa ndobis intervali ase ganieri? 

25. 20 Tinusze dakvirvebam aCvena, rom isini saSualod curaven 8.6 

mils saaTSi. SerCevis Sesworebuli standartuli gadaxraa 1.6. aageT 90%- 

iani ndobis intervali WeSmariti saSualosaTvis. 

26. SemTxveviT arCeuli 6 spilos saSualo wonaa 12200 funti (1 funti 

= 453.6 gr), xolo Sesworebuli standartuli gadaxra 200 funti. aageT 

95%-iani ndobis intervali WeSmariti saSualosaTvis. 

27. regionis 8 skolis saTadarigo maswavleblis yoveldRiuri xelfasebia: 

60l 56l 60l 55l 70l 55l 60l 55l. ipoveT wertilovani 

Sefasebebi da aageT 90%-iani ndobis intervali regionSi saTadarigo 

maswavleblis yoveldRiuri xelfasebis WeSmariti saSualosaTvis. 

28. 28 moqalaqis gamokiTxvis mixedviT maTi amJamindel misamarTze 

cxovrebis saSualo xangrZlivobam Seadgina 9.3 weli, xolo SerCevis Sesworebuli 

standartuli gadaxra aris 2 weli. aageT 90%-iani ndobis intervali 

populaciis saSualosaTvis. 

29. avtomaRaziis menejerma daadgina, rom mis 6 TanamSromels avtomobilis 

wylis tumbos Secvla SeuZlia saSualod 18 wuTSi, xolo SerCevis 

Sesworebuli standartuli gadaxra aris 3 wuTi. aageT 99%-iani ndobis 

intervali WeSmariti saSualosaTvis. 

102

30. SemTxveviT SerCeuli 25 avtomobiliani studenti kviraSi saSualod 

xarjavs 18.53 laris benzins. SerCevis Sesworebuli standartuli 

gadaxraa 3 lari. aageT 95%-iani ndobis intervali realuri saSualosa- 

Tvis. 

31. stresul situaciaSi myofi 10 kaciani jgufis guliscemis saSualo 

ricxvi wuTSi Seadgens 126-s, xolo Sesworebuli standartuli gadaxra 

aris 4. aageT 95%-iani ndobis intervali stresul situaciaSi myofi 

kacebis guliscemis realuri saSualosaTvis. 

32. stresul situaciaSi myofi 6 qali guliscemis saSualo ricxvi 

wuTSi Seadgens 115-s, xolo Sesworebuli standartuli gadaxra aris 6. 

aageT 95%-iani ndobis intervali stresul situaciaSi myofi qalebis guliscemis 

realuri saSualosaTvis. 

33. hospitalis 24 saoperacio oTaxSi xmauris donis saSualo iyo 

41.6 decibali, xolo SerCevis Sesworebuli standartuli gadaxra ki 7.5 

decibali. aageT 95%-iani ndobis intervali saoperacio oTaxebSi xmauris 

donis realuri saSualosaTvis. 

34. aageT 98%-iani ndobis intervali populaciis saSualosaTvis, Tu 

misgan aRebuli SerCevaa: 

22 24 120 382 50 38 297 

29 23 70 56 17 51 38 

35. erovnul gamocdaze maTematikaSi 20 abiturientis guliscemis 

saSualo iyo 96 dartyma wuTSi, xolo SerCevis Sesworebuli standartuli 

gadaxra ki 5. aageT 95%-iani ndobis intervali guliscemis 


36. dedaqalaqis 28 axaldaqorwinebulis saSualo wliuri Semosavali 

Seadgens 58219 lars, xolo SerCevis Sesworebuli standartuli gadaxraa 

56 lari. aageT 95%-iani ndobis intervali Semosavlebis realuri 


pasuxebi: 

1. 2.58; 2.33; 1.96; 1.65; 1.88. 2. 32.03; 11.01; (28.1, 35.97); 2.01. 3. 46.9709; 14.3582; 

(41.8329, 52.1089). 4. (77, 87); (75, 89); II, vinaidan saimedooba ufro didia. 5. 

196; 617.3; (52, 340). 6. (11.9,13.3); es Zalian mcire albaTobis mqonea, radgan 30 

bevrjer aRemateba 13.3-s. 7. (184, 188); (185, 187); II, vinaidan 100 > 40. 8. (18.13, 

18.87). 9. (5.6, 6.2). 10. 3222.4; 3480.1; (2341.5, 4130.3). 11. (37, 39); (35, 41); vinaidan 

8 4. 12. (59.5, 62.9). 13. 106. 14. 45. 15. (57.4, 58.6). 16. 25. 17. 10. 18. 5. 19. 2.898; 

2.624; 2.093; 2.074; 1.833. 20. 563.2; 87.9; (500.4, 626); 27.8. 21. 81.72; 11.58; (75.96, 87.48). 

22. (15, 17). 23. (17, 19). 24. 33.4; 28.7; (21.2, 45.6); monacemi 132 araCveulebrivad 

didia (“amovardnili” – “outlier” monacemia). 25. (8, 9.2). 26. (11990, 12410). 27. 

58.9; 5.1; (55.5, 62.3). 28. (8.7, 9.9). 29. (13, 23). 30. (17.29, 19.77). 31. (123, 129). 32. (109, 

121). 33. (38.4, 44.8). 34. (7.9, 165.9). 35. (94, 98). 36. (58197, 58241). 

103

Tavi II 

ndobis intervali bernulis sqemaSi 

ndobis intervali da SerCevis moculoba bernulis sqemisaTvis 

(proporciisaTvis): bernulis sqemaSi ucnobi p albaTobis (populaciis 

proporciis) wertilovani Sefasebaa fardobiTi sixSire w S / n, sadac 

S 

n 

n 

Xi 

( 

i 

1 

i1 

X , Tu i -ur cdaSi moxda warmateba, da X 0, Tu i -ur cda- 

Si moxda marcxi) – warmatebaTa raodenobaa n damoukidebel cdaSi. 

mniSvnelovnebis donis mqone asimptoturi ndobis intervals ucnobi 

p albaTobisaTvis (proporciisaTvis) aqvs Semdegi saxe: 

wn (1 wn ) wn (1 wn 

) 

( wn z/2 , wn z/2 

) . 

n 

n 

populaciis proporciis l sizustis intervaluri SefasebisaTvis 

saWiro SerCevis minimaluri moculobaa: 

* z 

/2 2 

n [ wn(1 wn)( ) ] 1 

l 

(amasTanave, roca wn 

-s aproqsimacia ucnobia, maSin unda gamoviyenoT 

wn 

0.5 , vinaidan namravlis wn(1 wn) 

maqsimaluria, roca wn 

0.5 ). 

dazustebuli mniSvnelovnebis donis mqone asimptoturi ndobis 

intervali ucnobi p albaTobisaTvis aris: 

2 2 2 2 

z /2 

wn (1 wn ) z /2 

z /2 

wn (1 wn 

) z 

/2 

wn z/2 w 

2 n z/2 

 

2 

( 

2n n 4n , 

2n n 4n 

) . 

2 2 

z/2 z/2 

1 

1 

n 

n 

mniSvnelovnebis donis mqone asimptoturi ndobis intervali puasonis 

populaciis ucnobi parametrisaTvis aris: 

X 

n 

X 

n 

( X 

nz/2 , X 

nz/2 

) . 

n 

n 


magaliTi 1. moxucebis 500 momvlelisagan Semdgar SerCevaSi 60 mamakacia. 

ipoveT 90%-iani saimedoobis ndobis intervali moxucebis movlis 

programaSi monawile mamakacTa WeSmariti proporciisaTvis. 

amoxsna. vinaidan 10.90 0.10 , amitom z /2 

z0.05 1.65 . 

garda amisa, Sn 

60 , n 500 . Sesabamisad, wn 

60/500 0.12 da 

1w n 

10.12 0.88 . 

amitom ndobis intervali iqneba: 

i 

n 

n 

104

0.12 0.88 0.12 0.88 

(0.12 1.65 ,0.12 1.65 ), (0.096,0.144) 

500 500 

anu 

9.6% p 14.4% . 

amrigad, moxucebis movlis programaSi monawile mamakacebis realuri 

proporcia 90%-iani saimedoobiT moTavsebulia 9.6%-sa da 14.4%-s Soris. 

magaliTi 2. mkvlevars surs 95%-iani saimedoobiT Seafasos im adamianebis 

proporcia, romlebsac saxlSi aqvT sakuTari kompiuteri. wina 

gamokvlevam aCvena, rom gamokiTxulTa 40%-s saxlSi aqvs sakuTari kompiuteri. 

mkvlevars surs 2%-iani sizuste WeSmariti proporciisaTvis. vipovoT 

SerCevis moculobis aucilebeli minimumi. 

amoxsna. gvaqvs: z 

/2 

z(1 0.95)/2 z0.025 1.96 , l 0.02 , wn 

0.40 , 1w n 

0.60 . 

amitom SerCevis mnimimaluri moculoba iqneba: 

1.96 2 

n [0.40 0.60 ( ) ] 1 [2304.96] 1 2305 . 

0.02 

e. i. unda gamoikiTxos 2305 adamiani. 

magaliTi 3. mkvlevars surs 90%-iani saimedoobiTa da 5%-iani sizustiT 

Seafasos aRmasrulebeli xelisuflebis im warmomadgenelTa proporcia, 

romlebsac aqvT manqanis telefoni. 

amoxsna. vinaidan ucnobia wn 

-is Sefaseba, statistikosi irCevs 

wn 

0.5 da 1w n 

0.5 mniSvnelobebs. 

garda amisa, am SemTxvevaSi l 0.05 , 10.90 0.1 , z /2 

z0.05 1.65 . 

Sesabamisad, 

1.65 2 

n [0.5 0.5 ( ) ] 1 [272.25] 1 273 . 

0.05 

e. i. unda gamoikiTxos aRmasrulebeli xelisuflebis 273 warmomadgeneli. 

amocanebi 

1. 1500 gamokiTxuli axalgazrdidan 39%-ma Tqva, rom isini apireben 

momaval wels aiRon ufro xangrZlivi Svebuleba. aageT 95%-iani ndobis 

intervali im axalgazrdebis realuri proporciisaTvis, romlebic apireben 

momaval wels ufro xangrZlivi Svebulebis aRebas. 

2. dedaqalaqis 100 SemTxveviT SerCeul mcxovrebs Soris 27 msuqania. 

aageT 90%-iani ndobis intervali dedaqalaqis mosaxleobaSi msuqani 

adamianebis realuri proporciisaTvis. 

3. 100 gamokiTxul umuSevars Soris 65% araa dainteresebuli dabrundes 

Zvel samsaxurSi. aageT 95%-iani ndobis intervali im umuSevrebis 

realuri proporciisaTvis, romlebsac ar surT Zvel samsaxurSi dabruneba. 

105

4. gamokiTxuli 1015 axalgazrdidan 13% dadebiTad afasebs sakanonmdeblo 

organos saqmianobas. aageT 95%-iani ndobis intervali im axalgazrdebis 

realuri proporciisaTvis, romlebic ase fiqroben. 

5. 200 gamokiTxuli muSidan 168 muSam ganacxada, rom 1 saaTis ganmavlobaSi 

satelefono zarma aranakleb 3-jer gaawyvetina muSaoba. aageT 

95%-iani ndobis intervali im muSebis realuri proporciisaTvis, romlebsac 

satelefono zarma aranakleb 3-jer gaawyvetina muSaoba. 

6. 120 gamokiTxul pirvelkursel gogonas Soris 18 gogonas ar 

surs muSaoba gaTxovebis Semdeg. aageT 95%-iani ndobis intervali im pirvelkurseli 

gogonebis realuri proporciisaTvis, romlebsac ar surT 

muSaoba gaTxovebis Semdeg. 

7. jandacvis saministros monacemebiT monacemebiT 13-14 wlis mozardebidan 

yoveli mexuTe zogjer eweva sigarets. mTeli qveynis masStabiT 

13-14 wlis mozardebSi sigaretis mwevelebis proporciasTan Sesadareblad 

ganaTlebis saministrom gamokiTxa 200 13-14 wlis skolis moswavle da 

daadgina, rom maTi 23% zogjer eweva sigarets. aageT 99%-iani ndobis intervali 

skolis 13-14 wlis moswavleebSi sigaretis mwevelTa realuri 

proporcisasTvis da SeadareT is jandacvis saministros monacemebs. 

8. 80 dakvirvebuli avtosagzao SemTxvevidan 46-is gamomwvevi mizezi 

iyo mZRolis mier alkoholis miReba. aageT 95%-iani ndobis intervali im 

avtosagzao SemTxvevebis realuri proporciisaTvis, romelTa gamomwvevi 

mizezi iyo mZRolis mier alkoholis miReba. 

9. 90 gamokiTxuli ojaxidan 40 flobs erT mainc iaraRs. aageT 95%- 

iani ndobis intervali im ojaxebis realuri proporciisaTvis, romlebic 

floben erT mainc iaraRs. 

10. 500 gamokiTxuli muSidan 45% profkavSirebis wevria. aageT 90%- 

iani ndobis intervali im muSebis realuri proporciisaTvis, romlebic 

Sedian profkavSirebSi. 

11. garkveuli asakobrivi jgufis 100 gardacvlili adamianis kvlevam 

aCvena, rom maTgan 25% gardaicvala kiboTi. aageT 98%-iani ndobis intervali 

am asakobrivi jgufis gardacvlil adamianebSi im adamianebis 

realuri proporciisaTvis, romlebic daiRupnen kiboTi. 

12. mkvlevars surs 95%-iani saimedoobiTa da 0.04-is toli sizustiT 

Seafasos realuri proporcia im adamianebis, romlebsac umogzauriaT sazRvargareT. 

ipoveT amisaTvis saWiro SerCevis aucilebeli moculoba, Tu 

cnobilia, rom 200 adamianisagan Semdgar SerCevaSi gasul wels sazRvargareT 

imogzaura 80 adamianma. 

13. medikoss surs 99%-iani saimedoobiTa da 2%-is sizustiT Seafasos 

im qalebis realuri proporcia, romlebic Rebuloben vitamins. adre 

Catarebuli kvlevis Tanaxmad 180 qalidan 25% Rebulobda vitamins. a). ra 

moculobis unda iyos saWiro SerCeva? b). Tu ar iqneboda xelmisawvdomi 

SerCeviTi proporcia, maSin ra moculobis SerCevis aReba mogviwevda? 

14. gamokvlevam aCvena, rom 55 welze meti asakis 100 qalidan 29% 

qvrivia. a). ra moculobis SerCevaa saWiro, raTa 90%-iani saimedoobiTa 

da 0.05-is sizustiT SevafasoT 55 welze meti asakis im qalebis realuri 

106

proporcia, romlebic qvrivebi arian? b). ra moculobis SerCeva unda avi- 

RoT im SemTxvevaSi, roca ara xelmisawvdomi SerCeviTi proporcia? 

15. mkvlevars surs 90%-iani saimedoobiTa da 5%-is sizustiT Seafasos 

im mamakacebis reluri proporcia, romelTa simaRle 5 futsa (1 futi 

= 30.48 sm) da 5 diumze (1 diumi = 2.54 sm) naklebia. a). ra moculobis Ser- 

Cevaa amisaTvis aucilebeli, Tu cnobilia, rom 300 mamakacidan 30-is sima- 

Rle 5 futsa da 5 diumze naklebia; b). b). ra moculobis SerCeva unda aviRoT 

im SemTxvevaSi, roca ar iqneba xelmisawvdomi SerCeviTi proporcia? 

16. maswavlebels surs 98%-iani saimedoobiTa da 0.03-is sizustiT 

Seafasos maRali klasis im moswavleebis realuri proporcia, romlebic 

yovel saRamos 1 saaTs mainc mecadineoben. a). ra moculobis unda iyos 

saWiro SerCeva, Tuki cnobilia, rom 250 gamokiTxuli moswavlidan 60% 

yovel saRamos 1 saaTs mainc mecadineobda; b). ra moculobis SerCeva unda 

aviRoT im SemTxvevaSi, roca ar iqneba xelmisawvdomi SerCeviTi proporcia? 

17. ipoveT saimedoobis xarisxi, Tu SerCevis moculobaa 600 da 

mkvlevars surs, rom realuri proporciis Sefasebis maqsimaluri cdomileba 

iyos 4%. amasTanave, cnobilia, rom adrindeli kvlevisas proporcia 

iyo 50%. 

18. 1015 gamokiTxulidan 68%-s sjera, rom respublikelebi upiratesobas 

aniWeben ganaTlebas. ra saimedoobis ndobis intervali unda gamoviyenoT, 

Tuki Sefasebis cdomileba ar unda aRematebodes 3%-s. 

pasuxebi: 

1. (0.365, 0.415) anu (36.5%,41.5%). 2. (0.197, 0.343). 3. (0.557, 0.743). 4. (0.109, 0.151). 5. 

(0.797, 0.883). 6. (0.086, 0.214). 7. (0.153, 0.307); es Sedegi eTanxmeba jandacvis 

saministros monacemebs. 8. (0.467, 0.683). 9. (0.337, 0.543). 10. (0.413, 0.487). 11. 

(0.149, 0.351). 12. 577. 13. 3121; 4161. 14. 225; 273. 15. 99; 273. 16. 1448; 1509. 17. 95%. 

18. 96%. 

107

Tavi III 

ndobis intervali dispersiisaTvis 


dispersiisaTvis cnobili saSualos SemTxvevaSi aris: 

n 

n 

 

2 2 

( xi 

a) ( xi 

a) 

 

i1 i1 

, , 

2 2 

n, /2 n,1 /2 

 

 

 

 

xolo ucnobi saSualos SemTxvevaSi ki: 

 

'2 '2 

( n 1) s ( n 1) 

s 

, 

2 2 

. 

n 1, /2 

 

n1,1 /2 

 

2 2 

zogjer gamoiyeneba aRniSvnebi: marjv. : n 1, /2 

da 2 2 marcx. 

: n 1,1 /2. 


standartuli gadaxrisaTvis SerCevis didi moculobis SemTxveva- 

Si daaxloebiT aris: 

' 

' 

' s ' s 

( s z/2 , s z/2 

). 

2n 

2n 

SerCeviTi dispersia ewodeba sidides: 

k 

n 

2 

2 n 1 

2 1 

n 

i 

s xi 

x xi 

x 

( x x i1 n n 

i ). 

i1 

n i1 

Sesworebuli SerCeviTi dispersia ewodeba sidides: 

n 

' n 

2 2 1 

s s x 

2 

i 

x . 

n1 n1 

i1 

SerCeviTi standartuli gadaxra (Sesabamisad, Sesworebuli SerCevi- 

Ti standartuli gadaxra) ewodeba ariTmetikul kvadratul fesvs SerCeviTi 

dispersiidan 

(Sesabamisad, Sesworebuli SerCeviTi dispersiidan 

' 

s 

2 

' 

s ) 

s 

s 

2 


magaliTi 1. ipoveT TambaqoSi nikotinis Semcvlelobis dispersiisa 

da standartuli gadaxrisaTvis 95%-iani ndobis intervali, Tu cnobilia, 

rom 20 sigaretisagan Sedgenil SerCevaSi nikotinis Semcvlelobis Sesworebuli 

standartuli gadaxra 1.6 miligramis tolia. 

amoxsna. vinaidan am SemTxvevaSi 10.95 0.05 , amitom xi-kvadrat 

ganawilebis ori kritikuli mniSvneloba, romelic Seesabameba / 2 0.025 - 

sa da 1 

/ 2 0.975 

-s da Tavisuflebis xarisxis ricxvs, romelic tolia 

2 

n1 20 1 19 , Sesabamisad iqneba: 19,0.025 32.852 da 

saZiebeli ndobis intervali iqneba: 

. amitom 

2 

19,0.975 

8.907 

108

2 

19,0.01 

36.2 

2 2 

(20 1) (1.6) (20 1) (1.6) 

( , ) , anu (1.5,5.5) 

32.852 8.907 

amrigad, Cven SegviZlia 95%-iani saimedoobiT vigulisxmoT, rom 

sigaretSi nikotinis Semcvlelobis realuri dispersia moTavsebulia 1.5 

miligramsa da 5.5 miligrams Soris. 

ndobis intervali standartuli gadaxrisaTvis iqneba: 

( 1.5, 5.5), anu (1.2,2.3) . 

maSasadame, sigaretSi nikotinis Semcvlelobis realuri standartuli 

gadaxris 95%-iani ndobis intervalia (1.2mg, 2.3mg). 

magaliTi 2. CavTvaloT, rom xmauri Sveulmfrenis kabinaSi, garkveul 

reJimSi momuSave Zravis dros, normaluri kanoniT ganawilebuli Sem- 

TxveviTi sididea. SemTxveviT SerCeul 20 SveulmfrenSi gazomes xmis done 

(decibalebSi). xmis donis Sesworebuli SerCeviTi dispersia aRmoCnda 

22.5-is toli. vipovoT ndobis intervali, romelic faravs mocemuli 

tipis Sveulmfrenebis kabinaSi xmauris sididis ucnob standartul gadaxras 

98%-iani saimedoobiT. 

amoxsna. Tavisuflebis 19-is toli xarisxiTa da (1 0.98)/ 2 0.01 

albaTobis saSualebiT 2 –is ganawilebis cxrilidan vpoulobT sidides: 

. analogiurad, (1 + 0.98)/2 = 0.99 albaTobis saSualebiT vpoulobT: 

2 19,0.99 

7.63 . Sesabamisad, ucnobi saSualos SemTxvevaSi standartuli 

gadaxris ndobis intervalis 

 

'2 '2 

( n 1) S ( n 1) 

S 

, 

 

2 2 

n 1, /2 

 

n1,1 /2 

 

formulis gamoyenebiT vRebulobT, rom saZiebeli ndobis intervalia: 

(3.44, 7.49). 

amocanebi 

1. cxrilis saSualebiT ipoveT 

109 

2 

marcx. 

da 

2 

marjv. 

, Tu: a). 0.05 , 

n 16 ; b). 0.1, n 5 ; g). 0.01, n 23 ; d). 0.05 , n 29 ; e). 0.1, n 14 . 

qvemoT moyvanil amocanebSi igulisxmeT, rom yvela sidide ganilebulia 

normalurad: 

2. aageT 95%-iani ndobis intervali elementis sicocxlis xangr- 

Zlivobis dispersiisa da standartuli gadaxrisaTvis, Tu cnobilia, rom 

20 elementiani SerCevis Sesworebuli standartuli gadaxraa 1.7 Tve. 

3. aageT 90%-iani ndobis intervali avtobusis usafrTxoebis Sesamowmeblad 

saWiro drois dispersiisa da standartuli gadaxrisaTvis, Tu 

cnobilia, rom 20 avtobusis usafrTxoebis Sesamowmeblad saWiro drois 

Sesworebuli standartuli gadaxraa 6.8 wT. 

4. aageT 99%-iani ndobis intervali Zravis zeTis 25 litriani konteineris 

wonis dispersiisa da standartuli gadaxrisaTvis, Tu 

cnobilia, rom 14 konteinerisaTvis Sesworebuli dispersia aris 3.3. 

5. qvemoT moyvanilia vaSlis wvenSi Saqris Semcveloba gramebSi:

18.6 19.5 20.2 20.4 19.3 

21 20.3 19.6 20.7 18.9 

22.1 19.7 20.8 18.9 20.7 

21.6 19.5 20.1 20.3 19.9 

aageT 99%-iani ndobis intervali populaciis dispersiisa da standartuli 

gadaxrisaTvis. 

6. aageT 90%-iani ndobis intervali universitetis meoTxe kursis 

studentebis asakis dispersiisa da standartuli gadaxrisaTvis, Tu cnobilia, 

rom 24 studentis asakis Sesworebuli standartuli gadaxraa 2.3 

w. 

7. aageT 98%-iani ndobis intervali drois im Sualedis dispersiisa 

da standartuli gadaxrisTvis, romelic sWirdeba satelefono kompanias 

zaris swori mimarTulebiT gadarTvisaTvis, Tu cnobilia, rom 15 zaris 

SemTxvevaSi am drois Sesworebuli standartuli gadaxra tolia 1.6 wT. 

8. aageT 90%-iani ndobis intervali aqciis fasis dispersiisa da 

standartuli gadaxrisaTvis Semdegi SerCevis mixedviT: 

26.69 13.88 28.37 12 75.37 

7.5 47.5 43 3.81 53.81 

13.62 45.12 6.94 28.25 28 

60.5 40.25 10.87 46.12 14.75 

9. avtoteqmomsaxurebis centri akeTebda reklamas, rom zeTis Secvlisas 

klients ar dasWirdeboda 30 wuTze meti lodini. zeTis Secvlis 

28 SemTxvevaSi drois Sesworebuli standartuli gadxra iyo 5.2 wuTi. 

aageT 95%-iani intervali populaciis standartuli gadaxrisaTvis. 

10. aageT 95%-iani ndobis intervali 12-unciani yavis paketis dispersiisa 

da standartuli gadaxrisaTvis, Tu Semowmebul 8 paketSi yavis wona 

iyo: 12.03 12.1 12.02 11.98 12 12.05 11.97 11.99. 

11. aageT 95%-iani ndobis intervali elementebis muSaobis xangrZlivobis 

standartuli gadaxrisaTvis, Tu cnobilia, rom 200 elementisaTvis 

Sesworebuli standartuli gadaxraa 18 Tve. 

savarjiSoebi gameorebisaTvis 

12. statistikis departamentis monacemebiT dedaqalaqis macxovreblebs 

kviraSi saSualod 54 saaTi aqvT televizori CarTuli. mkvlevarma 

gamokiTxa 50 moqalaqe da daadgina, rom maT saSualod 43.7 sT aqvT 

televizori CarTuli. SerCevis Sesworebuli standartuli gadaxraa 6.2 

sT. aageT 90%-iani ndobis intervali WeSmariti saSualosaTvis. rogor 

esadageba es Sedegi statistikis departamentis monacemebs? 

13. parkSi moseirne 36 adamianze dakvirvebam aCvena, rom isini saSualod 

gadaadgildebian 2.6 km/sT siCqariT. SerCevis Sesworebuli standartuli 

gadaxraa 0.4 km/sT. aageT 95%-iani ndobis intervali realuri saSualosaTvis. 

110

14. saSualo wona 60 SemTxveviT SerCeuli mikroavtobusis aris 2627 

kg, xolo Sesworebuli standartuli gadaxra ki 400 kg. aageT 99%-iani 

ndobis intervali WeSmariti saSualo wonisaTvis. 

15. 1500 gamokiTxuli adamianidan Svebulebis ganmavlobaSi saSualod 

7.5 Rame gaatara sastumroSi, Sesabamisi Sesworebuli standartuli 

gadaxra iyo 0.8 Rame. aageT 95%-iani ndobis intervali realuri 


16. SemTxveviT SerCeuli 25 qalaqgareT macxovreblis gamokiTxvis 

mixedviT isini saSualod 18.23 lars xarjaven transportSi, xolo 

Sesworebuli standartuli gadaxraa 3 lari. CaTvaleT, rom transportis 

xarjebi normaluradaa ganawilebuli da aageT 95%-iani ndobis intervali 


17. qalaqis garkveul nawilSi 5 Tviani dakvirvebis Sedegad TveSi 

saSualod 28 fostalioni iyo dakbenili mawanwala ZaRlebis mier. SerCevis 

Sesworebuli standartuli gadaxraa 3. CaTvaleT, rom dakbenili fostalionebis 

ricxvi normaluradaa ganawilebuli da aageT 90%-iani ndobis 

intervali TveSi dakbenili fostalionebis realuri saSualosaTvis. 

18. mkvlevars surs Seafasos profesoris saSualo xelfasi 95%-iani 

saimedoobiTa da 200 l. sizustiT. ra moculobis SerCeva unda iqnas ganxiluli, 

Tu cnobilia, rom xelfasis standartuli gadaxraa 1050 lari? 

19. mkvlevars surs Seafasos universitetis yovelwliuri saSualo 

danaxarjebi safosto momsaxurebaze 90%-iani saimedoobiTa da 25 laris 

sizustiT. ra minimaluri moculobis SerCeva unda iqnas ganxiluli, Tu 

cnobilia, rom safosto danaxarjebis standartuli gadaxraa 80 lari? 

20. 1500 gamokiTxuli axalgazrdidan 42% upiratesobas aniWebs istoriuli 

adgilebis daTvalierebas SvebulebaSi yofnisas. aageT 95%-iani 

ndobis intervali im axalgazrdebis realuri proporciisaTvis, vinc upiratesobas 

aniWebs istoriuli adgilebis daTvalierebas. 

21. 200 ubeduri SemTxvevidan, romelic saWiroebs gadaudebel samedicino 

daxmarebas, 40% moxda saxlSi. aageT 90%-iani ndobis intervali saxlSi 

momxdari ubeduri SemTxvevebis realuri proporciisaTvis. 

22. 100 gamokiTxulidan 85 kmayofilia meriis muSaobiT. aageT 90%- 

iani ndobis intervali im adamianebis realuri proporciisaTvis, romlebic 

kmayofilia meriis muSaobiT. 

23. dietologs surs 95%-iani saimedoobiTa da 2%-iani sizustiT Seafasos 

im mozardebis realuri proporcia, romlebic Wamen Zilis win. ra 

moculobis SerCeva iqneba amisaTvis aucilebeli, Tu adrindeli kvlevis 

Tanaxmad 100 gamokiTxulidan 18% Wamda Zilis win? 

24. 200 gamokiTxuli mozardidan 15% regularulad TamaSobda kalaTburTs. 

ra moculobis SerCeva unda aiRos mkvlevarma, Tu misi 

survilia 99%-iani saimedoobiTa da 1%-is sizustiT Seafasos im mozardebis 

realuri proporcia, vinc regularulad TamaSobs kalaTburTs. ) 

25. 28 forToxlis Sesworebuli standartuli gadaxraa 0.34 sm. aageT 

99%-iani ndobis intervali forToxlis diametris WeSmariti standartuli 


111

26. 22 balaxis sakreWi manqanis mier 1 litri benziniT gakreWili 

balaxis zolis sigrZis Sesworebuli SerCeviTi dispersiaa 2.6. aageT 90%- 

iani ndobis intervali realuri dispersiisaTvis. 

27. SemTxveviT SerCeuli 15 Tovlmavali manqanis batareebis muSaobis 

xangrZlivobis (TveebSi) Sesworebuli dispersia iyo 8.6. aageT 90%-iani 

ndobis intervali realuri dispersiisaTvis. 

28. dedaqalaqis 28 policiis oficris simaRlis Sesworebuli SerCeviTi 

standartuli gadaxra iyo 1.83 diumi (1 diumi = 2.54sm). aageT 95%-iani 

ndobis intervali policiis oficrTa simaRlis realuri standartuli 



29. 49 SemTxveviT SerCeuli wignis moyvaruli kviris ganmavlobaSi 

wignis SeZenaze saSualod xarjavs 23.45 lars. Sesabamisi Sesworebuli 

standartuli gadaxraa 2.8 lari. aageT 90%-iani ndobis intervali realuri 


30. 20 gamokiTxul pacients eqimTan erT viziti saSualod ujdeba 

44.8 lari. SerCevis Sesworebuli standartuli gadaxraa 3.53 lari. CaTvaleT, 

rom vizitis Rirebuleba ganawilebulia normalurad da aageT 95%- 

iani ndobis intervali eqimTan erTi vizitis Rirebulebis realuri saSualosaTvis. 

31. 40 SemTxveviT SerCeuli saskolo avtobusis saSualo wonaa 4150 

funti, xolo Sesworebuli standartuli gadaxra ki 480 funti. aageT 

99%-iani ndobis intervali avtobusis wonis realuri saSualosaTvis. 

32. SemTxveviT SerCeuli 10 sadazRvevo jgufis saSualo asaki iyo 

48.6 weli, xolo Sesworebuli standartuli gadaxra ki 4.1 weli. aageT 

95%-iani ndobis intervali sadazRvevo jgufebis asakis realuri saSualosaTvis. 

33. respublikur saavadmyofoSi, SemTxveviT SerCeuli 8 kviraze 

dakvirvebisas, kviraSi saSualod 438 pacienti mimarTavda saswrafo 

daxmarebis ganyofilebas. Sesworebuli standartuli gadaxra iyo 16. 

CaTvaleT, rom aRniSnuli kategoriis pacientTa ricxvi ganawilebulia 

normalurad da aageT 99%-iani ndobis intervali pacientTa realuri 


34. dedaqalaqis garkveul ubanSi, 4 Tviani dakvirvebis Sedegad 

dadginda, rom TveSi saSualod xdeboba 31 binis qurdoba, xolo Sesworebuli 

standartuli gadaxra iyo 4. aageT 90%-iani ndobis intervali 

TveSi binis qurdobis realuri saSualosaTvis. 

35. fakultetis dekans surs Seafasos saaTebis raodenoba kviris manZilze, 

romelsac pirvelkurselebi uTmoben swavlas. wina kvlevis standartuli 

gadaxra iyo 2.6 sT. ra minimaluri moculobis SerCeva unda iqnas 

ganxiluli, raTa 99%-iani saimedoobiT realuri saSualos gansxvaveba 

SerCeviTi saSualosagan ar arematebodes 0.5 saaTs. 

36. sagzao departaments surs 300 laris sizustiT Seafasos gzebis 

saremonto samuSaoebze yovelwlirad daxarjuli Tanxis saSualo 90%- 

112

iani saimedoobiT. ra minimaluri moculobis SerCevis ganxilva iqneba sa- 

Wiro, Tu cnobilia, rom standartuli gadaxraa 900 lari? 

37. 75 gamokiTxuli muSidan 53 samsaxurSi yoveldRiurad dadis avtobusiT. 

aageT 95%-iani ndobis intervali im muSebis realuri proporciisaTvis, 

vinc samsaxurSi dadis avtobusiT. 

38. 150 ubeduri SemTxvevidan, romelic moiTxovs gadaudebel samedicino 

daxmarebas, 36% modis 6 wlamde asakis bavSvebze. aageT 90%-iani 

ndobis intervali im ubeduri SemTxvevebis realuri proporciisaTvis, 

romlebSic moyvnen 6 wlamde bavSvebi. 

39. 90 gamokiTxuli ojaxidan 40 flobs erT mainc televizors. aageT 

95%-iani ndobis intervali im ojaxebis realuri proporciisaTvis, 

romlebic floben erT mainc televizors. 

40. dietologs surs 95%-iani saimedoobiTa da 3% sizustiT daadginos 

realuri proporcia im adamianebis, romlebic ar miirTmeven aranair 

vaxSams. ra minimaluri moculobis SerCeva dasWirdeba mas, Tu wina kvlevisas 

125 gamokiTxuli adamianidan 15% ar miirTmevda vaxSams. 

41. 25 SemTxveviT SerCeuli novelis gverdebis ricxvis Sesworebuli 

standartuli gadaxraa 9 gverdi. aageT 95%-iani ndobis intervali 

populaciis standartuli gadaxrisaTvis. 

42. aageT 90%-iani ndobis intervali policiis inspeqtoris mier avtomobilis 

usafrTxoebis Semowmebaze daxarjuli drois dispersiisa da 

standartuli gadaxrisaTvis, Tu 27 SemTxveviT SerCeuli avtomobilisaTvis 

am drois Sesworebuli standartuli gadaxra iyo 6.8 wuTi. CaTvaleT, 

rom dro ganawilebulia normalurad. 

43. 20 SemTxveviT SerCeuli avtomobilis mier 1 galoni (1 galoni = 

3.38 litri)benzinis gamoyenebis Sedegad gamoyofili mavne nivTierebebis 

raodenobis Sesworebuli standartuli gadaxraa 2.3 uncia. aageT 90%-iani 

ndobis intervali avtomobilebis mier gamoyofili mavne nivTierebebis 

standartuli gadaxrisaTvis. 

pasuxebi: 

1. 6.262, 27.488; 0.711, 9.488; 8.643, 42.796; 15.308, 44.461; 5.892, 22.362. 2. (1.7, 6.2); (1.3, 

2.5). 3. (30.9, 78.2); (5.6, 8.8). 4. (1.4, 11.7); (1.2, 3.4). 5. (0.4, 2.25); (0.63, 1.5). 6. (3.5, 9.3); 

(1.9, 3). 7. (1.2, 7.7); (1.1, 2.8). 8. (259.343, 772.724); (16.104, 27.798). 9. (4.1, 7.1). 10. 

(0.001, 0.008); (0.032, 0.089). 11. (16.2, 19.8). 12. (45.9, 48.7); 54 sT gacilebiT didia, 

vidre mosalodneli intervali. 13. (2.5, 2.7). 14. (2494, 2760). 15. (7.46, 7.54). 16. 

(16.99, 19.47). 17. (25, 31). 18. 106. 19. 28. 20. (0.395, 0.445) anu (39.5%, 44.5%). 21. 

(0.343, 0.457). 22. (0.791, 0.909). 23. 1418. 24. 8487. 25. (0.25, 0.51). 26. (1.5, 5.3). 27. (5.1, 

18.3). 28. (1.447, 2.491). 29. (22.79, 24.11). 30. (43.15, 46.45). 31. (3954, 4346). 32. (45.7, 

51.5). 33. (418, 458). 34. (26, 36). 35. 180. 36. 25. 37. (0.604, 0.810). 38. (0.295, 0.425). 39. 

(0.342, 0.547). 40. 545. 41. (7, 13). 42. (30.9, 78.2); (5.6, 8.8). 43. (1.8, 3.2). 

113

Tavi IV 

2 

hipoTezaTa Semowmeba saSualosaTvis ( cnobilia) 

ganawilebis parametrebis mniSvnelobebis Sesaxeb an ori ganawilebis 

parametrebis sidideebis Sedarebis hipoTezebs, parametruli hipoTezebi 

ewodeba. hipoTezebs ganawilebis saxis Sesaxeb ki araparametruli hipoTezebi 

ewodeba. hipoTezas, romelic wamoyenebulia SerCeviT monacemebTan 

misi Tanxmobis Sesamowmeblad, nulovani hipoTeza ewodeba da aRiniSneba 

H0 

-iT ( H 

0 

amtkicebs, rom ar arsebobs gansxvaveba parametrsa da mis 

konkretul mniSvnelobas Soris an ar arsebobs gansxvaveba or parametrs 

Soris). H 

0 

hipoTezasTan erTad ixilaven (wamoayeneben) alternatiul anu 

sawinaaRmdego hipoTezasac, romelsac H1-iT aRniSnaven ( H 

1 

amtkicebs, 

rom arsebobs gansxvaveba parametrsa da mis konkretul mniSvnelobas Soris 

an arsebobs gansxvaveba or parametrs Soris). nulovani hipoTeza moicavs 

tolobis niSans: 

kriteriumi: 

ormxrivi marjvena calmxrivi marcxena calmxrivi 

H0: 

0 

H0: 

0 

H0: 

0 

H : 

H : 

H : 

1 

 

0 

1 

 

0 

1 0 

samarTliani hipoTezis ukugdebas pirveli gvaris Secdoma ewodeba. 

pirveli gvaris Secdomis daSvebis albaTobas mniSvnelovnebis done ewodeba 

da -Ti aRiniSneba. arasamarTliani hipoTezis miRebas meore gvaris 

Secdoma ewodeba. misi albaToba aRiniSneba asoTi. ricxvs 1 

, romelic 

tolia albaTobis imisa, rom ar iqneba daSvebuli meore gvaris Secdoma, 

kriteriumis simZlavre ewodeba. 

statistikuri hipoTezis Semowmeba normaluri ganawilebis 

maTematikuri lodinis Sesaxeb cnobili dispersiis SemTxvevaSi: 

 

2 

N( , ) ; 

D 

2 

cnobilia; E ucnobia. 

hipoTeza: H 0 

: E a 0 

mniSvnelovnebis done: 

X a0 kriteriumis statistika: Z N(0,1) 

/ n 

xa0 

kriteriumis mniSvneloba T.V.: z 

/ n 

alternativa kritikuli are C.R. ( H0 

-is uaryofis are) 

H1 : E a1 a0 

z z , 

114

H1 : E a1 a0 

z z , 

H1: 

E a1 

z z /2 

an z 

z /2 

(sadac z 

aris N(0,1) 

-is zeda -kritikuli wertili anu kritikuli mniSvneloba 

C.V.). 

P{ Z z | H0}, Tu H1 : E 

a1 a0; 

 

P - mniSvneloba: P P{ Z z | H0}, Tu H1 : E 

a1 a0; 

= 

 

P{| Z | | z || H0}, Tu H1 : E 

a1. 

1 ( z), Tu H1 : E 

a1 a0; 

 

= ( z), Tu H1 : E 

a1 a0; 

2 [1 (| z |)], Tu H1: E 

a1. 

gadawyvetileba: Tu z C.R., maSin H 0 

hipoTezas ukuvagdebT, winaaRmdeg 

SemTxvevaSi amis safuZveli ara gvaqvs. 

P - mniSvnelobis meTodi: Tu P , maSin H 

0 

hipoTezas ukuvagdebT 

mniSvnelovnebis doniT, winaaRmdeg SemTxvevaSi amis safuZveli ara 

gvaqvs. 

SerCevis minimaluri raodenoba n , romlisTvisac I gvaris Secdomis 

albaTobaa , xolo II gvaris Secdomis albaToba naklebia -ze: 

2 2 2 

n 

[ ( z z ) /( a a 

) ] 1 

 

 

1 0 

SeniSvna: Tu SerCevis moculoba metia an toli 30-s da standartuli 

gadaxra ucnobia, kriteriumis statistikad ganixileba: 

X 

E 

Z 

s ' 

. 

/ n 


magaliTi 1. samecniero angariSis Tanaxmad sruli profesoris saSualo 

wliuri Semosavali aRemateba 42000 lars. SemTxveviT SerCeuli 30 

sruli profesoris wliuri Semosavlis saSualo aRmoCnda 43260 lari. 

0.05 mniSvnelovnebis donisaTvis SeamowmeT hipoTeza imis Sesaxeb, rom 

sruli profesoris saSualo xelfasi metia 42000 larze, Tu populaciis 

standartuli gadaxraa 5230 lari. 

amoxsna. vinaidan SerCevis moculoba 30 , amitom SegviZlia vigulisxmoT, 

rom saqme gvaqvs daaxloebiT normalur populaciasTan Na ( ,5230 ) . 

2 

CamovayaliboT ZiriTadi da alternatiuli hipoTezebi: H0 : a 42000 da 

H1 : a 42000 . vinaidan, 0.05 da kriteriumi marjvena calmxrivia, kritikuli 

mniSvneloba iqneba z 1.65 . gamovTvaloT kriteriumis mniSvneloba: 

115

xE 

43260 42000 z 1.32 . 

/ n 5230/ 30 

radganac 1.32 < 1.65 (anu kriteriumis mniSvneloba ar vardeba kritikul 

areSi), amitom ara gvaqvs safuZveli nulovani hipoTezis uaryofis. 

magaliTi 2. mkvlevaris azriT sportuli fexsacmlis saSualo fasi 

naklebia 80 larze. man katalogebidan SemTxveviT SearCia 36 wyvili sportuli 

fexsacmeli da amowera Sesabamisi fasebi: 

60 70 75 55 80 55 50 40 80 

70 50 95 120 65 80 85 85 45 

75 60 90 90 60 95 110 85 45 

90 70 70 90 75 85 80 60 110 

aris Tu ara sakmarisi safuZveli, rom 0.1 mniSvnelovnebis doniT gaviziaroT 

mkvlevaris azri? 

amoxsna. CamovayaliboT ZiriTadi da alternatiuli hipoTezebi: 

H0 : E 80 , H1 : E 80 . radganac, 0.1 da kriteriumi marcxena calmxrivia, 

kritikuli mniSvneloba iqneba z 1.28 . pirobaSi moyvanili nedli 

monacemebis mixedviT gamovTvaloT SerCeviTi saSualo da Sesworebuli 

n 

1 

SerCeviTi dispersia, gvaqvs: x 

xi 

75 

da ' n 

1 

s 2 

n ( xi 

x) 19.2 

i 1 

n 1 

. Sesabamisad, 

kriteriumis mniSvneloba iqneba: 

i1 

xE 

75 80 

z 1.56 . 

' 

s / n 19.2/ 36 

vinaidan kriteriumis mniSvneloba vardeba kritikul areSi, nulovani 

hipoTeza unda ukuvagdoT. Sesabamisad, Cven gvaqvs sakmarisi safuZveli 

rom gaviziaroT mkvlevaris mosazreba. 

magaliTi 3. jandacvis saministros angariSis mixedviT insultis 

mkurnalobis saSualo Rirebuleba Seadgens 24672 lars. imis gasarkvevad, 

konkretul saavadmyofoSi insultis mkurnalobis saSualo Rirebuleba 

aris Tu ar gansxvavebuli, mkvlevarma SemTxveviT SearCia am saavadmyofo- 

Si insultis mkurnalobis 35 SemTxveva da mkurnalobis saSualo Rirebuleba 

gamovida 25226 lari. populaciis standartuli gadaxraa 3251 lari. 

SeuZlia Tu ara mkvlevars 0.01 mniSvnelovnebis doniT amtkicos, rom 

am hospitalSi insultis mkurnalobis Rirebuleba gansxvavebulia 24672 

larisagan? 

amoxsna. H 0 

: E 24672 , H1 : E 24672 . kritikuli mniSvnelobebia: 2.58 

da 2.58 . kriteriumis mniSvneloba iqneba: 

xE 

25226 24672 z 1.01 . 

/ n 3251/ 35 

ramdenadac 1.01 < 2.58 (anu kriteriumis mniSvneloba ar vardeba kritikul 

areSi), amitom ar arsebobs sakmarisi safuZveli imisa, rom konkretul 

saavadmyofoSi mkurnalobis Rirebuleba gansxvavebulia. 

116

magaliTi 4. mkvlevars surs Seamowmos hipoTeza, rom maSvelebis sa- 

Sualo asaki 24 welze metia. man SemTxveviT SearCia 36 maSveli da daadgina, 

rom maTi saSualo asaki iyo 24.7 weli, xolo Sesworebuli standartuli 

gadaxra 2 weli. gvaqvs Tu ara sakmarisi safuZveli hipoTezis 

misaRebad 0.05 mniSvnelovnebis doniT? ipoveT P -mniSvneloba. 

amoxsna. H 0 

: E 24 , H1 : E 24 . kriteriumis mniSvneloba iqneba: 

24.7 24 z 2.1 . 

2/ 36 

gamovTvaloT P -mniSvneloba: P1 ( z) 10.9821 0.0179 . vinaidan 

P , amitom H 

0 

unda ukuvagdoT, anu sakmarisi safuZveli arsebobs imisa, 

rom maSvelis saSualo asaki metia 24 welze. 

SevniSnavT, rom Tu 0.01 , maSin ar gveqneboda H0 

-is uaryofis safuZveli. 

magaliTi 5. meteorologis azriT qalaqSi qaris saSualo siCqarea 

8km/sT. SemTxveviT SerCeuli 32 dRis monacemebiT qaris saSualo siCqare 

aRmoCnda 8.2 km/sT, xolo Sesworebuli standartuli gadaxra ki 0.6km/sT. 

0.05 mniSvnelovnebis doniT gvaqvs Tu ara safuZveli ar daveTanxmoT 

meteorologs? gamoiyeneT P -mniSvnelobis meTodi. 

amoxsna. H 0 

: E 8, H1 : E 8. gamovTvaloT kriteriumis mniSvneloba: 

8.2 8 

z 1.89 . 

0.6/ 32 

vipovoT P -mniSvneloba: P 2 [1 (| z|)] 2 (1 0.9706) 0.0294 . vinaidan, 

P , amitom H 

0 

ar unda uarvyoT, anu ara gvaqvs sakmarisi safuZveli 

rom ar gaviziaroT meteorologis azri. 

magaliTi 6. dakvirvebebis ra minimaluri raodenobaa saWiro imisaTvis, 

rom miRweul iqna 0.05-is toli mniSvnelovnebis done da 0.9-is toli 

simZlavre N( ,49) normaluri populaciis saSualos Sesaxeb ZiriTadi 

H0 : E 8 hipoTezis Semowmebisas H 1 

: E 11 alternativis winaaRmdeg. 

2 

amoxsna. am SemTxvevaSi 0.05 ; 10.9 0.1 ; a 0 

8 ; a 1 

11; 

49 . 

normaluri ganawilebis cxrilidan vpoulobT, rom z 1.64 da z 1.29 . 

amitom Sesabamisi formulis ZaliT n 48. 

amocanebi 

1. isargebleT standartuli normaluri ganawilebis cxriliT da 

ipoveT kritikuli mniSvneloba (an mniSvnelobebi) roca: a). 0.01 , krit. 

ormxrivia; b). 0.05 , krit. marj. calmxrivia; g). 0.05 , krit. marc. calmxrivia; 

d). 0.1, krit. marc. calmxrivia; e). 0.05 , krit. ormxrivia; 

v). 0.04 , krit. marj. calmxrivia; z). 0.01 , krit. marc. calmxrivia; 

T). 0.1, krit. ormxrivia; i). 0.02 , krit. marj. calmxrivia; k). 

0.02 , krit. ormxrivia. 

117

2. CamoayalibeT ZiriTadi da alternatiuli hipoTezebi Semdegi 

winadadebebisaTvis: a). avtobusis mZRolebis saSualo asaki 39 welia; b). 

eqimis saSualo wliuri Semosavali Seadgens 25000 lars; g). 

telewamyvanis saSualo asaki 25 welze metia; d). morbenalis saSualo 

guliscema naklebia 85 dartymaze wuTSi; e). studentis saSualo qula 

statistikaSi naklebia 56; v). aqciis saSualo fasi 250 larze metia; z). 

mamakacis saSualo pensia aRemateba 75 lars TveSi; T). 1000 gramian 

brinjis fuTaSi devs sul cota 950 grami brinji. 

3. mkvlevari fiqrobs, rom soflis saSualo biujeti Seadgens 25000 

lars. 0.05 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi safuZveli 

uarvyoT mkvlevaris mosazreba, Tu SemTxveviT SerCeuli 40 soflis biujeti 

aTasobiT larebSi Semdegia: 

16.7 17.6 26.5 6.3 16.5 11.9 23.7 14.3 94 4.7 

11.6 26.5 5.6 58.6 3.2 14.2 3.5 10.9 11.8 15.2 

30.1 19.7 11.7 38.8 36.3 4.8 7.9 14.2 18 24.5 

69.2 8.5 19.2 5 15.3 41 27.1 10.3 3.7 13.6 

4. statistikis departamentis monacemebiT dedaqalaqSi sastumros 

erTi nomris saSualo Rirebuleba Seadgens 69.21 lars. am hipoTezis Sesamowmeblad 

mkvlevarma SemTxveviT SearCia sastumros 30 nomeri da daadgina, 

rom maTi saSualo Rirebulebaa 68.43 lari, xolo Sesworebuli standartuli 

gadaxra ki 3.72 lari. 0.05 mniSvnelovnebis doniT gvaqvs Tu 

ara sakmarisi safuZveli uarvyoT hipoTeza? 

5. qarxnis menejeris azriT muSebis saaSualo saaTobrivi anazRaureba 

9.78 larze naklebia. SemTxveviT SerCeuli 18 muSis saSualo saaTobrivi 

xelfasi aRmoCnda 9.6 lari, xolo Sesworebuli standartuli gadaxra 

ki 1.42 lari. CaTvaleT, rom xelfasi normaluradaa ganawilebuli. 

0.1 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi safuZveli davadasturoT 

menejeris mosazreba? 

6. mkvlevaris SefasebiT didi biznesis saSualo Semosavali 24 milionze 

metia. 0.05 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi safuZveli 

davadasturoT mkvlevaris Sefaseba, Tu SemTxveviT SerCeuli 50 

kompaniis Semosavlebi (gazomili milionebSi) Semdegia: 

178 122 91 44 35 61 56 46 20 32 

30 28 28 20 27 29 16 16 19 15 

41 38 36 15 25 31 30 19 19 19 

24 16 15 15 19 25 25 18 14 15 

24 23 17 17 22 22 21 20 17 20 

7. gayinuli kerZis mwarmoebeli firmis direqtori acxadebs, rom kerZis 

saSualo kaloriuloba aris 800, xolo standartuli gadaxra ki 25. 

mkvklevarma Seamowma 12 kerZi da daadgina, rom maTi saSualo kaloriuloba 

iyo 873. gvaqvs Tu ara sakmarisi safuZveli 0.02 mniSvnelovnebis 

doniT uarvyoT direqtoris mtkicebuloba? CavTvaloT, rom kaloriuloba 

kerZSi ganawilebulia normalurad. 

8. samgzavro TviTmfrinavebis saSualo asaki Seadgens 14 wels. didi 

aviakompaniis SemTxveviT SerCeuli 36 TviTmfrinavis saSualo asaki aRmo- 

118

Cnda 11.8 w. , xolo Sesworebuli standartuli gadaxra ki 2.7 w. 0.01 

mniSvnelovnebis doniT SegviZlia Tu ara davaskvnaT, rom am kompaniis 

TviTmfrinavebis saSualo asaki naklebia vidre populaciis saSualo? 

9. dietologis gancxadebiT misi dietiT pacientebi 20 kviris manZilze 

saSualod ikleben 24 funts. Sesabamisi standartuli gadaxraa 5 

funti. dietologs surs miiRos ukeTesi Sedegi da amcirebs marilis moxmarebas. 

axali meTodis gamoyenebiT 40 SemTxveviT SerCeuli pacienti 20 

kviraSi saSualod iklebs 16.3 funts. 0.05 mniSvnelovnebis doniT 

SeiZleba Tu ara iTqvas, rom dieta Seicvala? 

10. statistikosis mtkicebiT latareis myidveli adamianebis saSualo 

asaki aris 70 weli. 0.05 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi 

safuZveli uarvyoT es mtkicebuleba, Tu SemTxveviT SerCeuli latareis 

bileTebis myidveli 30 adamianis asakis monacemebia: 

49 80 24 61 79 68 63 72 46 65 

76 71 90 56 70 71 71 67 52 82 

74 39 49 69 22 56 70 74 62 45 

11. gamokiTxvis Tanaxmad 55 welze meti asakis qalebi dReSi saSualod 

xarjaven 1660 kalorias. imis Sesamowmeblad, jandacvis sferoSi 

momuSave qalebi xarjaven Tu ara imave raodenobis kalorias, SemTxveviT 

SerCeul iqna jandacvis sferoSi momuSave 55 w. meti asakis 43 qali da 

aRmoCnda, rom maT mier dReSi daxarjuli kaloriebis saSualoa 1446, 

xolo standartuli gadaxra ki 56 kal. 0.05 mniSvnelovnebis doniT 

SeiZleba Tu ara imis mtkiceba, rom jandacvis sferoSi dasaqmebuli 

qalebis mier daxarjuli kaloriebis saSualo gasxvavdeba populaciis 

saSualosagan? 

12. firmis menejeris mtkicebiT ganaTebis muSaobis saSualo xangrZlivobaa 

36 Tve. standartuli gadaxra Seadgens 8 Tves. SemTxveviT SerCeuli 

50 ganaTebis muSaobis saSualo xangrZlivoba aRmoCnda 32 Tve. SeiZleba 

Tu ara am mtkicebis uaryofa 0.01 mniSvnelovnebis doniT? 

13. uZravi qonebis agentis mtkicebiT dedaqalaqSi saxlebis saSualo 

gasayidi fasia 60000 lari. 0.05 mniSvnelovnebis doniT gvaqvs Tu 

ara sakmarisi safuZveli uarvyoT agentis mtkiceba, Tu dedaqalaqSi Sem- 

TxveviT SerCeuli 36 gayiduli saxlis fasebia (aTasobiT larebSi): 

9.5 54 99 94 80 29 121.5 184.75 15 

164.45 6 13 188.4 121 308 42 7.5 32.9 

126.9 25.225 95 92 38 60 211 15 28 

53.5 27 21 76 85 25.225 40 97 284 

14. pacientebis garkveuli jgufis sisxlSi qolesterinis 

saSualo done Seadgens aranakleb 240 miligrams, xolo standartuli gadaxraa 

18 miligrami. SemTxveviT SerCeul 40 pacients misces axali 

preparati, romelic gankuTvnilia sisxlSi qolesterinis donis dasawevad. 

am preparatis miRebis Semdeg aRniSnuli pacientebis sisxlSi 

qolesterinis saSualo done aRmoCnda 229 miligrami. 0.01 mniSvnelovnebis 

doniT gvaqvs Tu ara sakmarisi safuZveli ganvacxadoT, rom 

axali preparati amcirebs qolesterinis dones? 

119

15. miuTiTeT unda ukuvagdoT Tu ara nulovani hiopoTeza mocemuli 

P -mniSvnelobis mixedviT, Tu: a). P -mniSv. = 0.258, 0.05 , krit. calmxrivia; 

b). P -mniSv. = 0.0684, 0.1, krit. ormxrivia; g). P -mniSv. = 0.0153, 

0.01 , krit. calmxrivia; d). P -mniSv. = 0.0232, 0.05 , krit. ormxrivia; 

e). P -mniSv. = 0.002, 0.01 , krit. calmxrivia. 

16. dekanatis mtkicebiT studentebis saxelmZRvaneloebis saSualo 

fasi metia 27.5 larze. SemTxveviT SerCeuli 50 saxelmZRvanelos saSualo 

fasi aRmoCnda 39.3 lari, xolo Sesworebuli standartuli gadaxra ki 5 

lari. ipoveT kriteriumis P -mniSvneloba. 0.05 mniSvnelovnebis doniT 

unda ukuvagdoT Tu ara nulovani hipoTeza? 

17. avtobusebis firmis menejeris mtkicebiT 50 mili/sT siCqariT mo- 

Zraobisas saskolo avtobusis saSualo samuxruWe manZili Seadgens 264 

futs (1 futi = 30.48sm). mkvlevaris mier SemTxveviT SerCeuli 20 saskolo 

avtobusis saSualo samuxruWe manZili aRmoCnda 262.3 futi, xolo Sesworebuli 

standartuli gadaxra ki 3 futi. mkvlevaris azriT saSualo samuxruWe 

manZili naklebia 264 futze. CaTvaleT, rom samuxruWe manZili 

ganawilebulia normalurad. ipoveT P -mniSvneloba. 0.01 mniSvnelovnebis 

doniT unda ukuvagdoT Tu ara nulovani hipoTeza? 

18. gasamravlebeli mowyobilobis (“qseroqsis”) maRaziis menejeris 

azriT erTi adamianis mier gamravlebuli gverdebis saSualo raodenoba 

dReSi naklebia 40 gverdze. qvemoT moyvanilia SemTxveviT SerCeuli 50 

momxmareblis mier dRis ganmavlobaSi gamravlebuli gverdebis raodenoba. 

0.01 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi safuZveli davadasturoT 

menejeris mosazreba? isargebleT hipoTezaTa Semowmebis P - 

mniSvnelobis meTodiT. 

2 2 2 5 32 5 29 8 2 49 

21 1 24 72 70 21 85 61 8 42 

3 15 27 113 36 37 5 3 58 82 

9 2 1 6 9 80 9 51 2 122 

21 49 36 43 61 3 17 17 4 1 

19. televizorebis firmis menejeris mtkicebiT maTi televizorebis 

muSaobis saSualo xangrZlivoba metia 84 Tveze. populaciis standartuli 

gadaxra aris 10 Tve. SemTxveviT SearCies 100 televizori da Seamowmes. 

SerCevis saSualo aRmoCnda 85.1 Tve. SeamowmeT hipoTeza, rom televizorebis 

muSaobis saSualo xangrZlivoba metia 84 Tveze da gamoTvaleT 

P -mniSvneloba. miRebuli P -mniSvnelobis mixedviT 0.01 mniSvnelovnebis 

doniT unda ukuvagdoT Tu ara nulovani hipoTeza? 

20. specialuri gayvanilobis mavTulis gamZleobis simtkice aris 

800 funti. populaciis standartuli gadaxraa 12 funti. mkvlevarma Seamowma 

SemTxveviT SerCeuli 20 mavTuli da daadgina, rom maTi saSualo simtkicea 

793 funti. SeiZleba Tu ara imis uaryofa, rom gamZleobis simtkicea 

800 funti? ipoveT P -mniSvneloba. miRebuli P -mniSvnelobis mixedviT 

0.01 mniSvnelovnebis doniT unda ukuvagdoT Tu ara nulovani hipoTeza? 

CaTvaleT, rom sidide normaluradaa ganawilebuli. 

120

21. gasul wels dedaqalaqis samedicino muSakebis saSualo saaTobrivi 

anazRaureba iyo 6.32 lari. populaciis standartuli gadaxraa 0.54 

lari. mimdinare wels SemTxveviT SerCeuli 50 samedicino muSakis saSualo 

saaTobrivi anazraureba Seadgens 6.51 lars. P -mniSvnelobis mixedviT, 

0.05 mniSvnelovnebis doniT, SeamowmeT hipoTeza imis Sesaxeb, rom saSualo 

ar Seicvala. 

22. avtomobilebis dileris rekomendaciiT 30000 km-is gavlis Semdeg 

gadacemaTa kolofi saWiroebs SekeTebas. imis gasarkvevad, momxmareblebi 

iTvaliswineben Tu ara dileris rekomendacias, man SemTxveviT SearCia 

40 momxmarebeli da daadgina, rom gadacemaTa kolofis SekeTebis momentisaTvis 

avtomobilebis saSualo ganarbeni iyo 30456 km. Sesworebuli 

standartuli gadaxra ki 1684 km. P -mniSvnelobis gamoTvliT, 0.1 mniSvnelovnebis 

doniT, daadgineT akeTeben Tu ara gadacemaTa kolofis Seke- 

Tebas 30000 km-is gavlis Semdeg avtomflobelebi. migaCniaT Tu ara Tqven, 

rom -s mniSvneloba 0.1 daaxloebiT mniSvnelovnebis donis tolia? 

23. avtomobilistis mtkicebiT is dReSi saSualod gadis 60 km-s. 

qvemoT moyvanilia avtomobilistis mier Tvis ganmavlobaSi yoveldRiurad 

gavlili manZilebi. davuSvaT, rom populaciis standartuli 

gadaxraa 13.42 km. gvaqvs Tu ara sakmarisi safuZveli uarvyoT avtomobilistis 

mtkicebuleba 0.05 mniSvnelovnebis doniT. isargebleT P - 


72 45 36 68 69 71 57 60 83 26 

60 72 58 87 48 59 60 56 64 68 

42 57 57 58 63 49 73 75 42 63 

121

pasuxebi: 

1. 2.58 ; 1.65; -2.58; -1.28; 1.96 ; 1.75; -2.33; 1.65 ; 2.05; 2.33 . 2. H0 : E 39 , 

H1 : E 39 ; H0 : E 25000 , H1 : E 25000 ; H0 : E 25 , H 1 

: E 25 ; H0 : E 85, 

H1 : E 85; H0 : E 56 , H1 : E 56 ; H0 : E 250 , H1 : E 250 ; H0 : E 75 , 

H1 : E 75 ; H0 : E 950 , H1 : E 950 . 3. H0 : E 25000 , H1 : E 25000 ; 

CV . . 1.96 ; T. V. z 1.59 ; H 

0 

. 4. H0 : E 69.21 , H1 : E 69.21 ; CV . . 1.96 ; 

T. V. z 1.15; H 

0 

. 5. H0 : E 9.78 , H1 : E 9.78 ; CV . . 1.28 ; T. V. z 0.54 ; 

H 

0 

. 6. H0 : E 24 , H1 : E 24 ; CV . . 1.65 ; T. V. z 

1.85 ; H 

1. 7. H0 : E 800 , 

H1 : E 800 ; CV . . 2.33 ; T. V. z 

10.12 ; H 

1. 8. H0 : E 14 , H1 : E 14 ; 

CV . . 2.33 ; T. V. z 4.89 ; H 

1. 9. H0 : E 24 , H1 : E 24 ; CV . . 1.96 ; 

T. V. z 9.73; H 

1. 10. H0 : E 70 , H1 : E 70 ; CV . . 1.96 ; T. V. z 2.59 ; H 

1. 

1. H0 : E 1660 , H1 : E 1660 ; CV . . 1.65 ; T. V. z 25.06 ; H 

1. 12. H0 : E 36 , 

H1 : E 36 ; CV . . 2.58 ; T. V. z 3.54 ; H 

1. 13. H0 : E 60000 , H1 : E 60000 ; 

CV . . 1.96 ; T. V. z 

1.78 ; H 

0 

. 14. H0 : E 240 , H1 : E 240 ; CV . . 2.33 ; 

T. V. z 3.87 ; H 

1. 15. ara; ara; ara; ki; ki. 16. H0 : E 27.5 , H1 : E 27.5 ; 

T. V. z 

2.55 ; P -mniSvneloba = 0.0054; ki. 17. H0 : E 264 , H1 : E 264 ; 

T. V. z 2.53 ; P -mniSvneloba = 0.0057; ki. 18. H0 : E 40 , H1 : E 40 ; 

T. V. z 2.45; P -mniSvneloba = 0.0071; ki. 19. H0 : E 84 , H1 : E 84 ; 

T. V. z 

1.1; P -mniSvneloba = 0.1357; ara. 20. H0 : E 800 , H1 : E 800 ; 

T. V. z 2.61; P -mniSvneloba = 0.009; ki. 21. H0 : E 6.32 , H1 : E 6.32 ; 

T. V. z 2.49 ; P -mniSvneloba = 0.0128; H 

1. 22. H0 : E 30000 , H1 : E 30000 ; 

T. V. z 

1.71; P -mniSvneloba = 0.0872; H 

1; ki. 23. H0 : E 60 , H1 : E 60 ; 

T. V. z 0.03; P -mniSvneloba = 0.976; ara. 

122

Tavi V 

2 

hipoTezaTa Semowmeba saSualosaTvis ( ucnobia) 


maTematikuri lodinis Sesaxeb ucnobi dispersiis SemTxvevaSi: 

N( , ); D ucnobia; E ucnobia. 

hipoTeza: H 0 

: E a 0 


X a0 

kriteriumis statistika: T T( n 1) 

' 

S / n 

xa0 

kriteriumis mniSvneloba T.V.: t 

' 

s / n 



H1 : E a1 a0 

t tn 

1, 

 

, 

H1 : E a1 a0 

t tn 

1, 

 

, 

H1: 

E a1 

t tn 

1, /2 

an t 

tn 

1, /2 

(sadac tn 1, 


zeda -kritikuli wertili anu kritikuli mniSvneloba C.V.). 

P{ T t | H0}, Tu H1 : E 

a1 a0; 

 

P - mniSvneloba: P P{ T t | H0}, Tu H1 : E 

a1 a0; 

= 

 

P{| T | | t || H0}, Tu H1 : E 

a1. 

1 FT 

( t), Tu H1 : E 

a1 a0; 

 

= FT 

( t), Tu H1 : E 

a1 a0; 

 

2 [1 FT 

(| t |)], Tu H1: E 

a1. 

gadawyvetileba: Tu t C.R., maSin H 0 




0 

hipoTezas ukuvagdebT, 

winaaRmdeg SemTxvevaSi amis safuZveli ara gvaqvs. 

rodis gamoiyeneba Z an T ganawileba 

cnobiliaTuara ? 

diax 

X E 

gamoiyene Z 

/ n 

ara 

aris Tu ara n 30? diax 

X E 

gamoiyene T 

' 

ara S / n 

123

X 

E 

gamoiyene T , T.x. n 1 

* 

' 

S / n 

* populacia unda iyos daaxloebiT normalurad ganawilebuli. 


magaliTi 1. jandacvis ministris mtkicebiT eqimis saSualo wliuri 

Semosavali Seadgens 24000 lars. SemTxveviT SerCeuli 10 eqimis wliuri 

Semosavlis saSualo aRmoCnda 23450 lari, xolo Sesworebuli 

standartuli gadaxra ki 400 lari. gvaqvs Tu ara sakmarisi safuZveli 

ministris mtkicebis uarsayofad 0.05 mniSvnelovnebis doniT? 


' 

H0 : E 24000 , H1 : E 24000 . pirobis Tanaxmad, gvaqvs: x 23450 , s 400 , 

n 10 . stiudentis ganawilebis cxrilidan, 0.05 -isa da Tavisuflebis 

n1 9 xarisxisaTvis, vpoulobT saWiro kritikul mniSvnelobebs: 

tn 

1, /2 

2.262 da tn 1, /2 

2.262 . kriteriumis mniSvneloba ki iqneba: 

xE 

23450 24000 

t 4.35 . 

' 

s / n 400/ 10 

vinaidan, 4.35 2.262 , amitom nulovani hipoTeza unda ukuvagdoT, 

anu Cven gvaqvs sakmarisi safuZveli ministris mtkicebis uarsayofad. 

magaliTi 2. ganaTlebis ministris mtkicebiT saTadarigo maswavleblis 

saSualo xelfasi dReSi naklebia 60 larze. gvaqvs Tu ara sakmarisi 

safuZveli ministris mtkicebis misaRebad 0.1 mniSvnelovnebis doniT, 

Tu SemTxveviT SerCeul 8 skolaSi dafiqsirda Semdegi monacemebi: 60 56 

60 55 70 55 60 55. 


H0 : E 60 , H1 : E 60 . SerCevis moculobaa n 8. stiudentis ganawilebis 

cxrilidan, 0.1-isa da Tavisuflebis n1 7 xarisxisaTvis, vpoulobT 

saWiro kritikul mniSvnelobas: tn 

1, 1.415 . kriteriumis mniSvnelobis 

gamosaTvlelad, winaswar gamovTvaloT SerCeviTi saSualo da 

n 

1 

Sesworebuli standartuli gadaxra: x 

xi 

58.88 

da 

1 

s ( x x) 5.08 . amitom: 

' 

n 

2 

i 

n1 

i1 

124 

n i 1 

xE 

58.88 60 

t 0.624 . 

' 

s / n 5.08/ 8 

vinaidan, kriteriumis mniSvneloba ar vardeba kritikul areSi, 

nulovani hipoTeza ar unda ukuvagdoT, anu Cven ar gvaqvs sakmarisi 

safuZveli ministris mtkicebis dasadastureblad. 

magaliTi 3. vipovoT P -mniSvneloba, Tu kriteriumis mniSvneloba 

stiudentis ganawilebisaTvis (T kriteriumis statistikis t kriteriumis

mniSvneloba) aris 2.056, SerCevis moculobaa 11 da kriteriumi marjvena 

calmxrivia. 

amoxsna. stiudentis ganawilebis zeda kritikuli wertilebis 

cxrilSi Tavisuflebis 111 10 

-is toli xarisxis gaswvriv movZebnoT 

iseTi ori mniSvneloba, romelTa Soris vardeba (moeqceva) 2.056. es 

mniSvnelobebia: 1.812 da 2.228. vinaidan, kriteriumi marjvena calmxrivia, 

SevxedoT striqons warweriT ―One tail, ‖ da vipovoT -s ori 

mniSvneloba, romelic Seesabameba 1.812-sa da 2.228-s. es mniSvnelobebia: 0.05 

da 0.025. Sesabamisad, P -mniSvneloba swored maT Sorisaa moTavsebuli: 

0.025 

ukuvagdoT H 

0 

, radganac P -mniSvneloba < 0.05; magram, Tu 0.01 , maSin 

H0 

-is uaryofis safuZveli ara gvaqvs, radganac P -mniSvneloba > 0.025 > 

0.01. 

SeniSvna: P -mniSvnelobis povna SeiZleba specialuri kalkulatoriT 

an kompiuteruli programiT. am magaliTSi kalkulatoriT gamoTvlili 

P -mniSvneloba = 0.033. 

magaliTi 4. vipovoT P -mniSvneloba, Tu kriteriumis mniSvneloba 

stiudentis ganawilebisaTvis (T kriteriumis statistikis t kriteriumis 

mniSvneloba) aris 2.983, SerCevis moculobaa 6 da kriteriumi ormxrivia. 

amoxsna. stiudentis ganawilebis zeda kritikuli wertilebis 

cxrilSi Tavisuflebis 615-is toli xarisxis gaswvriv movZebnoT 

iseTi ori mniSvneloba, romelTa Soris vardeba (moeqceva) 2.983. es mni- 

Svnelobebia: 2.571 da 3.365. vinaidan, kriteriumi ormxrivia, SevxedoT 

striqons warweriT ―Two tail, ‖ da vipovoT -s ori mniSvneloba, 

romelic Seesabameba 2.571-sa da 3.365-s. es mniSvnelobebia: 0.05 da 0.02. 

Sesabamisad, P -mniSvneloba swored maT Sorisaa moTavsebuli: 0.02 

mniSvneloba < 0.05. magaliTad, Tu 0.05 , maSin Cven unda ukuvagdoT H 

0 

, 

radganac P -mniSvneloba < 0.05; magram, Tu 0.01 , maSin H 

0 

-is uaryofis 

safuZveli ara gvaqvs, radganac P -mniSvneloba > 0.02 > > 0.01. 

SevniSnavT, rom aq kalkulatoriT gamoTvlili P -mniSvneloba aris 

0.031. 

magaliTi 5. eqimebis azriT morbenali adamiani ufro met Jangbads 

moixmars vidre saSualod yvela adamiani. SemTxveviT SerCeuli 15 morbenalisaTvis 

Jangbadis moxmarebis saSualo iyo 40.6 ml/kg, xolo 

Sesworebuli standartuli gadaxra ki 6ml/kg. Tu yvela adamianis 

saSualo moxmareba Seadgens 36.7ml/kg, gvaqvs Tu ara sakmao safuZveli 

davujeroT eqimebs 0.05 mniSvnelovnebis doniT? 


H0 : E 36.7 , H1 : E 36.7 . gamovTvaloT kriteriumis mniSvneloba: 

xE 

40.6 36.7 t 

' 

2.517 . 

s / n 6/ 15 

gamovTvaloT P -mniSvneloba: stiudentis ganawilebis cxrilSi 

n114 

-is gaswvriv 2.517 vardeba 2.145-sa da 2.624-s Soris, romelebic 

125

Seesabameba 0.025 -sa da 0.01 -s (kriteriumi marjvena calmxrivia). 

amitom 0.01 

0.025 < 0.05) da nulovani hipoTeza unda ukuvagdoT, anu Cven gvaqvs 

sakmarisi safuZveli gaviziaroT eqimebis mtkicebuleba. 

amocanebi 

1. isargebleT stiudentis ganawilebis cxriliT da ipoveT kritikuli 

mniSvneloba (an mniSvnelobebi) roca: a). n 10 , 0.05 , kriteriumi marjvena 

calmxrivia; b). n 18, 0.1, kriteriumi ormxrivia; g). n 6 , 

0.01 , kriteriumi marcxena calmxrivia; d). n 9 , 0.025 , kriteriumi 

marjvena calmxrivia; e). n 15, 0.05 , kriteriumi ormxrivia; v). n 23 , 

0.005 , kriteriumi marcxena calmxrivia; z). n 28 , 0.01 , kriteriumi 

ormxrivia; T). n 17 , 0.02 , kriteriumi ormxrivia. 

2. ipoveT P -mniSvnelobis sazRvrebi qvemoT moyvanili kriteriumis 

mniSvnelobebisaTvis: a). t 2.321 , n 15, kriteriumi marjvena calmxrivia; 

b). t 1.945 , n 28 , kriteriumi ormxrivia; g). t 1.267 , n 8, kriteriumi marcxena 

calmxrivia; d). t 1.562 , n 17 , kriteriumi ormxrivia; e). t 3.025 , 

n 24 , kriteriumi marjvena calmxrivia; v). t 1.145 , n 5, kriteriumi marcxena 

calmxrivia; z). t 2.179 , n 13, 0.01 , kriteriumi ormxrivia; T). 

t 0.665 , n 10 , 0.02 , kriteriumi marjvena calmxrivia. 

qvemoT moyvanil amocanebSi igulisxmeT, rom populacia daaxloebiT 

normaluradaa ganawilebuli. 

3. ekvatoris samxreTiT zafxulis TveebSi mosuli naleqebis saSualo 

Seadgens 11.52 diums (1 diumi = 2.54 sm). 2000 wels mkvlevarma SemTxveviT 

SeraCia ekvatoris samxreTiT mdebare 10 qalaqi da daadgina, rom mosuli 

naleqebis saSualo aris 7.42 diumi. SerCevis Sesworebuli standartuli 

gadaxra Seadgens 1.3 diums. 0.05 mniSvnelovnebis doniT SeuZlia 

Tu ara mkvlevars daaskvnas, rom 2000 wels mosuli naleqebis saSualo 

naklebia 11.52 diumze? 

4. mwvaneebis azriT msoflioSi daculi teritoriebis (nakrZalebis) 

farTobebis saSualo naklebia vidre 2000 akri (1 akri = 0.4 ha). SemTxveviT 

SerCeuli 5 daculi teritoriis farTobebi gazomili akrebSi Semdegia: 

959 1187 493 6249 541. 0.01 mniSvnelovnebis doniT gvaqvs Tu 

ara sakmarisi safuZveli daveTanxmoT mwvaneebis mosazrebas? 

5. uZravi qonebis agentis mtkicebiT dedaqalaqSi mcire biznesis 

ofisis xarjebis saSualoa 800 lari. SemTxveviT SerCeuli 10 mcire 

biznesisaTvis ofisis xarjebis saSualom Seadgina 863 lari, xolo 

Sesworebuli standartuli gadaxra iyo 20 lari. 0.05 mniSvnelovnebis 

doniT gvaqvs Tu ara sakmarisi safuZveli ukuvagdoT uZravi qonebis 

agentis mtkicebuleba? 

6. wlis ganmavlobaSi avadmyofobis gamo qarxnis erTi TanamSromlis 

mier gacdenili samuSao saaTebis saSualoa 48 saaTi. gacdenaTa raodenobis 

Sesamcireblad qarxnis direqtorma TanamSromlebs daavala monaw- 

126

ileoba mieRoT gajansaRebis programaSi. erTi wlis Semdeg, SemTxveviT 

SerCeuli 18 TanamSromlis gacdenili saaTebis saSualom Seadgina 41 sT, 

xolo Sesworebuli standartuli gadaxra iyo 5 sT. 0.01 mniSvnelovnebis 

doniT gaamarTla Tu ara gajansaRebis programam? 

7. mSeneblobis saministros mtkicebulebiT dedaqalaqSi Senobebis 

saSualo simaRlea sul cota 700 futia. 0.025 mniSvnelovnebis doniT 

gvaqvs Tu ara sakmarisi safuZveli ukuvagdoT mSeneblobis saministros 

mtkicebuleba, Tu SemTxveviT SerCeuli 10 Senobis simaRleebia: 

485 511 841 725 615 

582 616 635 535 520 

8. uZravi qonebis agentis mtkicebiT saofise farTis 1 kv. metris sa- 

Sualo saijaro gadasaxadi dedaqalaqSi Seadgens 17.63 lars. mkvlevarma 

SemTxveviT SearCia 15 ofisi da daadgina, rom maTi saijaro gadasaxadis 

saSualoa 18.72 lari, xolo Sesworebuli standartuli gadaxra ki 3.64 

lari. 0.05 mniSvnelovnebis doniT SeamowmeT hipoTeza imis Sesaxeb, 

rom ar arsebobs gansxvaveba saijaro gadasaxadebs Soris. 

9. uZravi qonebis agentis mtkicebiT samTaxiani binis saSualo saijaro 

gadasaxadi dedaqalaqSi Seadgens 750 lars. mkvlevarma SemTxveviT 

SearCia 12 samoTaxiani bina da daadgina, rom maTi saijaro gadasaxadis 

saSualoa 732 lari, xolo Sesworebuli standartuli gadaxra ki 17 lari. 

0.01 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi safuZveli ukuvagdoT 

agentis mtkicebuleba? 

10. mwarmoeblis mtkicebiT axal TovlisgadamamuSavebel manqanas 

SeuZlia danaxarjebis Semcireba. misi mtkicebiT 10 tona Tovlis axali 

manqaniT gadamuSavebisas 1 tonis gadamuSavebis Rirebulebam Seadgina 5.75 

lari, xolo Sesworebuli standartuli gadaxra ki aris 1.05 lari. Zveli 

manqanis SemTxvevaSi 1 tona Tovlis gadamuSavebis saSualo Rirebuleba 

iyo 6.62 lari. 0.1 mniSvnelovnebis doniT akeTebs Tu ara axali manqana 

fulis ekonomias? 

11. Tbilgazis mtkicebiT mcire kompaniebis gazis saSualo danaxarji 

TveSi ar aRemateba 350 lars. kompaniebis mflobelebi ki eWvoben, rom 

gazis danaxarji ufro metia. SemTxveviT SeirCa 12 mcire kompania da aRmoCnda, 

rom maTi saSualo danaxarji iyo 358 lari, xolo Sesworebuli 

standartuli gadaxra ki 16 lari. SeamowmeT hipoTeza imis Sesaxeb, rom 

0.05 mniSvnelovnebis doniT gazis saSualo danaxarji metia 350 larze. 

12. 0.05 mniSvnelovnebis doniT SeamowmeT hipoTeza imis Sesaxeb, 

rom raionebSi manqanis garecxvis saSualo fasia 3 lari, Tu cnobilia, 

rom SemTxveviT SerCeul 5 raionSi manqanis garecxvis saSualo fasi iyo 

3.7 lari, xolo Sesworebuli standartuli gadaxra ki 0.3 lari. 

13. gamokiTxvis Tanaxmad, martoxela adamianebis saxlSi telefoni 

TveSi saSualod 37-jer rekavs. am hipoTezis Sesamowmeblad mkvlevarma 

gamokiTxa 29 martoxela adamiani da daadgina, rom telefonis zarebis 

saSualo iyo 34.9, xolo Sesworebuli standartuli gadaxra ki 6. 0.05 

127

mniSvnelovnebis doniT gvaqvs Tu ara safuZveli ukuvagdoT aRniSnuli 

hipoTeza? gamoiyeneT P -mniSvnelobis meTodiT. 

14. studentebis azriT SabaTi saRamos barSi gatareba saSualod ar 

unda dajdes 30 larze meti. TavianTi mosazrebis Sesamowmeblad maT Sem- 

TxveviT SearCies 16 studenti da SeekiTxnen Tu ramdeni daxarjes gasul 

SabaT saRamos barSi. aRmoCnda, rom SerCeviTi saSualo aris 31.17 lari, 

xolo Sesworebuli standartuli gadaxra ki 5.51 lari. 0.05 mniSvnelovnebis 

doniT gvaqvs Tu ara safuZveli daveTanxmoT studentebis mosazrebas? 

isargebleT P -mniSvnelobis meTodiT. 

15. gasuli wlebis gamocdilebidan gamomdinare maswavlebels sjera, 

rom gamocdaze studentebis saSualo qula aris 75. gamoiyeneT P -mni- 

Svnelobis meTodi da 0.01 mniSvnelovnebis doniT SeamowmeT, rom 

studentebis saSualo qula isev aris 75, Tu cnobilia, rom mimdinare 

wels 20 SemTxveviT SerCeuli studentis qulebia: 

80 68 72 73 76 81 71 71 50 65 

63 71 70 76 75 69 70 72 70 74 

pasuxebi: 

1. 1.833 ; 3.365 ; 2.145 ; 2.771 ; 1.740 ; 2.306; -2.819; 2.583 . 2. (0.01,0.025); (0.05, 

0.1); (0.1, 0.25); (0.1, 0.2); P -mniSvneloba < 0.005); (0.1, 0.25); P -mniSvneloba = 

0.05; P -mniSvneloba > 0.25. 3. H0 : E 11.52 , H1 : E 11.52 ; CV . . 1.83 ; 

T. V. t 

9.97 ; ki. 4. H0 : E 2000 , H1 : E 2000 ; CV . . 3.747 ; T. V. t 0.104 ; 

ara. 5. H0 : E 800 , H1 : E 800 ; CV . . 2.262 ; T. V. t 

9.96 ; ki. 6. H0 : E 48 , 

H1 : E 48; CV . . 1.333 ; T. V. t 

5.94 ; ki. 7. H0 : E 700 , H1 : E 700 ; 

CV . . 2.262 ; T. V. t 2.71 ; ki. 8. H0 : E 17.63 , H1 : E 17.63 ; CV . . 2.145 ; 

T. V. t 

1.16 ; H 

0 

. 9. H0 : E 750 , H1 : E 750 ; CV . . 3.106 ; T. V. t 

3.67 ; ki. 

10. H0 : E 6.62 , H1 : E 6.62 ; n 10 ; CV . . 1.383 ; T. V. t 

2.62 ; ki. 11. 

H0 : E 350 , H1 : E 350 ; CV . . 1.796 ; T. V. t 

1.732 ; H 

0 

. 12. H0 : E 3, 

H1 : E 3; CV . . 2.776 ; T. V. t 

5.22 ; H 

1. 13. H0 : E 37 , H1 : E 37 ; 

T. V. t 1.88; 0.025 

0 

unda 

ukuvagdoT. 14. H0 : E 30 , H1 : E 30 ; T. V. t 

0.85 ; 0.2 

< 0.5; radganac P , amitom H 

0 

ar unda ukuvagdoT. 15. H0 : E 75 , 

H1 : E 75 ; T. V. t 2.83 ; 0.01 

H 0 

ar unda ukuvagdoT. 

128

Tavi VI 

hipoTezaTa Semowmeba dispersiisaTvis 


dispersiis Sesaxeb: 

2 

hipoTeza: H0: 

D 0 


n 

 

2 

( Xi 

E 

) 

i1 

2 

 

( n), Tu Ecnobilia 

; 

2 

kriteriumis statistika: 0 

2 

 

' 

( n 1) S 2 

 

 

 

( n1), Tu Eucnobia 

. 

2 

0 

n 

 

2 

( xi 

E 

) 

i1 

 

, Tu E 

cnobilia ; 

2 

kriteriumis mniSvneloba T.V.= 

0 

' 

( n 1) s 

2 

 

 

2 

, Tu E 

ucnobia . 

0 



2 

 

2 

[ n, 

, ), Tu Ecnobilia 

; 

H1: 

D 0 

 

2 

[ n 

1, , ), 

Tu Eucnobia, 

2 

2 

(0, n,1 

 

], Tu Ecnobilia 

; 

H1: 

D 

0 

 

2 

(0, n 

1,1 

], Tu Eucnobia, 

2 2 

 

2 

(0, n,1 /2 ] [ n, /2, ), Tu E 

cnobilia ; 

H1: 

D 0 

 

2 2 

(0, n 1,1 /2] [ n 

1, /2, ), 

Tu E 

ucnobia 

(sadac 2 n, 

aris 2 () n ganawilebis zeda -kritikuli wertili anu 

kritikuli mniSvneloba C.V.). 

alternativa 

P - mniSvneloba: 

2 

 

2 

P{ ( n) T. V.}, Tu Ecnobilia 

; 

H1: 

D 0 

P 

2 

P{ ( n 1) T. V.}, 

Tu Eucnobia, 

2 

 

2 

P{ ( n) T. V.}, Tu Ecnobilia 

; 

H1: 

D 

0 

P 

2 

P{ ( n 1) T. V.}, 

Tu Eucnobia, 

2 

2 

2 

2 

min( P{ 

( n) 

T. 

V.},min( 

P{ 

( n) 

T. 

V.}), 

TuE 

cnobilia; 

H1: 

D P 

0 

2 

2 

2 

min( P{ 

( n 1) 

T. 

V.},min( 

P{ 

( n 1) 

T. 

V.}), 

TuE 

ucnobia. 

129

gadawyvetileba: Tu T.V.C.R., maSin H 0 




0 




magaliTi 1. profesors surs gaarkvios, qulebis cvalebadoba Tavis 

jgufSi, sadac 23 studentia, aris Tu ara naklebi, vidre populaciis 

dispersia 225 . profesoris jgufSi qulebis Sesworebuli dispersiaa 198. 

0.05 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi safuZveli 

vamtkicoT, rom profesoris jgufSi qulebis cvalebadoba naklebia 

populaciis dispersiaze? CaTvaleT, rom qulebi normaluradaa 

ganawilebuli. 


H0 : D 225 , H1 : D 225 . vipovoT kritikuli mniSvneloba: vinaidan 

kriteriumi marcxena calmxrivia, 1 

0.95 da Tavisuflebis xarisxia 

n1 

22, amitom xi kvadrat ganawilebis zeda kritikuli wertilebis 

2 2 

cxrilidan vpoulobT, rom n 1,1 

22,0.95 12.338 . maSasadame, kritikuli 

area CR . . (0,12.338] . gamovTvaloT kriteriumis mniSvneloba: 

( n1) 

s 

 

2 

0 

130 

' 2 

(231) 198 

TV . . 19.36 

. 

225 

vinaidan, kriteriumis mniSvneloba ar vardeba kritikul areSi, 

amitom nulovan hipoTezas vRebulobT, anu ar arsebobs sakmarisi 

safuZveli imisaTvis, rom profesoris jgufSi qulebis cvalebadoba 

naklebia populaciis dispersiaze. 

magaliTi 2. saavadamyofos administratori darwmunebulia, rom 

standartuli gadaxra im pacientebis ricxvis, romlebic ambulatoriulad 

mkurnaloben dReSi 8-ze metia. pacientebis ricxvi CavTvaloT, 

rom ganawilebulia normalurad. 0.1 mniSvnelovnebis doniT SeiZleba 

Tu ara gaviziaroT administratoris mosazreba, Tu 15 SemTxveviT 

SerCeuli dRis monacemebia: 

25 30 5 15 18 

42 16 9 10 12 

12 38 8 14 27 


H0 : D 64 , H1 : D 64 . ramdenadac kriteriumis marjvena calmxrivia, 

0.1 da Tavisuflebis xarisxia n1 

14 , amitom kritikuli mniSvneloba 

2 

iqneba: n 

1, 21.0641 . gamovTvaloT Sesworebuli standartuli gadaxra, 

gvaqvs: 

1 1 1 

s ( x x) ( x x ) 11.2 

n n n 

' 2 2 

i i j 

n1 i1 n1 

i1 n j1 

.

amitom kriteriumis mniSvneloba iqneba: 

( n1) 

s 

 

2 

0 

' 2 

2 

(15 1) (11.2) 

TV . . 27.44 . 

2 

8 

vinaidan, kritikuli mniSvneloba vardeba kritikul areSi 

( 27.44 [21.0641, )), amitom H 

0 

unda ukuvagdoT, anu Cven gvaqvs sakmarisi 

safuZveli, raTa CavTvaloT, rom populaciis standartuli gadaxra metia 

8-ze. 

magaliTi 3. sigaretis kompanias surs Seamowmos hipoTeza, rom mis 

sigaretSi nikotinis Semcvlelobis dipersia aris 0.644. nikotinis Semcvleloba 

izomeba miligramebSi da igulisxmeba, rom is normalurad ganawilebulia. 

20 sigaretisgan aRebuli SerCevis Sesworebuli standartuli 

gadaxraa 1 miligrami. 0.05 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi 

safuZveli uarvyoT kompaniis hipoTeza? 


H0 : D 0.644 , H1 : D 0.644 . am SemTxvevaSi saqme gvaqvs ormxriv kriteriumTan. 

0.05 -isa da Tavisuflebis n1 19 xarisxisaTvis xi 

kvadrat ganawilebis zeda kritikuli weertilebis cxrilidan vpoulobT, 

2 2 

rom kritikuli mniSvnelobebia: n 1, /2 

19,0.025 32.852 da 

2 2 

n 1,1 

/2 

19.0.975 8.907 . kriteriumis mniSvneloba ki iqneba: 

( n1) 

s 

 

2 

0 

' 2 

2 

(20 1) (1) 

TV . . 29.5 . 

0.644 

kriteriumis mniSvneloba moTavsebulia or kritikul mniSvnelobas 

Soris (8. 907 < 29.5 < 32.852), anu is ar vardeba kritikul areSi. amitom 

H 

0 

ar unda ukuvagdoT. Sesabamisad, Cven ara gvaqvs sakmarisi safuZveli 

uarvyoT kompaniis mosazreba. 

magaliTi 4. vipovoT P -mniSvneloba, Tu kriteriumis mniSvneloba xi 

kvadrat ganawilebisaTvis aris 19.274, SerCevis moculobaa 8 da kriteriumi 

marjvena calmxrivia. 

amoxsna. xi kvadrat ganawilebis zeda kritikuli wertilebis cxrilSi 

Tavisuflebis 817-is toli xarisxis gaswvriv movZebnoT iseTi 

ori mniSvneloba, romelTa Soris vardeba (moeqceva) 19.274. es mniSvnelobebia: 

18.475 da 20.278. SevxedoT pirvel striqons da vipovoT -s ori mni- 

Svneloba, romelic Seesabameba 18.475-sa da 20.278-s. es mniSvnelobebia: 0.01 

da 0.005. Sesabamisad, P -mniSvneloba swored maT Sorisaa moTavsebuli: 

0.005 

H 

0 

, radganac P -mniSvneloba naklebia 0.01; magram, Tu 0.001 , 

maSin H0 

-is uaryofis safuZveli ara gvaqvs, radganac P -mniSvneloba > 

0.005 > 0.001. 


0.007. 

131

magaliTi 5. vipovoT P -mniSvneloba, Tu kriteriumis mniSvneloba xi 

kvadrat ganawilebisaTvis aris 3.823, SerCevis moculobaa 13 da kriteriumi 

marcxena calmxrivia. 

amoxsna. xi kvadrat ganawilebis zeda kritikuli wertilebis 

cxrilSi Tavisuflebis 131 12 

-is toli xarisxis gaswvriv movZebnoT 

iseTi ori mniSvneloba, romelTa Soris vardeba (moeqceva) 3.823. es mni- 

Svnelobebia: 3.571 da 4.404. SevxedoT pirvel striqons da vipovoT -s 

ori mniSvneloba, romelic Seesabameba 3.571 -sa da 4.404-s. es mniSvnelobebia: 

0.99 da 0.975. rodesac miRebuli mniSvnelobebi ganawilebis marcxena 

mxaresaa, 1-s unda gamovakloT -s es mniSvnelobebi da miviRebT P -mniSvnelobis 

sazRvrebs. es sazRvrebia: 1 – 0.99 =0.01 da 1 – 0.975 = 0.025. 

Sesabamisad, 0.01 


0.014. 

SeniSvna: rodesac xi kvadrat kriteriumi ormxrivia, calmxrivi 

kriteriumis Sesabamisi orive sazRvari unda gaormagdes. Sesabamisad, 

wina magaliTSi: 2 0.01 

< 0.05. 

magaliTi 6. wina gamokvlevebidan mkvlevarma icis, rom manqanis 

Semowmebis drois standartuli gadaxraa 16.8 wuTi. 24 SemTxveviT 

SerCeuli manqanis Semowmebis drois Sesworebuli standartuli gadaxra 

aRmoCnda 12.5 wuTi. 0.05 mniSvnelovnebis doniT SegviZlia Tu ara 

davaskvnaT, rom standartuli gadaxra Seicvala? isargebleT P - 

mniSvnelobis meTodi. 


H0 : D 16.8 , H1 : D 16.8 . gamovTvaloT kriteriumis mniSvneloba: 

( n1) 

s 

 

2 

0 

' 2 

2 

(24 1) (12.5) 

TV . . 12.733 

. 

2 

(16.8) 

gamovTvaloT P -mniSvneloba: Tavisuflebis n1 32 xarisxis 

gaswvriv xi kvadrat ganawilebis zeda kritikuli wertilebis cxrilSi 

12.733 vardeba 11.689-sa da 13.091-s Soris. -s Sesabamisi mniSvnelobebia: 

0.975 da 0.95. vinaidan miRebuli mniSvnelobebi ganawilebis marcxena 

mxaresaa, 1-s unda gamovakloT -s es mniSvnelobebi: 10.975 0.025 , 

10.95 0.05 . radganac kriteriumi ormxrivia, miRebuli ricxvebi unda 

gaormagdes da miviRebT P -mniSvnelobis sazRvrebs: 0.05 

< 0.1 (kalkulatoriT gamoTvlili P -mniSvneloba = 0.085). 

vinaidan, P -mniSvneloba metia -ze, amitom nulovani hipoTeza ar 

unda ukuvagdoT. Sesabamisad, Cven ar gvaqvs sakmarisi safuZveli, raTa 

davaskvnaT, rom standartuli gadaxra Seicvala. 

amocanebi 

1. CamoayalibeT SesaZlo ZiriTadi da alternatiuli hipoTezebi da 

gamoTvaleT kritikuli mniSvneloba, roca D 225, Tu: a). 0.05 , n 18, 

132

kriteriumi marjvena calmxrivia; b). 0.1, n 23 , kriteriumi marcxena 

calmxrivia; g). 0.05 , n 15, kriteriumi ormxrivia; d). 0.1, n 8, 

kriteriumi ormxrivia; e). 0.01 , n 17 , kriteriumi marjvena calmxrivia; 

v). 0.025 , n 20 , kriteriumi marcxena calmxrivia; z). 0.01 , n 13, 

kriteriumi ormxrivia; T). 0.025 , n 29 , kriteriumi marcxena 

calmxrivia. 

2 

2. ipoveT P -mniSvnelobis sazRvrebi qvemoT moyvanili kriteriumis 

mniSvnelobebisaTvis: a). 29.321 , n 16 , kriteriumi marjvena 

2 

2 

calmxrivia; b). 10.215 , n 25 , kriteriumi marcxena calmxrivia; g). 

2 

2 

24.672 , n 11, kriteriumi ormxrivia; d). 23.722 , n 9 , kriteriumi 

2 

marjvena calmxrivia; e). 13.974 , n 28 , kriteriumi ormxrivia; v). 

2 

2 

10.571 , n 19 , kriteriumi marcxena calmxrivia; z). 12.144 , n 6 , 

kriteriumi ormxrivia; T). 2 8.201 , n 23 , kriteriumi ormxrivia; 

qvemoT moyvanil amocanebSi igulisxmeT, rom sidideebi normaluria 

an daaxloebiT normaluradaa ganawilebuli. 

3. dietologis mtkicebiT sxvadasxva saxis erT magidis kovz sirof- 

Si kaloriebis ricxvis standartuli gadaxra aris 60. 0.1 mniSvnelovnebis 

doniT SeiZleba Tu ar am mtkicebulebis uaryofa, Tu SemTxveviT 

SerCeuli 18 sxvadasxva saxis sirofis erT kovzSi kaloriebis ricxvia: 

53 210 100 200 100 220 

210 100 240 200 100 210 

100 210 100 210 100 60 

4. SerCeul iqna 18 TiTo kologramiani Saqris fuTa. awonvis Sedegad 

aRmoCnda, rom Sesworebuli SerCeviTi dispersia Seadgens 6.5-s. 

0.01 mniSvnelovnebis doniT SeamowmeT hipopTeza imis Sesaxeb, rom 

populaciis dispersia metia 6.2-ze. 

5. kopaniis menejeris mtkicebiT maT mier gamoSvebul iogurtSi Saqris 

Semcvlelobis dispersia ar aRemateba 25 grams. SemTxveviT SerCeul 

20 iogurtSi gazomes Saqris Semcvleloba da aRmoCnda, rom Sesworebuli 

SerCeviTi dispersia tolia 36-is. 0.1 mniSvnelovnebis doniT SeiZleba 

Tu ar am mtkicebulebis uaryofa? 

6. wina kvlevebis Tanaxmad mkvlevari Tvlis, rom pirvelkursel 

studentTa wlovanebebis dispersia Seadgens 1.6-s. SemTxveviT SerCeuli 50 

pirvelkurselis wlovanebebis Sesworebuli SerCeviTi dispersia 

aRmoCnda 2.3. 0.05 mniSvnelovnebis doniT aris Tu ara pirvelkursel 

studentTa wlovanebebis dispersia 1.6-ze meti? 

7. kopaniis menejeris mtkicebiT im satelefono saubrebis xangrZlivobebis 

standartuli gadaxra, romelic mas sWirdeba firmis 

saqmeebis mosawesrigeblad, ar aRemateba 1.2 wuTs. menejeris SemTxveviT 

SerCeuli 15 satelefono saubris xangrZlivobis Sesworebuli 

standartuli gadaxra aRmoCnda 1.8 wuTi. 0.01 mniSvnelovnebis doniT 

SeamowmeT hipopTeza imis Sesaxeb, rom standartuli gadaxra ar 

aRemateba 1.2-s. isargebleT P -mniSvnelobis meTodiT. 

133

8. plastikatis boTlSi moTavsebuli 12 uncia (1 uncia = 28.3 gr) 

kaustikuri sodis wonis dasaSvebi standartuli gadaxra ar aRemateba 

0.03 uncias. isargebleT P -mniSvnelobis meTodiT da 0.05 

mniSvnelovnebis doniT SeamowmeT dasaSvebia Tu ara populaciis standartuli 

gadaxra, Tu SemTxveviT SerCeul 8 sodis boTlSi sodis 

wonebi iyo: 

12.03 12.1 12.02 11.98 

12 12.05 11.97 11.99 

9. qvemoT moyvanilia samSeneblo kompaniis mier Sesyiduli 12 

foladis mavTulis winaaRmdegobebis maCveneblebi. mavTulis partiis 

dawuneba xdeba im SemTxvevaSi, roca Sesworebuli standartuli gadaxra 

metia 2-ze. 0.01 mniSvnelovnebis doniT gvaqvs Tu ara safuZveli 

daviwunoT mavTulebis partia? 

2001 1998 2002 2000 1998 1999 

1997 2005 2003 2001 1999 2006 

pasuxebi: 

1. H0 : D 225 , H1 : D 225 , CV . . 27.587 ; H0 : D 225 , H1 : D 225 , 

CV . . 14.042 ; H0 : D 225 , H1 : D 225 , CV . . 5.629 , 26.119; H0 : D 225 , 

H1 : D 225 , CV . . 2.167 , 14.067; H0 : D 225 , H1 : D 225 , CV . . 32 ; 

H0 : D 225 , H1 : D 225 , CV . . 8.907 ; H0 : D 225 , H1 : D 225 , CV . . 3.074 , 

28.299; H0 : D 225 , H1 : D 225 , CV . . 15.308 . 2. (0.01, 0.025); (0.005, 0.01); (0.01, 

0.025): P -mniSvneloba < 0.005; (0.025, 0.05); (0.1, 0.2); (0.05, 0.1); P -mniSvneloba < 

2 

0.01. 3. H0 : 60 , H1 : 60 ; CV . . 8.672 , 27.587; TV . . 

19.707 ; H 

0 

. 4. 

2 

H0 : D 6.2 , H1 : D 6.2 ; CV . . 33.409 ; TV . . 

17.823 ; H 

0 

. 5. H0 : D 25 , 

2 

H1 : D 25 ; CV . . 27.204 ; TV . . 

27.36 ; ki. 6. H0 : D 1.6 , H1 : D 1.6 ; 

2 

2 

CV . . 55.758 ; TV . . 

70.438 ; ki. 7. H0 : 1.2 , H1 : 1.2 ; TV . . 

31.5 ; P - 

2 

mniSvneloba < 0.005 < ; H 

1. 8. H0 : 0.03 , H1 : 0.03 ; TV . . 

14.381 ; 

0.025 

2 

TV . . 

22.02 ; ara. 

134

Tavi VII 

hipoTezaTa Semowmeba bernulis sqemaSi 

statistikuri hipoTezis Semowmeba bernulis sqemaSi ucnobi 

p albaTobis Sesaxeb (didi moculobis SemTxvevaSi): 

hipoTeza: H 0 

: p p 0 


X p0 

kriteriumis statistika: Z 

p (1 p )/ n 

0 0 

S 

X wn 

) 

n 

( 

n 

x 

p0 


p (1 p )/ n 

0 0 



H1: 

p p0 

z z , 

H1: 

p p0 

z z , 

H1: 

p p0 

z z /2 

an z z /2, 

sadac z 

aris N(0,1) 

-is zeda -kritikuli wertili anu kritikuli mni- 

Svneloba C.V. (aq igulisxmeba, rom: np 5 da nq 5. Tuki am pirobebidan 

erTi mainc ar sruldeba, maSin kritikuli mniSvneloba unda vipovoT binomialuri 

ganawilebis cxrilidan). 

1 ( z), Tu H1: p p0; 

 

P - mniSvneloba: P ( z), Tu H1: p p0; 

 

2 [1 (| z |)], Tu H1: p p0. 





0 



SeniSvna. mcire moculobis SerCevisaTvis (kerZod, Tu n 30) kriteriumis 

statistikad gamoiyeneba Sn 

Bi( n, p0 

) . magaliTad, calmxrivi 

marjvena alternativis dros C.R. { Sn 

c }, sadac c mTeli dadebiTi 

ricxvi unda SeirCes pirobidan, rom: 

n 

k k nk 

P{ Sn 

c 

| H0} Cnp0 (1 p0) 

 

. 

kc 

c 

Tu aseTi c ar arsebobs, maSin c 

-s arCeven pirobidan 

P{ Sn 

c | H0} 

P{ Sn 

c 1| H0} 

, 

an rac igivea: 

n 

kc 

n 

k k nk k k nk 

n 0 

(1 

0) 

 

n 0 

(1 

0) 

kc1 

C p p C p p . 

135

hipoTezis Semowmeba puasonis populaciis parametris 

Sesaxeb (mcire moculobis SerCevebisaTvis): 

puasonis populaciis parametris Sesaxeb hipoTezis Semowmebisas 

mcire moculobis SerCevis SemTxvevaSi, iyeneben P -mniSvnelobis meTods. 

hipoTeza: H0: 

 

0 

H : 

 

alternativa: 

1 0 


kriteriumis statistika: Po( 0 

) 

kriteriumis mniSvneloba T.V.: (puasonis SemTxveviTi sididis dakvirvebuli 

mniSvneloba) 

P - mniSvneloba: 

 

 

k 

0 0 

2 P{ 

| H0} 2 

e 

, Tu 0; 

 

k0 

k! 

P 

1 k 

 

0 0 

2 P{ | H0} 2 (1 

e 

), Tu 0. 

 

k0 

k! 

Tu P , maSin amboben, rom Sedegi (laparakia -ze) statistikurad 

mniSvnelovania da H 

0 

hipoTezas uaryofen, winaaRmdeg SemTxvevaSi 

ki – is statistikurad umniSvneloa da nulovan hipoTezas ar uaryofen. 

didi moculobis SemTxvevaSi aqac iyeneben normalur aproqsimacias, 

kerZod, im faqts, rom 

2 2 2 

( ) / [ N(0,1)] (1) . 

0 0 


magaliTi 1. davuSvaT Catarebulia 50 damoukidebeli eqsperimenti, А 

xdomilebis moxdenis fardobiTi sixSire aRmoCnda 0.12. mniSvnelovnebis α 

= 0.01 donisaTvis SevamowmoT nulovani Н 0 : р = 0.1 hipoTeza alternatiuli 

Н 1 : р > 0.1 hipoTezis winaaRmdeg. 

amoxsna. vipovoT kriteriumis mniSvneloba 

(0.12 0.1) 50 

z 

0.471. 

0.1 

0.9 

kritikuli are iqneba marjvena calmxrivi, xolo kritikuli mniSvneloba 

unda vipovoT pirobidan 

( CV . .) 10.01 0.99 . 

normaluri ganawilebis funqciis cxrilidan vpoulobT, rom z 2.33 . 

vinaidan, z z , amitom miiReba hipoTeza imis Sesaxeb, rom p 0.1. 

magaliTi 2. ganaTlebis inspeqtoris SefasebiT maRali klasis 

moswavleebis 15% tovebs skolas. gasul wels, SemTxveviT SerCeuli 

maRali klasis 200 moswavlidan 38-ma datova skola. 0.05 mniSvnel- 

136

ovnebis doniT gvaqvs Tu ara sakmarisi safuZveli ukuvagdoT inspeqtoris 

mosazreba? 


H0 : p 0.15 , H1 : p 0.15 . vinaidan, 0.05 da kriteriumi ormxrivia, kritikuli 

mniSvnelobebia: z /2 

1.96 da z /2 

1.96 . garda amisa, gvaqvs: 

38 

p0 0.15 , n 200 , x w n 

0.19 . amitom kriteriumis mniSvneloba iqneba: 

200 

x 

p0 

0.19 0.15 

z 1.58 

. 

p0(1 p0)/ n 0.15 0.85/ 200 

rogorc vxedavT, kriteriumis mniSvneloba ar vardeba kritikul 

areSi, amitom nulovan hipoTezas ar ukuvagdebT. amrigad, Cven ara gvaqvs 

sakmarisi safuZveli inspeqtoris mosazrebis sawinaaRmdegod. 

magaliTi 3. prokuroris mtkicebiT advokatebis 25%-ze meti iyenebs 

reklamas. 200 SemTxveviT SerCeul advokatze dakvirvebam aCvena, rom 63 

maTgani ama Tu im formiT iyenebda reklamas. 0.05 mniSvnelovnebis 

doniT gvaqvs Tu ara sakmarisi safuZveli gaviziaroT prokuroris 

mtkicebuleba? 


63 

H0 : p 0.25 , H1 : p 0.25 . gvaqvs: p0 0.25 , n 200 , x w n 

0.315 . 

200 

gamovTvaloT kriteriumis mniSvneloba: 

x 

p0 

0.315 0.25 

z 2.12 . 

p0(1 p0)/ n 0.25 0.75/ 200 

gamovTvaloT P -mniSvneloba: 

P1 ( z) 1 (2.12) 19830 0.0170 . 

radganac P -mniSvneloba < (0.017 < 0.05), amitom H 

0 

unda ukuvagdoT. 

maSasadame, Cven gvaqvs sakmarisi safuZveli gaviziaroT prokuroris 

mosazreba imis Sesaxeb, rom advokatebis 25%-ze meti iyenebs reklamas. 

amocanebi 

1. satelefono kompaniis SefasebiT maTi momxmareblebis 40%-s gaaCnia 

lodinis reJimis mqone mowyobileba. am hipoTezis Seamowmeblad Seir- 

Ca 100 momxmarebeli da gairkva, rom maT 37%-s aqvs lodinis reJimis mqone 

mowyobileba. 0.01 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi safuZveli 

ukuvagdoT kompaniis mosazreba? 

2. saTamaSoebis mwarmoeblis azriT 14 wlamde asakis bavSebis sul 

cota 23%-s aqvs sasrialo gorgolaWebiani dafa. 14 wlamde asakis 40 bav- 

Svian SerCevaSi sasrialo dafa aRmoaCnda 7 bavSvs. gvaqvs Tu ara sakmarisi 

safuZveli 0.05 mniSvnelovnebis doniT uarvyoT saTamaSoebis mwarmoeblis 

mosazreba? 

137

3. samSobiaro saxlis warsuli Canawerebis mixedviT ara dRenakluli 

bavSvebis saSualo wona metia vidre 7 funti da 2 uncia (1 funti = 

453.6 gr; 1 uncia = 28.3 gr). mimdinare wels 100 dabadebuli bavSvidan 23-is 

wona meti iyo vidre 7 funti da 2 uncia. 0.01 mniSvnelovnebis doniT 

gvaqvs Tu ara sakmarisi safuZveli vamtkicoT rom proporcia Seicvala? 

4. kriminalistebis kvlevis mixedviT xanZrebis sul cota 40% gamowveulia 

21 wlamde axalgazrdebis mier. mkvlevaris dakvirvebis mixedviT 

80 xanZridan 30 gamowveuli iyo 21 wlamde axalgazrdis mier. 0.1 mniSvnelovnebis 

doniT gvaqvs Tu ara sakmarisi safuZveli ukuvagdoT kriminalistebis 

mosazreba? 

5. ukanaskneli kvlevis mixedviT aviakatastrofaSi moyolili adamianebis 

araumetes 32% iRupeba. 100 adamianisagan Semdgar SerCevaSi, romlebic 

moyvnen aviakatastrofaSi, 38 daiRupa. 0.05 mniSvnelovnebis doniT 

unda uarvyoT Tu ara kvlevis Sedegi? 

6. statistikis departamentis monacemebiT dedaqalaqis mcxovrebTa 

63%-s gaaCnia avtomopasuxe. kompania “magTis” 143 gamokiTxuli TanamSromlidan 

85-s aqvs avtomopasuxe. 0.05 mniSvnelovnebis doniT SeamowmeT 

hipoTeza, rom avtomopasuxis mflobelTa proporcia “magTis” TanamSromlebs 

Soris igivea rac dedaqalaqis mcxovreblebs Soris. 

7. statistikuri angariSis mixedviT srulwlovani mosaxleobis 17% 

gasul wels daeswro warmodgenas operaSi. am hipioTezis Sesamowmeblad 

mkvlevarma gamokiTxa 90 adamiani da daadgina, rom maTgan gasul wels 

operas daeswro 22. 0.05 mniSvnelovnebis doniT SeamowmeT hipoTeza, 

rom realuri proporcia Seadgens 17%-s. 

8. ukanaskneli kvlevis mixedviT pirvelkurseli studentebis sul 

cota 15% Warbwoniania. 80 SemTxveviT SerCeul pirvelkursels Soris 9 

aRmoCnda Warbwoniani. 0.05 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi 

safuZveli uarvyoT kvlevis Sedegi? 

9. kvlevis mixedviT universitetis studentebis araumetes 25% 

universitetamde misasvlelad gadis 10 km-ze mets. SeTxveviT SerCeuli 

100 studentidan aRmoCnda, rom 30 universitetamde misasvlelad gadis 10 

km-ze mets. 0.1 mniSvnelovnebis doniT SeamowmeT kvlevis Sedegebis 

siswore. isargebleT P -mniSvnelobis meTodiT. 

10. satelefono kompaniis gancxadebiT misi momxmareblebis 30%-ze 

mets aqvs 2-ze meti telefoni. am gancxadebis dasadastureblad kompaniam 

SemTxveviT SearCia 200 momxmarebeli da daadgina, rom maTgan 72-s aqvs 2 

telefonze meti. 0.05 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi 

safuZveli davadasturoT kompaniis gancxadeba? isargebleT P -mniSvnelobis 

meTodiT. 

11. kriminalistebis mtkicebiT mkvlelobebis 10% Cadenilia qalebis 

mier. gvaqvs Tu ara sakmarisi safuZveli 0.01 mniSvnelovnebis doniT 

ukuvagdoT es mtkicebuleba, Tu 67 SemTxveviT SerCeuli mkvlelobidan 10 

aRmoCnda qalebis mier Cadenili? gamoiyeneT P -mniSvnelobis meTodiT. 

12. monetis 9-jer agdebisas 3-jer movida gerbi. 0.1 mniSvnelovnebis 

doniT SegviZlia Tu ara davaskvnaT, rom moneta arawesieria (miTiTe- 

138

a: isargebleT binomialuri ganawilebis cxriliT da ipoveT 2 P{ 3} , 

roca p 0.5 da n 9 ). 

13. gasul wels TviTmfrinavis mgzavrebis 20% mgzavrobda pirveli 

klasiT. mimdinare wels SemTxveviT SerCeuli 15 mgzavridan 5-ma imgzavra 

pirveli klasiT. 0.1 mniSvnelovnebis doniT SegviZlia Tu ara davaskvnaT, 

rom proporcia Seicvala? 

pasuxebi: 

1. H0 : p 0.4 , H1 : p 0.4 ; CV . . 2.58 ; T. V. z 0.612 ; ara. 2. H0 : p 0.23 , 

H1 : p 0.23 ; CV . . 1.65 ; T. V. z 0.83 ; ara. 3. H 

0 

: p 0. 23, H 

1 

: p 0. 23; 

CV . . 2.58 ; T. V. z 2.9 ; ki. 4. H0 : p 0.4 , H1 : p 0.4 ; CV . . 1.28 ; 

T. V. z 0.457 ; ara. 5. H 0 

: p 0.32 , H1 : p 0.32 ; CV . . 1.65 ; T. V. z 1.29 ; ara. 

6. H0 : p 0.63 , H1 : p 0.63 ; CV . . 1.96 ; T. V. z 0.88 ; H 

0 

. 7. H0 : p 0.17 , 

H1 : p 0.17 ; CV . . 1.96 ; T. V. z 

1.88 ; H 

0 

. 8. H0 : p 0.15 , H1 : p 0.15 ; 

CV . . 1.65 ; T. V. z 0.94 ; H 

0 

. 9. H0 : p 0.25 , H1 : p 0.25 ; T. V. z 

1.15 ; P - 

mniSvneloba = 0.1251 > ; H 

0 

. 10. H0 : p 0.3 , H1 : p 0.3; T. V. z 

1.85 ; P - 

mniSvneloba = 0.0322 < ; H 

1. 11. H0 : p 0.1 , H1 : p 0.1; T. V. z 

1.34 ; P - 

mniSvneloba = 0.1802 > ; H 0 

. 12. ara. 13. ara. 

139

Tavi VIII 

ndobis intervali da hipoTezaTa Semowmeba 

Tu hipoTezis Semowmebis amocanaSi xdeba nulovani hipoTezis 

uaryofa, maSin imave mniSvnelovnebis donis mqone ndobis intervali 

populaciis parametrisaTvis ar moicavs parametris hipoTetur 

mniSvnelobas. 

Tu hipoTezis Semowmebis amocanaSi ar xdeba nulovani hipoTezis 

uaryofa, maSin imave mniSvnelovnebis donis mqone ndobis 

intervali populaciis parametrisaTvis moicavs parametris 

hipoTetur mniSvnelobas. 


mgaliTi 1. Saqari dafasoebulia 5 funtian (1 funti = 453.6 gr) 

fuTebSi. kontroliors eWvi aqvs, rom fuTaSi ar aris 5 funti Saqari. 

50 Saqris fuTisagan Semdgari SerCevis saSualo aRmoCnda 4.6 

funti, xolo Sesworebuli standartuli gadaxra ki 0.7 funti. 

0.05 mniSvnelovnebis doniT gvaqvs Tu ara safuZveli davaskvnaT, 

rom fuTaSi saSualod ar aris 5 funti Saqari? aageT 95%-iani ndobis 

intervali WeSmariti saSualosaTvis. 


H0 : E 5, H1 : E 5. am ormxrivi kriteriumis Sesabamisi kritikuli 

mniSvnelobebia: z 

/2 

z0.025 1.96 da z /2 

z0.025 1.96 . kriteriumis 

mniSvneloba iqneba: 

x E 

4.5 5 0.4 

z 4.04 . 

' 

n 50 0.099 

s / 0.7/ 

radganac 4.04 1.96 , amitom nulovani hipoTeza unda ukuvagdoT, 

anu Cven gvaqvs sakmarisi safuZveli, raTa davaskvnaT, rom fu- 

TaSi saSualod ar aris 5 funti Saqari. 

avagoT 95%-iani ndobis intervali saSualosaTvis: 

' 

s 

x z/2 E 

x z/2 

, 

n 

n 

140 

' 

s 

0.7 0.7 

4.6 1.96 E 4.6 1.96 

, 

50 50 

4.4 E 

4.8 . 

rogorc vxedavT, 95%-iani ndobis intervali (anu ndobis intervali 

mniSvnelovnebis doniT 0.05 ) saSualosaTvis E ar moicavs 

saSualos hipoTetur mniSvnelobas E 5. 

magaliTi 2. mkvlevari amtkicebs, rom specialuri dietiT gamokvebili 

zrdasruli Rori saSualod iwonis 200 funts. 10 aseTi 

Rorisagan Sedgenili SerCevis saSualo wona aRmoCnda 198.2 funti, 

xolo Sesworebuli standartuli gadaxra ki 3.3 funti. 0.05 mni-

Svnelovnebis doniT gvaqvs Tu ara safuZveli davujeroT mkvlevars? 

aageT 95%-iani ndobis intervali realuri saSualosaTvis. 


H0 : E 200 , H1 : E 200 . radganac populaciis standartuli gadaxra 

ucnobia da n 30, amitom unda gamoviyenoT T testi. am SemTxvevaSi 

ormxrivi kriteriumis Sesabamisi kritikuli mniSvnelobebia: 

tn 

1, /2 

t9,0.025 2.262 da tn 1, /2 

t9,0.025 2.262 . kriteriumis mniSvneloba 

iqneba: 

x E 

198.2 200 1.8 

t 1.72 . 

' 

s / n 3.3/ 

10 1.0436 

Sesabamisad, nulovani hipoTeza ar unda ukuvagdoT, anu ara 

gvaqvs sakmarisi safuZveli uarvyoT mkvlevaris mosazreba, rom Roris 

saSualo wonaa 200 funti. 

avagoT 95%-iani ndobis intervali saSualosaTvis: 

' 

s 

' 

s 

x t/2 E 

x t/2 

, 

n 

n 

3.3 3.3 

198.2 2.262 E 198.2 2.262 , 

10 10 

195.8 E 

200.6 . 

rogorc vxedavT, 95%-iani ndobis intervali (anu ndobis intervali 

mniSvnelovnebis doniT 0.05 ) saSualosaTvis E moicavs 

saSualos hipoTetur mniSvnelobas E 200 . 

amocanebi 

1. Txilamurebis maRaziis menejeris mtkicebiT zamTris 

Tveebis ganmavlobaSi misi maRaziis saSualo dRiuri brunva 

Seadgens 1800 lars. SemTxveviT SerCeuli zamTris 10 dRis saSualo 

dRiuri brunva aRmoCnda 1830 lari. populaciis standartuli 

gadaxraa 200 lari. 0.05 mniSvnelovnebis doniT gvaqvs Tu ara 

safuZveli uarvyoT menejeris mtkicebuleba? aageT 95%-iani ndobis 

intervali saSualosaTvis. aris Tu ara ndobis intervalis 

interpretacia TanxvedraSi hipoTezis Semowmebis SedegTan? igulisxmeT, 

rom sidide ganawilebulia normalurad. 

2. samgzavro avtobuss saSualod gadayavs 42 mgzavri. gasul 

wlebSi populaciis standartuli gadaxra Seadgenda 8-s. mimdinare 

wels SemTxveviT SerCeuli 10 avtobusis mgzavrTa saSualo 

aRmoCnda 48. 0.1 mniSvnelovnebis doniT SegviZlia Tu ara 

davaskvnaT, rom saSualo igivea? aageT 90%-iani ndobis intervali 

saSualosaTvis. aris Tu ara ndobis intervalis interpretacia 

TanxvedraSi hipoTezis Semowmebis SedegTan? igulisxmeT, rom 

sidide ganawilebulia normalurad. 

141

3. uZravi qonebis agentis mtkicebiT sakuTari saxlebis 

komunaluri gadasaxadebis saSualo TveSi Seadgens 86 lars. 

warsuli kvlevis Tanaxmad populaciis standartuli gadaxraa 6 

lari. SemTxveviT SerCeuli 15 saxlis mflobelis saSualo 

gadasaxadi aRmoCnda 84 lari. 0.01 mniSvnelovnebis doniT 

SeamowmeT agentis mtkicebuleba. aageT 99%-iani ndobis intervali 


TanxvedraSi hipoTezis Semowmebis SedegTan? igulisxmeT, rom 

sidide ganawilebulia normalurad. 

4. erTadgiliani naviT mdinaris dinebis mimarTulebiT 

gacurvis saSualo dro sportsmenebisaTvis Seadgens 47 wuTs. 

vinaidan gazafxulze dinebis siCqare ufro swrafia, SemTxveviT 

SerCeuli 10 sportsmenis saSualo dro aRmoCnda 42 wuTi. wina 

kvlevebidan cnobilia, rom standartuli gadaxra Seadgens 7 wuTs. 

0.1 mniSvnelovnebis doniT SevamowmoT hipoTeza, rom swrafi 

dinebis dros saSualo gansxvavebulia. aageT 90%-iani ndobis 

intervali saSualosaTvis. aris Tu ara ndobis intervalis 

interpretacia TanxvedraSi hipoTezis Semowmebis SedegTan? igulisxmeT, 

rom sidide ganawilebulia normalurad. 

5. wina kvlevebis Tanaxmad pirvelkurseli studentebi kviraSi 

saSualod 22 saaTs xarjaven swavlaze. standartuli gadaxra 

Seadgens 4 saaTs. mimdinare wels gamokiTxes 60 studenti da 

aRmoCnda, rom maT mier kviris ganmavlobaSi swavlaze daxarjuli 

drois saSualo Seadgenda 20.8 saaTs. 0.01 mniSvnelovnebis 

doniT SeamowmeT hipoTeza, rom swavlaze daxarjuli drois 

saSualo Seicvala. aageT 99%-iani ndobis intervali 


TanxvedraSi hipoTezis Semowmebis SedegTan? 


6. meteorologis mtkicebiT aSS-s qalaqebSi umaRlesi 

temperaturis saSualo Seadgens 98-s (farengeitiT). 0.05 mniSvnelovnebis 

doniT gvaqvs Tu ara safuZveli ukuvagdoT meteorologis 

mosazreba, Tu cnobilia, rom SemTxveviT SerCeul 50 qalaqSi 

umaRlesi temperaturis monacemebia: 

97 94 96 105 99 96 80 95 101 97 

101 87 88 97 94 98 95 88 94 94 

99 99 98 96 96 97 98 99 92 97 

99 108 97 98 114 91 96 102 99 102 

100 93 88 102 99 98 80 95 101 61 

7. kvlevis Tanaxmad mweveli adamiani saSualod dReSi eweva 14 cal 

sigarets. am hipoTezis Sesamowmeblad SemTxveviT SeirCa 40 mweveli 

da aRmoCnda, rom isini dReSi saSualod 18 cal sigarets eweodnen. 

142

SerCevis Sesworebuli standartuli gadaxra iyo 6. 0.05 mniSvnelovnebis 

doniT gvaqvs Tu ara safuZveli CavTvaloT, rom 

mwevelebis mier dReSi moweuli sigaretis ricxvi sinamdvileSi 

gansxvavebulia 14-sagan? 

8. skolis konsultants surs Seamowmos mosazreba imis 

Sesaxeb, rom im moswavleebis saSualo asaki, romlebic Tavs 

anebeben swavlas, Seadgens 16.3 wels. SemTxveviT SeirCa 32 moswavle, 

romlebmac Tavi daanebes swavlas da aRmoCnda, rom maTi saSualo 

asaki iyo 16.9 weli. populaciis standartuli gadaxra Seadgens 0.3- 

s. 0.01 mniSvnelovnebis doniT gvaqvs Tu ara safuZveli uarvyoT 

aRniSnuli mosazreba? isargebleT P -mniSvnelobis meTodiT. 

9. mkvlevars surs Seamowmos aris Tua ara dedaqalaqis 

mosaxleobis saSualo asaki 61.2 weli. 22 SemTxveviT SerCeuli 

moqalaqis saSualo asaki aRmoCnda 59.8 weli, xolo Sesworebuli 

SerCeviTi standartuli gadaxra ki 1.5 weli. 0.01 mniSvnelovnebis 

doniT gansxvavdeba Tu ara realurad saSualo asaki 61.2 

wlisagan? igulisxmeT, rom sidide daaxloebiT normaluradaa 

ganawilebuli. 

10. zafxulis TveebSi temperaturis saSualo aSS-s CrdiloeT 

nawilSi Seadgens 67-s (farengeitiT). mimdinare wels SemTxveviT 

SerCeuli 10 qalaqis saSualo temperatura aRmoCnda 69.6, xolo 

Sesworebuli SerCeviTi standartuli gadaxra ki 1.1. 0.1 mniSvnelovnebis 

doniT SegviZlia Tu ara davaskvnaT, rom mimdinare wlis 

zafxuli ufro Tbili iyo vidre saSualod? 

11. winandeli kvlevis mixedviT mkvlelobis msxverplTa saSualo 

asaki ar aRemateba 23.2 wels. mimdinare wels SemTxveviT 

SerCeuli 18 mkvlelobis msxverplTa saSualo asaki aRmoCnda 22.6 

weli, xolo Sesworebuli standartuli gadaxra ki 2 weli. 0.05 

mniSvnelovnebis doniT SegviZlia Tu ara davaskvnaT, rom mimdinare 

wels msxverplTa saSualo asaki gaizarda? igulisxmeT, rom sidide 

daaxloebiT normaluradaa ganawilebuli. isargebleT P - 


12. gaeros monacemebiT msoflios 10 popularuli erovnuli 

muzeumis damTvalirebelTa saSualo ricxvi gasul wels Seadgenda 

6 milion adamians. mimdinare wels, qvemoT moyvanili am 10 muzeumis 

damTvalirebelTa ricxvis monacemebis mixedviT, 0.02 mniSvnelovnebis 

doniT, SegviZlia Tu ara davaskvnaT, rom mimdinare wels 

damTvalirebelTa saSualo Seicvala (monacemebi gamosaxulia 

milionebSi): 

4.7 17.2 14 6.1 6.1 

4.9 9.3 9.4 6.4 6.1 

13. universitetis finansuri daxmarebis departamenti 

imedovnebs, rom studentTa sul cota 30% miiRebs ama Tu im saxis 

143

finansur daxmarebas. imis gasarkvevad, sworia Tu ara es mosazreba, 

SemTxveviT SerCeul iqna 60 studenti da aRmoCnda, rom maTgan 15 

miiRo finansuri daxmareba. 0.05 mniSvnelovnebis doniT 

SevamowmoT hipoTeza, rom studentTa sul cota 30% miiRebs 

finansur daxmarebas. 

14. wina kvlevis Tanaxmad srulwlovani adamianebis sul cota 

60% sauzmeze miirTmevs kvercxs kviraSi oTxjer mainc. am 

mosazrebis Sesamowmeblad dietologma SemTxveviT SearCia 100 

srulwlovani adamiani da daadgina, rom maT Soris 54% sauzmeze 

miirTmevs kvercxs kviraSi oTxjer mainc. 0.1 mniSvnelovnebis 

doniT eTanxmeba Tu ara dietologis Sedegi wina kvlevis Sedegs? 

15. samSeneblo kompaniis mtkicebiT binis myidvelTa 80%-s 

surs binaSi qondes buxari. am mosazrebis Sesamowmeblad 

mkvlevarma SemTxveviT SearCia 30 binis myidveli da daadgina, rom 

maTgan 20-s surda binaSi qonoda buxari. 0.02 mniSvnelovnebis 

doniT SeamowmeT kompaniis mtkicebulebis siswore. 

16. “nokias” distributoris mtkicebiT 13 – 16 wlis asakis mozardebis 

65%-s aqvs maTi firmis telefoni. am mosazrebis 

Sesamowmeblad mkvlevarma SemTxveviT SearCia 80 mozardi da 

daadgina, rom maTgan 57-s hqonda “nokias” firmis telefoni. 0.05 

mniSvnelovnebis doniT SegviZlia Tu ara ukuvagdoT 

distributoris mosazreba? isargebleT P -mniSvnelobis meTodiT. 

17. fexburTis federaciis prezidentis mtkicebiT 

fexburTelebis saSualo wona Seadgens 225 funts (1 funti = 453.6 

gr). am mosazrebis Sesamowmeblad, mkvlevarma SemTxveviT SearCia 50 

fexburTeli da daadgina, rom maTi saSualo wona iyo 230 funti, 

xolo Sesworebuli standartuli gadaxra ki 15 funti. 0.01 mni- 

Svnelovnebis doniT SeamowmeT prezidentis mtkicebuleba. 


18. filmebis redaqtoris mtkicebiT videofilmebis 

xangrZlivobis standartuli gadaxra Seadgens 3.4 wuTs. SemTxveviT 

SerCeuli 24 videofilmis xangrZlivobis Sesworebuli 

standartuli gadaxra aRmoCnda 4.2 wuTi. 0.05 mniSvnelovnebis 

doniT, SegviZlia Tu ara davaskvnaT, rom SerCevis standartuli 

gadaxra gansxvavebulia redaqtoris mtkicebulebisagan? 

19. “Jigulis” mier moxmarebuli sawvavis standartuli 

gadaxra Seadgens 4.3 litrs. SemTxveviT SerCeuli 20 “Jigulis” mier 

moxmarebuli sawvavis Sesworebuli standartuli gadaxra aRmoCnda 

2.6 litri. 0.05 mniSvnelovnebis doniT, SegviZlia Tu ara 

davaskvnaT, rom SerCevis standartuli gadaxra naklebia 4.3 

litrze? isargebleT P -mniSvnelobis meTodiT. 

20. uZravi qonebis agentis mtkicebiT dedaqalaqis centrSi 1 

kv. metri saofise farTis saijaro gadasaxadis standartuli 

gadaxra Seadgens 95 lars. 0.02 mniSvnelovnebis doniT, SegviZ- 

144

lia Tu ara agentis mtkicebulebis uaryofa, Tu SemTxvevT 

SerCeuli 18 saofise farTis 1 kv. metris saijaro gadasaxadebia: 

400 345 325 395 400 300 

375 435 495 525 290 460 

425 250 200 525 375 390 

21. saRebavebis firmis menejeris mtkicebiT garkveuli 

saRebavis gaSrobis drois standartuli gadaxra Seadgens 18 wuTs. 

xuTma sxvadasxva testirebam aCvena rom saRebavis gaSrobis drois 

Sesworebuli standartuli gadaxra Seadgens 21 wuTs. 0.05 mniSvnelovnebis 

doniT SeamowmeT hipoTeza standartuli gadaxris 

Sesaxeb? 

22. sociologs surs Seamowmos sworia Tu ara, rom garkveuli 

profesiis qalebi pirvel Svils aCenen 28.6 wlis asakSi. 0.05 mni- 

Svnelovnebis doniT ra daskvna unda gamoitanos sociologma, Tu 

mas eqneba SemTxveviT SerCeuli 36 qalis mier pirveli Svilis 

gaCenis asakis Semdegi monacemebi: 

32 28 26 33 35 34 29 24 22 

25 26 28 28 34 33 32 30 29 

30 27 33 34 28 25 24 33 25 

37 35 33 34 36 38 27 29 26 

23. uZravi qonebis agentis mtkicebiT axali saxlis yidvisas 

damatebiTi xarjebis saSualo Seadgens 6500 lars. SemTxveviT 

SerCeuli 40 gayiduli saxlis myidvelis damatebiTi xarjebis 

saSualom Seadgina 6600 lari. populaciis standartuli gadaxraa 

120 lari. 0.05 mniSvnelovnebis doniT SeamowmeT agentis 

mtkicebuleba. 

24. kolejis administrators surs Seamowmos aris Tu ara 

studentis mier swavlisaTvis Tavis danebebis saSualo asaki 21 

weli. man aiRo bolo 17 wlis monacemebi da yoveli wlisaTvis 

daTvala swavlisaTvis Tavis danebebis saSualo asaki, romelic 

moyvanilia qvemoT. 0.05 mniSvnelovnebis doniT SeiZleba Tu ara 

hipoTezis uaryofa? 

12 18 24 16 21 20 18 19 19 

22 25 16 18 19 19 20 23 

25. taqsebis firmis menejeris mtkicebiT maTi taqsistebis 

muSaobis staJis saSualo Seadgens sul cota 12.4 wels. SemTxveviT 

SerCeuli 15 taqsistis muSaobis staJis saSualo aRmoCnda 11.2 weli, 

xolo Sesworebuli standartuli gadaxra ki 2 weli. 0.1 mniSvnelovnebis 

doniT aris Tu ara sinamdvileSi staJis saSualo 

ufro naklebi, vidre amas amtkicebs menejeri? 

145

pasuxebi: 

1. H0 : E 1800 , H1 : E 1800 ; CV . . 1.96 ; T. V. z 

0.47 ; H 

0 

; 

1706.04 E 

1953.96 ; 1800 (1706.04,1953.96) . 2. H0 : E 42 , 

H1 : E 42 ; CV . . 1.65 ; T. V. z 

0.47 ; H 

1; 43.83 E 

52.17 ; 

42 (43.83,52.17) . 3. H0 : E 86 , H1 : E 86 ; CV . . 2.58 ; T. V. z 1.29 ; 

H 

0 

; 80 E 

88 ; 86 (80,88) . 4. H0 : E 47 , H1 : E 47 ; CV . . 1.65 ; 

T. V. z 2.26 ; H 

1; 38.35 E 

45.65 ; 47 (38.35,45.65) . 5. H0 : E 22 , 

H1 : E 22; CV . . 2.58 ; T. V. z 2.32 ; H 

0 

; 19.47 E 

22.13 ; 

22 (19.47,22.13) . 6. H0 : E 98 , H1 : E 98 ; CV . . 1.96 ; T. V. z 2.02 ; 

H 

1. 7. H0 : E 14 , H1 : E 14 ; CV . . 1.96 ; T. V. z 

4.22 ; H 

1. 8. 

H0 : E 16.3 , H1 : E 16.3 ; T. V. z 

11.3 ; P -mniSvneloba < 0.01; ki. 9. 

H0 : E 61.2 , H1 : E 61.2 ; CV . . 2.831 ; T. V. t 

4.378 ; ki. 10. 

H0 : E 67 , H1 : E 67 ; CV . . 1.383 ; T. V. t 

7.47 ; ara. 11. H0 : E 23.2 , 

H1 : E 23.2; T. V. t 

1.27 ; 0.75 

H0 : E 6, H1 : E 6; CV . . 2.821 ; T. V. t 

1.835 ; ara. 13. H0 : p 0.3 , 

H1 : p 0.3; CV . . 1.65 ; T. V. z 0.85 ; H 

0 

. 14. H0 : p 0.6 , H1 : p 0.6 ; 

CV . . 1.28 ; T. V. z 1.22 ; ki. 15. H0 : p 0.8 , H1 : p 0.8 ; CV . . 2.33 ; 

T. V. z 1.83 ; H 

0 

. 16. H0 : p 0.65 , H1 : p 0.65 ; T. V. z 

1.17 ; P - 

mniSvneloba = 0.242 > ; H 

0 

. 17. H0 : E 225 , H1 : E 225 ; 

T. V. z 

2.36 ; P -mniSvneloba = 0.0182; H 

0 

. 18. H0 : 3.4 , H1 : 3.4 ; 

CV . . 11.689 da 38.076; 

TV 

2 

. . 6.95 

TV 

; ara. 19. H0 : 4.3 , H1 : 4.3 ; 

2 

. . 35.1 

; 0.005 

oramokrefiani amocanebi populaciaTa saSualoebisaTvis 

H 

ori damoukidebeli populaciis saSualos Soris 

gansxvavebis hipoTezaTa Semowmeba I 

Na ( , ) da 

2 

1 1 

2 

2 2 

Na ( , ) an orive SerCevis moculoba metia an toli 

30-ze; da damoukidebelia; 2 1 

da 2 2 

cnobilia. X 1 

,..., X 

n 

da Y 1 

,..., Y m 

ori SerCevaa Sesabamisad da populaciebidan. 

kriteriumi: 


H0 : a1 a2 

0 H0 : a1 a2 

0 H0 : a1 a2 

0 

an H0 : a1 a2 

0 an H0 : a1 a2 

0 

H1 : a1 a2 

0 H1 : a1 a2 

0 

H1 : a1 a2 

0 

hipoTeza: H 0 

: a 1 

a 

2 

0 


( X nY m) ( a1 

a2) 


N(0,1) 

2 2 

/ n 

/ m 

147 

1 2 

( xn 

ym) ( a1 

a2) 


2 2 

1 / n2 

/ m 



H1 : a1 a2 

0 

z z , 

H1 : a1 a2 

0 

z z , 

H1 : a1 a2 

0 

z z /2 

an z 

z /2 

(sadac z aris N(0,1) 

-is zeda -kritikuli wertili anu kritikuli 

mniSvneloba C.V.). 

P{ Z z | H0}, Tu H1 : a1 a2 

0; 

 

P - mniSvneloba: P P{ Z z | H0}, Tu H1 : a1 a2 

0; = 

 

P{| Z | | z || H0}, Tu H1 : a1 a2 

0. 

1 ( z), Tu H1 : a1 a2 

0; 

 

= ( z), Tu H1 : a1 a2 

0; 

2 [1 (| z |)], Tu H1 : a 1 

a 2 

0. 





0 



gvaqvs. 

(1 ) 

saimedoobis ndobis intervali saSualoTa sxvaobisaTvis:

( xn y ) z / n / m a a ( x y ) z / n / m . 

m 

2 2 2 2 

/2 1 2 1 2 n m /2 1 2 


magaliTi 1. erTi normaluri populaciidan, romlis standartuli 

gadaxraa 5, aRebulia 40 moculobis mqone SerCeva. SerCeviTi saSualo Seadgens 

102-s. pirvelisagan damoukidebeli meore normaluri populaciidan, 

romlis standartuli gadaxraa 6, aRebulia 50 moculobis mqone SerCeva. 

am ukanasknelis SerCeviTi saSualoa 99. 0.04 mniSvnelovnebis doniT 

SeamowmeT aris Tu ara gansxvaveba saSualoebs Soris. 

amoxsna. gvaqvs: N( ,25) , N( ,36) , n 40 , m 50, xn 

102 , ym 

99 . 

CamovayaliboT ZiriTadi da alternatiuli hipoTezebi: H0 : a1 a2 

0, 

H1 : a1 a2 

0. alternativa ormxrivia, amitom kritikuli are iqneba 

C. R. ( , z0.02 ] [ z0.02 

, ) ( , 2.055] [2.055, ) 

. gamovTvaloT kriteriumis 

mniSvneloba. gvaqvs: 

( xn 

ym) ( a1 a2) 102 99 

T. V. z 2.59 . 

2 2 

 

25/ 40 36/50 

1 

/ n2 

/ m 

vinaidan, 2.59 > 2.055 (anu T. V. C. R. 

), amitom ZiriTad hipoTezas ukuvagdebT 

0.04 mniSvnelovnebis doniT. 

gamovTvaloT P -mniSvneloba. P -mniSvneloba = 2[1 (2.59)] 0.0016 . 

vinaidan, P -mniSvneloba < 0.04 , amitom P -mniSvnelobis meTodiTac 

igive daskvna gamogvaqs: 0.04 mniSvnelovnebis doniT ukuvagdebT H 

0 

hipoTezas. 

magaliTi 2. mdinaris marjvena sanapiroze SemTxveviT SerCeuli 100 

ojaxis saSualo Semosavali aRmoCnda x 100 29980 lari, xolo marcxena 

sanapiroze SemTxveviT SerCeuli 100 ojaxis saSualo Semosavali ki 

y100 28650 lari. pirveli populaciis standartuli gadaxraa 1 4740 , 

xolo meoris 2 

5365 . 0.05 mniSvnelovnebis doniT SegviZlia Tu ara 

davaskvnaT, rom mdinaris marjvena sanapiroze saSualo Semosavlebi ufro 

maRalia, vidre marcxenaze? aageT 95%-iani ndobis intervali saSualoTa 



H0 : a1 a2 

0, H1 : a1 a2 

0. gvaqvs: nm 100, z z0.05 1.645 . kriteriumis 

mniSvneloba iqneba: 

( xn 

ym) ( a1 a2) 29980 28650 

T. V. z 1.86 

. 

2 2 

/ n 

/ m 4740/100 5365/100 

1 2 

vinaidan, z 

z0.05 

, amitom ZiriTadi hipoTeza unda ukuvagdoT, anu 

0.05 mniSvnelovnebis doniT SegviZlia davaskvnaT, rom mdinaris marjvena 

sanapiroze saSualo Semosavlebi ufro maRalia, vidre marcxenaze? 

148

avagoT ndobis intervali. tolobidan 90% (1 ) 100% , vaskvniT, 

rom 0.1. Sesabamisad, z /2 

z0.05 1.645 . amitom, gvaqvs: 

( xn y ) z / n / m a a ( x y ) z / n / m , 

m 

2 2 2 2 

/2 1 2 1 2 n m /2 1 2 

4740 5365 4740 5365 

29980 28650 1.645 a a 

29980 28650 1.645 

, 

1 2 

100 100 

1330 1.645 10.05 a1 a2 

1330 1.645 10.05 , 

1313.5 a1 a2 

1346.5 . 

vinaidan ndobis intervali ar moicavs nuls, Cveni unda miviRoT gadawyvetileba 

nulovani hipoTezis ukugdebis Sesaxeb, rac TanxvedraSia 

hipoTezis Semowmebisas miRebul SedegTan. 

Hori damoukidebeli populaciis saSualos Soris gansxvavebis 

hipoTezaTa Semowmeba II 

2 

2 

Na ( 

1, 1) 

da Na ( 

2, 1) 

; da damoukidebelia; 2 2 2 

1 

2 

ucnobia. 

X 1 

,..., X 

n 

da Y 1 

,..., Y m 


hipoTeza: H 0 

: a 1 

a 

2 

0 


( X nY m) ( a1 

a2) 

kriteriumis statistika: T T( nm2) 

, 

' 

S 1/ n1/ 

m 

aq 

nm , 

1 1 

[( 1) ( 1) ] [ ( ) ( ) ] 

n 

m 

'2 '2 '2 2 2 

Sn, m 

n S1 m S2 

Xi X n Yi 

Y 

n 

n m 2 n m 2 i1 i1 

2 

S nm , 2 

 

2 

, 

( n m2) ( n m 2) . 

 

( xn 

ym) ( a1 

a2) 


' 

s 1/ n1/ 

m 



H1 : a1 a2 

0 

t tnm 

2, 

 

, 

H1 : a1 a2 

0 

t tnm 

2, 

 

, 

H1 : a1 a2 

0 

t tnm 

2, /2 

an t 

tnm 

2, /2 

(sadac tn m 2, 

aris Tavisuflebis nm 2 xarisxis mqone stiudentis 

ganawilebis zeda -kritikuli wertili anu kritikuli mniSvneloba 

C.V.). 

nm , 

149

P{ T t | H0}, Tu H1 : a1 a2 

0; 

 

P - mniSvneloba: P P{ T t | H0}, Tu H1 : a1 a2 

0; = 

 

P{| T | | t || H0}, Tu H1 : a1 a2 

0. 

1 FT 

( t), Tu H1 : a1 a2 

0; 

 

= FT 

( t), Tu H1 : a1 a2 

0; 

 

2 [1 FT 

(| t |)], Tu H1 : a1 a2 

0. 





0 



gvaqvs. 

(1 ) 

saimedoobis ndobis intervali saSualoTa sxvaobisaTvis: 

( xn y ) t s 1/ n1/ m a a ( x y ) t s 1/ n 1/ m. 

m 

' 

' 

nm2, /2 n, m 1 2 n m nm2, /2 n, 

m 


magaliTi 1. erTi normaluri populaciidan aRebulia n 10 moculobis 

SerCeva, romlisTvisac xn 

23, s ' 1 

4 da misgan damoukidebeli meore 

normaluri populaciidan aRebulia m 8 moculobis SerCeva, romlisTvisac 

ym 

26, s ' 2 

5. orive populaciis dispersiebi tolia. 0.05 mniSvnelovnebis 

doniT SeamowmeT aris Tu ara gansxvaveba saSualoebs Soris. 


H0 : a1 a2 

0, H1 : a1 a2 

0. kriteriumi ormxrivia. kritikuli mniSvnelobebia: 

CV . . tn m2, /2 

t16,0.025 

2.12 . vipovoT saerTo dispersiis Sefaseba: 

'2 1 '2 '2 1 

s nm , 

[( n 1) s1 ( m 1) s2 

] (9 16 7 25) 19.94 

nm2 10 82 

. 


( xn 

ym) ( a1 a2) (2326) 0 

T. V. t 1.42 

. 

' 

s 1/ n1/ m 19.94 1/10 1/8 

nm , 

vinaidan T. V. C. R. 

, amitom ZiriTadi hipoTezis uaryofis safuZveli 

ara gvaqvs, anu saSualoebs Soris gansxvaveba ar aris arsebiTi. 

magaliTi 2. ori damoukidebeli populaciidan, romelTa dispersiebi 

tolia, aRebulia 15-is toli moculobis SerCevebi da gamoTvlilia 

saSualoebi da Sesworebuli SerCeviTi dispersiebi: xn 

132.45 , s '2 

1 

123; 

ym 

128.06 , s '2 

2 

95. igulisxmeT, rom populaciebi normaluradaa ganawilebuli 

da SeadareT erTmaneTs maTi saSualoebi. aageT 95%-iani ndobis intervali 

saSualoTa sxvaobisaTvis. 


H0 : a1 a2 

0, H 1 

: a 1 

a 

2 

0. kriteriumi marcxena calmxrivia, amitom kriti- 

150

kuli mniSvneloba iqneba: CV . . tn m2, t28,0.05 

1.7 . vipovoT saerTo dispersiis 

Sefaseba: 

'2 1 '2 '2 1 

s nm , 

[( n 1) s1 ( m 1) s2 

] (14 123 15 95) 109 

nm2 1515 2 

. 


( xn 

ym) ( a1 a2) (132.45 128.06) 0 

T. V. t 1.15 

. 

' 

snm 

, 

1/ n1/ m 109 1/15 1/15 

radganac 1.15 > - 1.7, amitom nulovani hipoTezis uaryofis 

safuZveli ar gagvaCnia, anu ara gvaqvs sakmarisi safuZveli CavTvaloT, 

rom meore populaciis saSualo ufro metia vidre pirvelis. 

avagoT ndobis intervali saSualoTa sxvaobisaTvis. SevitanoT monacemebi 

Sesabamis formulaSi: 

' 

' 

(( xn y ) t s 1/ n1/ m,( xn y ) t s 1/ n 1/ m) 

, 

m 

nm2, /2 n, m m nm2, /2 n, 

m 

(4.39 1.7 109 1/15 1/15,4.39 1.7 109 1/15 1/15) , 

( 2.09,10.87) . 

ramdenadac ndobis intervali moicavs nuls, Cveni ara gvaqvs safuZveli 

nulovani hipoTezis ukugdebis, rac eTanxmeba hipoTezis Semowmebisas 

miRebul Sedegs. 


hipoTezaTa Semowmeba III 

populaciebi araa normaluri, magram orive SerCevis moculoba metia 

an toli 30-ze ( n 30, m 30 ) da dispersiebic ucnobia. X1,..., X 

n 

da 

Y1,..., Y m 


hipoTeza: H 0 

: a 1 

a 

2 

0 


( X ) ( 

1 2) 

as 

nY m a a 


N(0,1) 

'2 '2 

S / n S / m 

1 2 

(simbolo as niSnavs, rom sidide miaxloebiT normaluradaa ganawilebuli; 

'2 n 

1 

S 2 

1 

: ( Xi 

X 

n) 

, '2 m 

1 

S 2 

2 

: ( Yi 

Y 

m) 

) 

n1 

m1 

i1 

( xn 

ym) ( a1 

a2) 


'2 '2 

s1 / n s2 

/ m 



H1 : a1 a2 

0 

z z , 

H1 : a1 a2 

0 

z z , 

H1 : a1 a2 

0 

z z /2 

an z 

z /2 

i1 

151

(sadac z 

aris N(0,1) 

-is zeda -kritikuli wertili anu kritikuli mni- 

Svneloba C.V.). 

1 ( z), Tu H1 : a1 a2 

0; 

 

P - mniSvneloba: P ( z), Tu H1 : a1 a2 

0; 

 

2 [1 (| z |)], Tu H1 : a1 a2 

0. 





0 



gvaqvs. 

(1 ) 


'2 '2 '2 '2 

( xn ym) z/2 s1 / ns2 / m a1 a2 ( xn ym) z/2 s1 / n s2 

/ m . 

SeniSvna: im SemTxvevaSic, roca Cven vamowmebT hipoTezas imis Sesaxeb, 

rom saSualoTa Soris gansxvaveba aris ara nuli, aramed raime konkretuli 

ricxvi ( a 1 

a 2 

c ), kriteriumis statistikad isev ganixileba statistika: 

( X ) ( 

1 2) 

as 

nY m a a 

Z 

N(0,1) 

. 

S / n S / m 

'2 '2 

1 2 


magaliTi 1. saerTaSoriso statistikuri asociaciis monacemebiT 

londonSi sastumros oTaxis saSualo Rirebuleba aris $88.42, xolo parizSi 

ki $80.61. davuSvaT, rom TiToeul SemTxvevaSi aRebulia 50 sxvadasxva 

sastumros monacemi da Sesworebuli SerCeviTi standartuli gadaxrebia 

Sesabamisad $5.62 da $4.83. 0.05 mniSvnelovnebis doniT aris Tu ara 

arsebiTi gansxvaveba saSualoebs Soris? aageT 95%-iani ndobis intervali 



H0 : a1 a2 

0, H 1 

: a 1 

a 

2 

0. vipovoT kritikuli mniSvnelobebi: 

CV . . z 

/2 

z0.025 

1.96 . 


( xn 

ym) ( a1 a2) (88.42 80.61) 0 

z 7.45 . 

'2 '2 2 2 

s1 / n s2 

/ m 5.62 /50 4.83 /50 

vinaidan, 7.45 > 1.96, amitom nulovani hipoTeza unda ukuvagdoT, anu 

saSualoebs Soris gansxvaveba arsebiTia. 

avagoT ndobis intervali. am SemTxvevaSi 95% (1 ) 100% , saidanac 

0.025 . Sesabamisad, z /2. amitom ndobis intervali iqneba: 

2 2 

(88.42 80.61) 1.96 5.62 /50 4.83 /50 a1 a2 

 

152

2 2 

(88.42 80.61) 1.96 5.62 /50 4.83 /50 , 

5.76 a1 a2 

9.86 . 

ramdenadac ndobis intervali ar moicavs nuls, Cveni unda miviRoT 

gadawyvetileba nulovani hipoTezis ukugdebis Sesaxeb, rac eTanxmeba hipoTezis 

Semowmebisas miRebul Sedegs. 

magaliTi 2. mkvlevars surs Seamowmos hipoTeza imis Sesaxeb, rom 

kolejSi sportuli Sejibrebebis saSualo ricxvi vaJebisaTvis ufro metia 

vidre gogonebisaTvis. 0.1 mniSvnelovnebis doniT gvaqvs Tu ara 

sakmarisi safuZveli gaviziaroT es mosazreba, Tu SejibrebaTa raodenobebis 

monacemebia (isargebleT P -mniSvnelovnebis meTodiT): 

vaJebi 

gogonebi 

6 11 11 8 15 6 8 11 13 8 

6 14 8 12 18 7 5 13 14 6 

6 9 5 6 9 6 5 5 7 6 

6 9 18 7 6 10 7 6 5 5 

15 6 11 5 5 16 10 7 8 5 

9 9 5 5 8 7 5 5 6 5 

8 9 6 11 6 9 18 13 7 10 

7 7 5 10 7 11 4 6 8 7 

10 7 10 8 11 14 12 5 8 5 

9 5 11 5 8 7 8 5 7 10 


H0 : a1 a2 

0, H 1 

: a 1 

a 

2 

0. TiToeuli monacemebisaTvis gamovTvaloT SerCeviTi 

saSualo da Sesworebuli SerCeviTi standartuli gadaxrebi: 

50 

50 

1 

1 

x50 

xi 

8.6 

, y50 

yi 

7.9 

; 

50 

50 

i1 

50 

' 1 

2 

s1 

: 50 

' 1 

2 

( xi 

8.6) 3.3 , s2 

: ( yi 

7.9) 3.3 . 

50 1 

i1 

50 1 

i1 

Sesabamisad, kriteriumis mniSvneloba iqneba: 

( xn 

ym) ( a1 a2) (8.6 7.9) 0 

z 1.06 

. 

'2 '2 2 2 

s1 / n s2 

/ m 3.3 /50 3.3 /50 

vipovoT P -mniSvnelovneba: P -mniSvnelovneba = 1 (1.06) 0.1446 . vinaidan 

0.1446 > 0.1, amitom ZiriTadi hipoTezis uaryofis safuZveli ara 

gvaqvs, anu ara gvaqvs sakmarisi safuZveli vamtkicoT, rom vaJebisaTvis 

sportuli Sejibrebebis saSualo metia vidre gogonebisaTvis. 

amocanebi 

1. mkvlevars surs Seamowmos aris Tu ara amerikis mTavari mdinareebis 

saSualo sigrZe igive rac evropis mTavari mdinareebis saSualo sigrZe. 

0.01 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi safuZveli 

uarvyoT es mosazreba, Tu cnobilia, rom SemTxveviT SerCeuli mTavari 

mdinareebis sigrZeTa monacemebia: 

153 

i1

amerika 

evropa 

729 329 450 330 481 532 1776 1224 

329 600 1243 525 877 447 824 634 

850 532 710 300 565 675 724 357 

560 332 2315 410 1122 634 326 580 

800 1310 605 926 567 932 1124 405 

310 375 545 470 454 820 505 496 

434 360 865 1036 230 626 210 252 

447 652 360 722 600 1575 2290 

430 1979 259 425 

2. Semowmebul iqna gafantuli skleroziT daavadebul adamianTa 

ori 120 da 34 kaciani jgufis unarebi. pirvel jgufSi Sediodnen is adamianebi 

visac uvlida sakuTari meuRle, xolo meoreSi ki is adamianebi visac 

uvlida ucxo adamiani. miRebuli Sedegebia: x 120 2, s ' 1 

0.6 ; y 34 

1.7 , 

' 

s2 0.7 . 0.1 mniSvnelovnebis doniT aris Tu ara gansxvaveba am ori 

jgufis saSualoebs Soris? 

3. mkvlevars surs gaarkvios aris Tu ara mweveli adamianis pulsi 

ufro maRali vidre aramwevelis. SemTxveviT SerCeuli 100 mwevelisa da 

100 aramwevelis gamokvlevis Sedegebia Sesabamisad: x 100 90 , s ' 1 

5 ; y 100 

88 , 

' 

s2 6. SeuZlia Tu ara mkvlevars 0.05 mniSvnelovnebis doniT daaskvnas, 

rom mweveli adamianis pulsi ufro maRalia vidre aramwevelis? 

4. statistikosis mtkicebiT testis Sesrulebisas fsiqologiis 

mimarTulebis studentebis saSualo qula ufro maRalia vidre 

ekonomikis mimarTulebis studentebis. SemTxveviT SerCeuli TiToeuli 

mimarTulebis 50 – 50 studentis testirebis Sedegad Sesabamisad aRmo- 

Cnda, rom: x 50 118 , s ' 1 

15 ; y 50 

115 , s ' 2 

15. gvaqvs Tu ara sakmarisi safu- 

Zveli 0.01 mniSvnelovnebis doniT davadasturoT es mtkicebuleba? 


rom saavadmyofoebis derefnebSi xmauris done (gazomili decibalebSi) 

ufro maRalia, vidre oTaxebSi, Tu Sesabamisi SerCeviTi maxasiaTeblebia: 

x84 61.2 , s ' 1 

7.9 ; y 34 

59.4 , s ' 2 

7.9 . 

6. uZravi qonebis agents surs Seadaros saxlebis gayidvis fasebi 

qalaqis CrdiloeT da samxreT nawilebSi. kvlevis Sedegad miRebuli Ser- 

CeviTi maxasiaTeblebia: x 35 63255, s ' 1 

5602 ; y 40 

59102 , s ' 2 

4731. gvaqvs Tu 

ara sakmarisi safuZveli 0.01 mniSvnelovnebis doniT ukuvagdoT hipo- 

Teza, rom saxlebis gayidvis fasebis saSualo erTi da igivea? 

7. Semowmebul iqna qalTa ori jgufis codnis done. pirvel jgufSi 

Sediodnen isini vinc institutis damTavrebidan ramodenime TveSi Tavi 

daanebes samuSaos TavianTi profesiis mixedviT, xolo meore jgufSi ki 

isini vinc darCnen samuSaoze TavianTi profesiis mixedviT. kvlevis Sedegad 

miRebuli SerCeviTi maxasiaTeblebia: x103 3.16 , 

' 

s1 0.52 ; y225 3.28 , 

154

' 

s2 0.46 . 0.05 mniSvnelovnebis doniT SegviZlia Tu ara davaskvnaT, rom 

codnis done pirvel jgufSi ufro maRalia? 

8. qvemoT moyvanilia dedaqalaqidan da regionebidan SemTxveviT SerCeuli 

30 – 30 abiturientis mier erovnul gamocdebze mogrovili qulebi. 

gvaqvs Tu ara sakmarisi safuZveli vamtkicoT, rom dedaqalaqis abiturientebis 

qulebis saSualo metia vidre regionebis? 

regionebi 

dedaqalaqi 

656 594 636 642 683 626 531 643 582 637 

642 682 617 721 701 624 630 630 540 606 

614 632 753 698 649 570 623 710 587 543 

631 626 584 609 642 547 609 673 699 598 

532 597 698 705 636 703 525 624 615 730 

580 691 715 634 749 504 515 613 527 518 

9. kolejis administratoris mtkicebiT kolejis wliuri danaxarji 

vaJebis sportze metia vidre qalebis sportze. qvemoT moyvanili monacemebis 

mixedviT, gvaqvs Tu ara sakmarisi safuZveli 0.01 mniSvnelovnebis 

doniT davadasturoT administratoris mtkicebuleba? 

vaJebi 

13351 22220 11456 12244 8383 8796 7551 5254 6576 3377 

10128 14029 10160 16175 623 5544 5811 7550 1664 7656 

10652 15048 11480 10126 8811 7120 9119 5472 9505 8033 

11015 12371 11267 12703 9732 8605 2063 6670 6797 7040 

12919 13763 12403 5286 9571 9544 8725 9463 7723 9626 

qalebi 

10333 11248 14698 7654 7054 6869 933 9883 6959 6030 

12745 16249 11597 9331 7300 7874 6989 5232 8478 7058 

12016 10248 13371 5468 6407 6909 6502 6815 9959 9907 

10082 11041 10353 7055 8324 9110 9277 5933 3991 5832 

11324 10127 7435 8917 8411 7235 8903 6925 5922 6502 

10. aris Tu ara gansxvaveba SemTxveviT SerCeuli kviris ganmavloba- 

Si taqsebis ori kompaniis TiToeuli taqsis mier gavlili manZilebis sa- 

Sualoebs Soris, Tu Sesabamisi SerCevebis maxasiaTeblebia: x35 837 , 

' 

s1 30 ; y 40 

753 , s ' 2 

40. CaTvaleT, rom populaciebi normaluria; mniSvnelovnebis 

doned aiReT 0.05 da isargebleT P -mniSvnelobis meTodiT. 

11. qalTa ori jgufs SesTavazes kiTxvari, raTa gaerkviaT sakuTari 

Tavisadmi pativiscemis xarisxi. pirvel jgufSi Sediodnen isini vinc institutis 

damTavrebidan ramodenime TveSi Tavi daanebes samuSaos Tavian- 

Ti profesiis mixedviT, xolo meore jgufSi ki isini vinc darCnen samuSaoze 

TavianTi profesiis mixedviT. kvlevis Sedegad miRebuli SerCeviTi 

maxasiaTeblebia: x103 3.05 , s ' 1 

0.75 ; y 225 

2.96 , s ' 2 

0.75 . 0.05 mniSvnelovnebis 

doniT SegviZlia Tu ara davaskvnaT, rom am jgufebSi TviTpativiscemis 

xarisxi sxvadasxvaa? isargebleT P -mniSvnelobis meTodiT. 

12. dekanats ainteresebs aris Tu ara adgilobriv da Camosul 

studentTa asakis saSualoebs Soris gansxvaveba. qvemoT moyvanilia 

155

TiToeuli jgufidan SemTxveviT SerCeuli 50 - 50 studentis asakTa 

monacemebi. mniSvnelovnebis doned aiReT 0.05 da isargebleT P - 


adgilobrivi studentebi 

22 25 18 19 26 22 19 25 20 30 

19 19 21 23 27 26 26 18 19 19 

23 24 18 29 21 21 26 27 18 19 

23 21 28 26 19 22 18 22 26 18 

32 18 20 18 24 19 23 22 18 20 

Camosuli studentebi 

18 23 26 19 29 20 20 20 18 30 

26 23 21 25 19 23 22 35 21 18 

18 22 19 19 19 22 28 22 19 36 

23 25 25 21 18 27 20 24 20 18 

19 27 19 35 24 20 32 20 19 22 

13. ori jgufis testirebisas miRebuli qulebis mixedviT gamoTvlili 

Sesabamisi SerCeviTi maxasiaTeblebia: x36 83.6 , s1 4.3 ; y36 79.2 , 

' 

' 

s2 3.8 . aageT 90%-iani ndobis intervali saSualoTa sxvaobisaTvis. 

14. ori sxvadasxva dasaxelebis sigaretis SemTxveviT SerCeul 30 – 

30 RerSi nikotinis Semcvlelobis donis mixedviT gamoTvlili Sesabamisi 

SerCeviTi maxasiaTeblebia: x 30 28.6 , s ' 1 

5.1; y30 32.9 , s ' 2 

4.4 . aageT 

99%-iani ndobis intervali saSualoTa sxvaobisaTvis. 

15. ori saxis elementis SemTxveviT SerCeuli partiebis Semowmebis 

mixedviT gamoTvlili Sesabamisi SerCeviTi maxasiaTeblebia: x27 9.2 , 

' 

s1 0.3 ; y 30 

8.8 , s ' 2 

0.1. CaTvaleT, rom sidideebi normaluradaa ganawilebuli 

da aageT 95%-iani ndobis intervali saSualoTa sxvaobisaTvis. 

pasuxebi: 

1. H0 : a1 a2 

, H1 : a1 a2 

; CV . . 2.58 ; T. V. z 0.856 ; ki. 2. H0 : a1 a2 

0, 

H1 : a1 a2 

0; CV . . 1.65 ; T. V. z 

2.27 ; ara. 3. H0 : a1 a2 

, H1 : a1 a2 

; 

CV . . 1.65 ; T. V. z 

2.56 ; ki. 4. H0 : a1 a2 

, H1 : a1 a2 

; CV . . 2.33 ; T. V. z 

1; 

ara. 5. H0 : a1 a2 

, H1 : a1 a2 

; CV . . 2.05 ; T. V. z 

1.12 ; H 

0 

. 6. H0 : a1 a2 

, 

H1 : a1 a2 

; CV . . 2.58 ; T. V. z 

3.44 ; ki. 7. H0 : a1 a2 

, H1 : a1 a2 

; CV . . 1.65 ; 

T. V. z 2.01; ara. 8. H0 : a1 a2 

, H1 : a1 a2 

; CV . . 1.65 ; T. V. z 

3.3 ; ara. 9. 

H0 : a1 a2 

, H1 : a1 a2 

; CV . . 2.33 ; T. V. z 

1.09 ; ara. 10. H0 : a1 a2 

, H1 : a1 a2 

; 

T. V. z 

10.36 ; P -mniSvneloba < 0.002 < ; ki. 11. H0 : a1 a2 

, H1 : a1 a2 

; 

T. V. z 

1.01; P -mniSvneloba = 0.3124; ara. 12. H0 : a1 a2 

, H1 : a1 a2 

; 

T. V. z 0.76 ; P -mniSvneloba = 0.4472; H 

0 

. 13. 2.8 a1 a2 

6 . 14. 

7.3 a1 a2 

1.3 . 15. 0.3 a1 a2 

0.5 . 

156

Tavi X 

oramokrefiani amocanebi populaciaTa saSualoebisaTvis 

(mcire SerCevebi). satertvaitis meTodi 


hipoTezaTa Semowmeba (mcire SerCevebi) IV 

X1,..., X 

n 

da Y 1 

,..., Y m 

ori SerCevaa damoukidebeli da populaciebidan, 

romelTa dispersiebi gansxvavebulia; erT-erTi an orive SerCevis 

moculoba naklebia 30-ze. 

hipoTeza: H 0 

: a 1 

a 

2 

0 


( X nY m) ( a1 

a2) 

kriteriumis statistika: T 

T() 

k , 

'2 '2 

S1 / n S2 

/ m 

sadac Tavisuflebis xarisxi k min( n1, m 1) . 

( xn 

ym) ( a1 

a2) 


'2 '2 

s1 / n s2 

/ m 



H1 : a1 a2 

0 

t tk, 

 

, 

H1 : a1 a2 

0 

t tk, 

 

, 

H1 : a1 a2 

0 

t tk, /2 

an t 

tk, /2 

(sadac t k, 

 

aris Tavisuflebis k xarisxis mqone stiudentis ganawilebis 


P{ T t | H0}, Tu H1 : a1 a2 

0; 

 

P - mniSvneloba: P P{ T t | H0}, Tu H1 : a1 a2 

0; = 

 

P{| T | | t || H0}, Tu H1 : a1 a2 

0. 

1 FT 

( t), Tu H1 : a1 a2 

0; 

 

= FT 

( t), Tu H1 : a1 a2 

0; 

 

2 [1 FT 

(| t |)], Tu H1 : a1 a2 

0. 





0 



gvaqvs. 

(1 ) 


( x 

n 

y 

m 

'2 '2 

'2 '2 

tk. / 2 

s1 

/ n s2 

/ m, 

xn 

y tk. 

/ 2 

s1 

/ n s2 

/ m) 

m 

. 

157

SeniSvna: saSualoTa Sesaxeb tolobis Sesamowmeblad T kriteriumis 

gamoyenebamde winaswar unda gamoviyenoT F kriteriumi, raTa gavarkvioT 

aris Tu ara dispersiebi toli. am ukanasknelis gamosayeneblad ki aucilebelia 

populaciebi iyos normaluri an daaxloebiT normaluri. 

satertvaitis (Satterthwaite) meTodi 

X1,..., X 

n 

da Y 1 

,..., Y m 

ori SerCevaa damoukidebeli da populaciebidan, 

romelTa dispersiebi gansxvavebulia. 

hipoTeza: H 0 

: a 1 

a 

2 

0 


( X n Ym 

) ( a1 

a2) 

kriteriumis statistika: T , 

'2 '2 

S / n S / m 

( xn 

ym) ( a1 

a2) 


'2 '2 

s1 / n s2 

/ m 

Tu mocemuli mniSvnelovnebis donisaTvis T statistikis dakvirvebuli 

mniSvneloba t akmayofilebs Semdeg utolobas 

t[ c]. 

/ 2 

t t[ 

c]. 

/ 2 

, 

sadac [] c aRniSnavs c ricxvis mTel nawils, xolo c gamoiTvleba 

Semdgi wesiT 

'2 '2 2 

( s1 

/ n s2 

/ m) 

c 

, 

'2 2 

'2 2 

( s1 

/ n) 

/( n 1) 

( s2 

/ m) 

/( m 1) 

maSin mniSvnelovnebis doniT H 0 

: a 1 

a 

2 

0 hipoTezis uaryofis safuZveli 

ara gvaqvs, winaaRmdeg SemTxvevaSi, mas uarvyofT. 

1 saimedoobis mqone ndobis intervals aqvs Semdegi saxe: 

( x 

n 

y 

m 

1 

'2 '2 

'2 '2 

t[ c]. 

/ 2 

s1 

/ n s2 

/ m, 

xn 

y t[ 

c]. 

/ 2 

s1 

/ n s2 

/ m) 

m 

. 


magaliTi 1. fermeruli meurneobebis saSualo farTobi indianas 

StatSi Seadgens 191 akrs (1 akri = 0.4 ha), xolo aiovas StatSi ki 199 

akrs. davuSvaT, rom es monacemebi miRebulia Sesabamisad 8-isa da 10-is 

toli SerCevebidan da Sesabamisi Sesworebuli standartuli gadaxrebia 

38 akri da 12 akri. 0.05 mniSvnelovnebis doniT SeiZleba Tu ara davaskvnaT, 

rom fermeruli meurneobebis saSualo farTobi am or StatSi gansxvavebulia? 

igulisxmeba, rom populaciebi normaluria. aageT 95%-iani 

ndobis intervali saSualoTa sxvaobisaTvis. 

amoxsna. winaswar SevamowmoT hipoTeza imis Sesaxeb aris Tu ara populaciaTa 

dispersiebi toli. CamovayaliboT ZiriTadi da alternatiuli 

2 2 

2 2 

2 '2 '2 2 

hipoTezebi: H0 : 1 2 

, H1 : 1 2 

. am SemTxvevaSi 38 s1 s2 

12 . Sesabamisad, 

Tavisuflebis xarisxebia: 8 – 1 = 7 da 10 – 1 = 9. amitom kritikuli 

2 

. 

158

mniSvneloba iqneba: CV . . Fkl 

, , /2 

F7,9,0.025 

4.2 . gamovTvaloT kriteriumis mni- 

'2 '2 2 2 

Svneloba: T. V. f s1 / s2 

38 /12 10.03 . radganac 10.03 > 4.2, amitom 

nulovani hipoTeza unda ukuvagdoT, anu populaciaTa dispersiebi 

gansxvavebulia. Sesabamisad, saSualoTa Sesaxeb tolobis Sesamowmeblad 

unda visargebloT T kriteriumiT. 

CamovayaliboT ZiriTadi da alternatiuli hipoTezebi: H0 : a1 a2 

, 

H1 : a1 a2 

. radganac kriteriumi ormxrivia da dispersiebi gansxvavebulia, 

amitom Tavisuflebis xarisxi tolia k min(8 1,10 1) 7 . Sesabamisad, kritikuli 

mniSvnelobebia CV . . tk 

, /2 

t7,0.025 

2.365 . gamovTvaloT kriteriumis 

mniSvneloba: 

( xn 

ym) ( a1 a2) (191 100) 0 

T. V. t 0.57 

. 

'2 '2 2 2 

s1 / n s2 

/ m 38 /812 /10 

rogorc vxedavT kriteriumis mniSvneloba ar Sedis kritikul are- 

Si, amitom nulovan hipoTezis uaryofis safuZveli ara gvaqvs, anu Cven 

ara gvaqvs sakmarisi safuZveli raTa davaskvnaT, rom saSualoebi gansxvavebulia. 

avagoT ndobis intervali. amisaTvis saWiro sidideebi SevitanoT 

Sesabamis formulaSi. gvaqvs: 

2 2 2 2 

38 12 38 12 

(191 199) 2.365 a a 

(191 199) 2.365 

, 

1 2 

8 10 8 10 

41.02 a1 a2 

25.02 . 

magaliTi 2. mkvlevars surs Seamowmos aris Tu ara kerZo 

klinikaSi eqimis wliuri Semosavali ufro maRali vidre saxelmwifo 

klinikaSi. man SemTxveviT SearCia TiToeuli kategoriis 10 da 8 eqimi da 

' 

Sesabamisi SerCeviTi maxasiaTeblebi aRmoCnda: x10 26800 , s1 600 ; 

y8 25400 , s ' 2 

450. 0.01 mniSvnelovnebis doniT SeiZleba Tu ara davaskvnaT, 

rom kerZo klinikaSi Semosavali ufro maRalia vidre 

saxelmwifoSi? igulisxmeT, rom populaciebi daaxloebiT normaluria. 


amoxsna. winaswar gavarkvioT aris Tu ara populaciaTa dispersiebi 

2 2 

toli. CamovayaliboT ZiriTadi da alternatiuli hipoTezebi: H : , 

H : . gamovTvaloT kriteriumis mniSvneloba: 

2 2 

1 1 2 

0 1 2 

'2 '2 2 2 

T. V. f s1 / s2 

600 / 450 1.78 . 

vipovoT P -mniSvneloba. 10 – 1 = 9 da 8 – 1 = 7 Tavisuflebis xarisxis 

mixedviT fiSeris ganawilebis cxrilidan vpoulobT, rom P -mniSvneloba 

> 0.2. Sesabamisad, P -mniSvneloba > 0.01 , amitom nulovani hipoTezis 

uaryofis safuZveli ara gvaqvs, anu populaciaTa dispersiebi tolia. 

159

gadavideT saSualoebis Sesaxeb hipoTezaTa Semowmebaze. CamovayaliboT 

ZiriTadi da alternatiuli hipoTezebi: H 0 

: a 1 

a 2 

, H 1 

: a 1 

a 2 

. gamovTvaloT 

kriteriumis mniSvneloba: 

( xn 

ym) ( a1 

a2) 

T. V. 

t 

 

'2 '2 

( n 1) s1 ( m1) s2 

1 1 

 

n m2 

n m 

(26800 25400) 0 

 

5.47 

. 

2 2 

(10 1) 600 (8 1) 450 1 1 

 

10 82 10 8 

vipovoT P -mniSvneloba. Tavisuflebis 10 + 8 – 2 = 16 xarisxis mixedviT 

stiudentis ganawilebis cxrilidan vpoulobT, rom P -mniSvneloba 

naklebia 0.005-ze. Sesabamisad, P -mniSvneloba < 0.01 . amitom Cven ukuvagdebT 

nulovan hipoTezas, anu Cven gvaqvs sakmarisi safuZveli 

davaskvnaT, rom kerZo klinikaSi eqimis saSualo Semosavali ufro didia 

vidre saxelmwifo klinikaSi. 

magaliTi 3. davuSvaT, rom Seiswavleba qolesterinis Semcveloba 

gulis daavadebiT gardacvalebuli adamianebis memkvidreebSi. davuSvaT, 

rom 100 aseT bavSvze dakvirvebam mogvca qolesterinis saSualo done 

207.3 mg/dl, xolo standartuli gadaxraa 35.6 mg/dl. garda amisa, daakvirdnen 

qolesterinis doneebs im bavSvebSi, romelTa mSoblebic cocxlebia 

da ara aqvT gulis daavadeba. aseTnairad SerCeuli 74 bavSvis monacemebis 

mixedviT qolesterinis saSualo done aRmoCnda 193.4 mg/dl, xolo 

standartuli gadaxra –17.3 mg/dl. Cveni amocanaa = 0.05 mniSvnelovnebis 

donisaTvis SevamowmoT hipoTeza populaciebis saSualoebis tolobis Sesaxeb. 

amoxsna. davuSvaT, rom populaciebi ganawilebulia normalurad, 

magram maTi parametrebi ucnobia. es amocana gavyoT or etapad: pirvel 

etapze SevamowmoT hipoTeza dispersiebis tolobis Sesaxeb. Tu 

hipoTezas dispersiebis tolobis Sesaxeb ver uarvyofT, Semdeg 

SevamowmoT hipoTeza populaciebis saSualoebis tolobis Sesaxeb toli 

dispersiebis SemTxvevaSi. im SemTxvevaSi Tu ar dadasturda dispersiaTa 

tolobis hipoTeza, saSualoTa tolobis hipoTeza SevamowmoT 

satertvaitis meTodiT. 

ganvixiloT SerCeviT dispersiaTa Sefardeba da SevadaroT is fiSeris 

ganawilebis zeda kritikuli wertilis mniSvnelobas F 99,73,0.025 

. F 100,74 

statistikis dakvirvebuli mniSvneloba tolia 

2 2 

f100,74 35.6 /17.3 4.23 . 

vinaidan cxrilebSi ar aris fiSeris ganawilebis zeda kritikuli 

wertilis mniSvneloba -- F 

99,73,0.025 

, amitom gamovTvaloT p -mniSvneloba, 

romelic alternativis ormxrivobis gamo tolia 

p 2 P{ F 4.23} 0.0002 , 

100.74 

160

ac imis maCvenebelia, rom Sedegi statistikurad mniSvnelovania da maSasadame, 

hipoTeza dispersiaTa tolobis Sesaxeb unda uarvyoT. 

gadavideT meore etapze. radgan davadgineT, rom populaciaTa dispersiebi 

gansxvavebulia, viyenebT satertvaitis meTods saSualoTa tolobis 

Sesaxeb hipoTezis Sesamowmeblad. gamovTvaloT statistikis 

dakvirvebuli mniSvneloba, gvaqvs: 

207.3 193.4 

13.9 

t 

100,74 

 

3.4 . 

2 

2 

35.6 /10017.3 

/74 4.089 

axla gamovTvaloT Tavisuflebis xarisxis miaxloebiTi mniSvneloba 

c : 

2 

2 2 

2 

(35.6 /10017.3 

/ 74) 16.718 

c 

151.4 

, 

2 

2 2 

(35.6 /100)/99 

(17.3 / 74) / 73 1.8465 

da maSasadame, [ c] 151 . 

vinaidan t 100,74 

3.4 t 120,0.025 

1.98 

t 151,0.025 

, amitom 0.05 mniSvnelovnebis 

doniT populaciaTa saSualoebis tolobis hipoTezas uarvyofT. 

amocanebi 

qvemoTmoyvanil amocanebSi yvela sidide normaluria an daaxloebiT 

normaluri 

1. uZravi qonebis agentis mtkicebiT sagadasaxado inspeqtoris 

mier Sefasebuli saxlis Rirebuleba Seesabameba saxlis gayidvis fass. 

SemTxveviT SerCeuli 10 - 10 saxlisaTvis sagadasaxado inspeqtoris 

Sefasebisa da gayidvis fasis mixedviT gamoTvlili Sesabamisi SerCeviTi 

maxasiaTeblebi aRmoCnda: x 10 83256, s ' 1 

3256 ; y 10 

88354, s ' 2 

2341. 0.05 

mniSvnelovnebis doniT aris Tu ara saSualoebs Soris gansxvaveba 

mniSvnelovani? aageT 95%-iani ndobis intervali saSualoTa 


2. mkvlevaris azriT mamrobiTi sqesis eqimis Semosavali metia 

vidre mdedrobiTis. SemTxveviT SerCeuli 16 mamrobiTi sqesis eqimisa da 

20 mdedrobiTi sqesis eqimis Semosavlebis mixedviT gamoTvlili 

SerCeviTi maxasiaTeblebia: x16 23800, s ' 1 

300 ; y 18 

23750, s ' 2 

250. 0.05 

mniSvnelovnebis doniT SeiZleba Tu ara davadasturoT mkvlevaris 

mosazreba? 

3. programirebis pedagogis azriT maTematikis mimarTulebis students 

kompiuteruli programis dawera da Sesworeba SeuZlia ufro swrafad 

vidre biznesis mimarTulebis students. SemTxveviT SerCeuli 12 maTematikis 

(Sesabamisad, 18 biznesis) mimarTulebis studentis mier konkretuli 

programis daweraze da Sesworebaze daxarjuli drois saSualo aRmoCnda 

36 wuTi (Sesabamisad, 39 wuTi), xolo Sesabamisi Sesworebuli standartuli 

gadaxrebia: 4 wuTi da 9 wuTi. 0.1 mniSvnelovnebis doniT 

SeiZleba Tu ara davadasturoT pedagogis mosazreba? 

161

4. mkvlevaris SefasebiT studenti gogonebi saswavlo wlis ganmavlobaSi 

ufro met dReebs acdenen vidre vaJebi. SemTxveviT SerCeuli 16 

studenti gogonas (Sesabamisad, 22 studenti vaJis) mier wlis ganmavloba- 

Si gacdenili dReebis saSualom Seadgina 3.9 dRe (Sesabamisad, 3.6 dRe), 

xolo Sesabamisi Sesworebuli standartuli gadaxrebia: 0.6 dRe da 0.8 

dRe. 0.01 mniSvnelovnebis doniT SeiZleba Tu ara daveTanxmoT mkvlevaris 

Sefasebas? 

5. klinikis administrators ainteresebs aris Tu ara binaze eqimis 

yoveldRiuri gamoZaxebebis saSualo ufro meti vidre klinikaSi mosuli 

pacientebis saSualo. qvemoTmoyvanili SerCevebis monacemebis mixedviT, 

0.01 mniSvnelovnebis doniT, SeiZleba Tu ara davadasturoT, rom eqimis 

binaze gamoZaxebaTa saSualo metia klinikaSi mosul pacientTa saSualoze? 

binaze gamoZaxeba 

klinikaSi mosvla 

------------------------------------- -------------------------------------- 

25 42 57 5 28 37 

21 34 44 16 16 48 

6. kriminalists surs Seamowmos hipoTeza imis Sesaxeb, rom ugzoukvlod 

dakargul arasrulwlovan adaminTa saSualo metia vidre srulwlovanTa 

saSualo. qvemoTmoyvanili ukanaskneli 5 wlis monacemebis mixedviT, 

0.1 mniSvnelovnebis doniT, gvaqvs Tu ara sakmarisi safuZveli 

davadasturoT kriminalistis hipoTeza? 

arasrulwlovanebi 65513 65934 64213 61954 59167 

srulwlovanebi 31364 34478 36937 35946 38209 

7. finansuri deklaraciis mimRebi saxelmwifo firma (Sesabamisad, 

kerZo firma) saSualod 10 adamianis momsaxurebas andomebs 21 wuTs (Sesabamisad, 

14 adamianis momsaxurebas andomebs 27 wuTs), xolo Sesworebuli 

SerCeviTi standartuli gadaxraa 5.6 wuTi (Sesabamisad, 4.3 wuTi). 0.02 

mniSvnelovnebis doniT aris Tu ara gansxvaveba firmebis momsaxurebis 

saSualo droebs Soris? aageT 98%-iani ndobis intervali saSualoTa 


8. A tipis 7 sarecxi manqanis saSualo fasia 815 lari, xolo B tipis 

9 sarecxi manqanis saSualo fasia 845 lari. Sesworebuli SerCeviTi 

standartuli gadaxrebi ki Sesabamisad aris 19 lari da 9 lari. 0.05 

mniSvnelovnebis doniT SeiZleba Tu ara davaskvnaT, rom B tipis sarecxi 

manqana ufro Zviria? 

9. 12 eqimis (Sesabamisad, 27 meddis) dazRvevis saSualo Tviuri premia 

Seadgens 56 lars (Sesabamisad, 63 lars), xolo Sesworebuli SerCevi- 

Ti standartuli gadaxra eqimisa da meddisaTvis Sesabamisad tolia 3 

larisa da 5.75 laris. 0.05 mniSvnelovnebis doniT SeiZleba Tu ara 

davaskvnaT, rom meddis sadazRvevo premia ufro metia vidre eqimis? isargebleT 

P -mniSvnelobis meTodiT. 

10. jandacvis saministros monacemebiT dazRveuli (Sesabamisad, 

dauzRvevuli) mSobiare qalebi samSobiaroSi rCebian saSualod 2.3 dRis 

(Sesabamisad, 1.9 dRis) ganmavlobaSi. 16 - 16 mSobiare qalisagan Sedgenili 

162

ori jgufis samSobiaroSi gatarebuli dReebis Sesworebuli standartuli 

gadaxra qalebis orive kategoriisaTvis Seadgens 0.6 dRes. 0.01 

mniSvnelovnebis doniT SeamowmeT hipoTeza saSualoTa tolobis Sesaxeb. 

aageT 99%-iani ndobis intervali saSualoTa sxvaobisaTvis. isargebleT 


11. qvemoT moyvanilia dro, romelic sWirdeba 6 – 6 TeTr da yavisfer 

Tagvs, raTa iswavlos martiv labirinTSi garbena. 0.05 mniSvnelovnebis 

doniT gaarkvieT iwvevs Tu ara Tagvis feri labirinTSi garbenis 

Sesaswavlad saWiro droSi gansxvavebas? aageT 95%-iani ndobis intervali 


TeTri Tagvi 18 24 20 13 15 12 

yavisferi Tagvi 25 16 19 14 16 10 

pasuxebi: 

2 2 

2 2 

1. I). H0 : 1 2 

, H1 : 1 2 

; CV . . 4.03 ; f 1.93 ; H 

0 

; II). H0 : a1 a2 

, H1 : a1 a2 

; 

CV . . t 

2.101 ; t 4.2 ; ki -- H 

1; III). 7762 a1 a2 

2434 . 2. I). 

18,0.025 

2 2 

2 2 

H0 : 1 2 

, H1 : 1 2 

; CV . . 2.62 ; f 1.44 ; H 

0 

; II). H0 : a1 a2 

, H1 : a1 a2 

; 

2 2 

2 2 

CV . . t 

1.65 ; t 0.55 ; ara -- H 

0 

. 3. I). H : , H : ; CV . . 2.72 ; 

34,0.05 

0 1 2 

1 1 2 

f 5.06 ; H 1 

; II). H 0 

: a 1 

a 2 

, H 1 

: a 1 

a 2 

; CV . . t11,0.1 

1.363 ; t 1.24 ; ara -- H 0 

. 4. 

2 2 

2 2 

I). H0 : 1 2 

, H1 : 1 2 

; CV . . 3.88 ; f 1.78 ; H 

0 

; II). H0 : a1 a2 

, H1 : a1 a2 

; 

2 2 

2 2 

CV . . t 

2.33 ; t 1.26 ; ara -- H 

0 

. 5. I). H : , H : ; CV . . 14.94 ; 

36,0.01 

0 1 2 

1 1 2 

f 1.41 ; H 

0 

; II). H0 : a1 a2 

, H1 : a1 a2 

; CV . . t10,0.01 

2.764 ; t 1.45 ; ara -- H 

0 

. 6. 

2 2 

2 2 

I). H0 : 1 2 

, H1 : 1 2 

; CV . . 6.39 ; f 1.139 ; H 

0 

; II). H0 : a1 a2 

, H1 : a1 a2 

; 

2 2 

2 2 

CV . . t 

1.397 ; t 16.252 ; ki -- H 

1. 7. I). H : , H : ; CV . . 4.19 ; 

8,0.1 

0 1 2 

1 1 2 

f 1.7 ; H 0 

; II). H 0 

: a 1 

a 2 

, H 1 

: a 1 

a 2 

; CV . . t22,0.01 

2.508 ; t 2.97 ; ki -- H 1 

; III). 

2 2 

2 2 

11.1 a1 a2 

0.93 . 8. I). H0 : 1 2 

, H1 : 1 2 

; CV . . 4.65 ; f 4.46 ; H 

0 

; II). 

2 2 

H0 : a1 a2 

, H1 : a1 a2 

; CV . . t14,0.05 

1.761 ; t 4.2 ; ki -- H 

1. 9. I). H : , 

0 1 2 

2 2 

H1 : 1 2 

; f 3.67 ; 0.02 

1; II). H0 : a1 a2 

, 

H1 : a1 a2 

; df . . 11; t 4.98 ; P -mniSvneloba < 0.005 < ; ki -- H 

1. 10. I). 

2 2 

2 2 

H0 : 1 2 

, H1 : 1 2 

; f 1; P -mniSvneloba > 0.2 > 0.01 ; H 

0 

; II). 

H0 : a1 a2 

, H1 : a1 a2 

; df . . 30 ; t 1.89 ; 0.05 

0 

; III). 

2 2 

2 2 

0.15 a1 a2 

0.95 . 11. I). H0 : 1 2 

, H1 : 1 2 

; CV . . 7.15 ; f 1.23 ; H 

0 

; II). 

H0 : a1 a2 

, H1 : a1 a2 

; CV . . t10,0.025 

2.228 ; t 0.119 ; ara -- H 

0 

; III). 

5.9 a a 

6.5. 

1 2 

163

Tavi XI 

oramokrefiani amocanebi populaciaTa dispersiebisaTvis 

hipoTezaTa Semowmeba ori damoukidebeli normaluri 

populaciis dispersiebis Sesaxeb 

2 

2 

Na ( 

1, 1) 

da Na ( 

2, 2) 

damoukidebelia; orive populaciis parametrebi 

ucnobia. 

kriteriumi: 


2 2 

H : / 1 

2 2 

H : / 1 

2 2 

H : / 1 

0 1 2 

2 2 

0 1 2 

H : / 1 

0 1 2 

2 2 

0 1 2 

H : / 1 

164 

0 1 2 

2 2 

0 1 2 

H : / 1 

2 2 

hipoTeza: H0 : 1 / 2 

1 


'2 2 

S1 / 1 

kriteriumis statistika: F F( n 1, m1) 

-- fiSeris 

'2 2 

S2 / 2 

ganawileba n 1 da m 1 Tavisuflebis xarisxebiT (sadac n da m 

SerCevebis moculobebia) 

'2 

s1 

kriteriumis mniSvneloba T.V.: f 

'2 

s2 



2 2 

H1 : 1 / 2 

1 

f Fn 

1, m1, 

 

, 

2 2 

H1 : 1 / 2 

1 

0 f F n 1, 

m1, 

1 

, 

2 2 

H1 : 1 / 2 

1 

0 f F n 1, 

m1,1 

 

/ 2 an f Fn 

1, m1, /2 

(sadac Fkl 

,, 

aris k da l Tavisuflebis xarisxis mqone fiSeris ganawilebis 

zeda -kritikuli wertilia anu kritikuli mniSvneloba C.V.). 

(1 ) 

saimedoobis ndobis intervali dispersiaTa 

2 2 

1 

/ 

2 

fardobisaTvis 

'2 '2 2 2 '2 '2 

F s / s 

/ F s / s . 

n1, m1,1 /2 2 1 1 2 n1, m1, /2 2 1 

dispersiaTa tolobis Sesaxeb hipoTezaTa Semowmebis 

gamartivebuli procedura: 

2 2 

hipoTeza: H0 : 1 / 2 

1 


'2 '2 

max{ s1 , s2} 

kriteriumis statistikis mniSvneloba T.V.: f 

'2 '2 

min{ s1 , s2} 


-is uaryofis are)

H : / 1 

f Fkl 

,, 

, 

2 2 

1 1 2 

2 2 

H1 : 1 / 2 

1 

f Fkl 

, , /2 

(sadac Fkl 

,, 

aris k da l Tavisuflebis xarisxis mqone fiSeris ganawilebis 

zeda -kritikuli wertilia anu kritikuli mniSvneloba C.V.; 

'2 '2 

'2 '2 

k n 1 da lm 1, Tu s s , da piriqiT, km 1 da ln 1, Tu s s ). 

1 2 

(1 ) 

saimedoobis ndobis intervali dispersiaTa 

2 2 

1 

/ 

2 

fardobisaTvis 

2 2 

f F 

 

f / F ) 

f F . 

l. k,1 

/ 2 

( 

k, 

l, 

/ 2 1 

/ 

2 k, 

l, 

/ 2 

1 2 


magaliTi 1. mkvlevars ainteresebs aris Tu ara gansxvaveba mweveli 

da aramweveli adamianebis pulsis ricxvTa dispersiebs Soris. SemTxveviT 

SerCeuli 26 mwevelisagan da 18 aramwevelisagan Semdgari ori SerCevis 

Sesabamisi Sesworebuli SerCeviTi dispersiebia: s '2 

1 

36, s '2 

2 

10. 0.05 

mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi safuZveli davaskvnaT, 

rom dispersiebi gansxvavebulia? CaTvaleT, rom sidide ganawilebulia 

normalurad. 


2 2 

2 2 

H : , H : . visargebloT gamartivebuli proceduriT. vinaidan 

0 1 2 

1 1 2 

'2 '2 

kriteriumi ormxrivia, s1 s2 

da Tavisuflebis xarisxebia 26 – 1 = 25 , 18 

– 1 =17, amitom kritikuli mniSvneloba iqneba: CV . . Fkl 

, , /2 

F25,17,0.025 

2.56 . 

'2 '2 

gamovTvaloT kriteriumis mniSvneloba: TV . . s1 / s2 

36/10 3.6 . radganac 

3.6 > 2.56, amitom ZiriTadi hipoTeza unda ukuvagdoT, anu Cven gvaqvs sakmarisi 

safuZveli davaskvnaT, rom dispersiebi gansxvavebulia. 

magaliTi 2. sahaero gadazidvebis centris mtkicebiT amerikis aeroportebSi 

mgzavrTa ricxis dispersia ufro metia vidre evropis aeroportebSi 

mgzavrTa ricxis dispersia. 0.01 mniSvnelovnebis doniT gvaqvs 

Tu ara sakmarisi safuZveli davadasturoT es mosazreba? qvemoT moyvanilia 

SemTxveviT SerCeul aeroportebSi mgzavrebis ricxvi (gazomili milionebSi). 

CaTvaleT, rom sidide ganawilebulia normalurad. isargebleT 


amerikis aeroportebi evropis aeroportebi 

------------------------------------------- ------------------------------------------- 

36.8 73.5 60.7 51.2 

72.4 61.2 42.7 38.6 

60.5 40.1 


2 2 

2 2 

H : , H : . vipovoT TiToeuli jgufis Sesworebuli SerCeviTi 

0 1 2 

1 1 2 

dispersia: s '2 

1 

246.38 , s '2 

2 

95.87 . amitom kriteriumis mniSvneloba iqneba: 

165

'2 '2 

'2 '2 

TV . . s1 / s2 

246.38/95.87 2.57 . vinaidan, s1 s2 

, amitom k 61 5 da 

l 41 3. fiSeris ganawilebis cxrilidan amoviRoT Semdegi fragmenti: 

0.1 0.05 0.025 0.01 0.005 

F 

5.31 9.01 14.88 28.24 45.39 

5,3, 

vinaidan 2.57 < 5.31, amitom P -mniSvneloba > 0.1. Sesabamisad, P -mniSvneloba 

> 0.01 , amitom ZiriTadi hipoTeza ar unda ukuvagdoT, anu 

0.01 mniSvnelovnebis doniT Cven ara gvaqvs sakmarisi safuZveli davaskvnaT, 

rom amerikis aeroportebSi mgzavrTa ricxis dispersia ufro metia 

vidre evropis aeroportebSi mgzavrTa ricxis dispersia. 

magaliTi 3. ori damoukidebeli normaluri populaciidan aRebulia 

'2 

15-is toli moculobis SerCevebi, romelTaTvisac: xn 

132.45 , s1 123; 

'2 

ym 

128.06 , s2 95. SeamowmeT hipoTeza dispersiaTa tolobis Sesaxeb 

0.05 mniSvnelovnebis doniT da aageT 95%-iani ndobis intervali dispersiaTa 

fardobisaTvis. 


2 2 

2 2 

H : / 1, 

H : / 1. visargebloT gamartivebuli proceduriT. 

0 1 2 

vinaidan, 

s 

'2 '2 

1 2 

1 1 2 

s , amitom k n1 14 da l m1 14. gamovTvaloT kriteriumis 

mniSvneloba: f 123/95 1.3. radganac saqme gvaqvs marjvena calmxriv 

alternativasTan, amitom kritikuli area [ Fkl 

, , , ) [ F14,14,0.05, ) [2.48, ) . 

ramdenadac 1.3 < 2.48, amdenad nulovani hipoTezis uaryofis safuZveli 

ara gvaqvs, anu 0.05 mniSvnelovnebis doniT SegviZlia davaskvnaT, rom 

populaciaTa dispersiebi tolia. 

avagoT ndobis intervali. Tanafardobidan (1 ) 100% 95% gamomdinare 

0.05 , / 2 0.025 . amitom saZiebeli intervali iqneba: 

( f / F14,14,0.025 , f F14,14,0.025 

) , 

(1.3/ 2.9,1.3 2.9) anu (0.45,3.77) . 

magaliTi 4. Sesadarebelia ori dazgis muSaobis sizuste. pirvel 

dazgaze Catarebulia 25 dakvirveba, xolo meoreze ki 20. s '2 

1 

3.1, s '2 

2 

1.4 . 

SegviZlia Tu ara 0.05 mniSvnelovnebis doniT davaskvnaT, rom meore 

2 2 

dazga ufro zustia, vidre pirveli ( )? aageT 90%-iani ndobis inte- 

1 2 

rvali 

2 2 

1 

/ 

2 

fardobisaTvis. 


2 2 

2 2 

'2 '2 

H0 : 1 / 2 

1, 

H1 : 1 / 2 

1. gvaqvs: n 25 , m 20 ; s1 s2 

, k n1 24 da 

l m1 19; kritikuli mniSvneloba CV . . Fkl 

, , F24,19,0.05 

2.04 ; kritikuli 

C. R. [ F , ) [2.04, ); kriteriumis mniSvneloba: 

are 

24,19,0.05 

TV . . f 3.1/1.4 2.214 . 

166

vinaidan T. V. C. R. 

, amitom nulovan hipoTezas ukuvaggdebT da 

0.05 mniSvnelovnebis doniT vaskvniT, rom meore dazga ufro zustia, 

vidre pirveli. 

90%-iani ndobis intervali 

2 / 2 fardobisaTvis iqneba: 

1 2 

( f / F , f F ) = (1.08,4.51). 

24,19,0.05 24,19,0.05 

amocanebi 

1. isargebleT fiSeris ganawilebis cxrilebiT da ipoveT kritikuli 

mniSvnelobebi Tu mocemulia ori damoukidebeli normaluri SerCevis 

Sesworebuli SerCeviTi dispersiebi: a). s '2 

1 

128, n 23 ; s '2 

2 

162, m 16 ; 

kriteriumi ormxrivia da 0.01 . b). s '2 

1 

37 , n 14 ; s '2 

2 

89, m 25; kriteriumi 

marjvena calmxrivia da 0.01 . g). s1 232 , n 30 ; s2 387 , 

'2 

'2 

m 46 ; kriteriumi ormxrivia da 0.05 . d). s '2 

1 

164, n 21; s '2 

2 

53, m 17 ; 

kriteriumi ormxrivia da 0.1. e). s '2 

1 

92.8 , n 11; s '2 

2 

43.6 , m 11; kriteriumi 

marjvena calmxrivia da 0.05 . 

2. ipoveT P -mniSvneloba F kriteriumis TiToeuli mniSvnelobisaTvis: 

a). F 2.97, n 9, m 14 , kriteriumi marjvena calmxrivia; b). F 3.32 , 

n 6 , m 12 , kriteriumi ormxrivia; g). F 2.28, n 12, m 20 , kriteriumi marjvena 

calmxrivia; d). F 3.51, n 12, m 21, kriteriumi marjvena calmxrivia; 

e). F 4.07, n 6, m 10 , kriteriumi ormxrivia; v). F 1.65, n 19, m 28, 

kriteriumi marjvena calmxrivia; z). F 1.77, n 28, m 28 , kriteriumi marjvena 

calmxrivia; T). F 7.29, n 5, m 8, kriteriumi ormxrivia. 

qvemoT moyvanil amocanebSi igulisxmeba, rom yvela sidide ganawilebulia 

normalurad: 

3. pedagogis mtkicebiT, im SemTxvevaSi, roca leqciebis kursi Serwymulia 

kompiuteruli damuSavebis komponentTan, studentTa sagamocdo 

qulebis dispersia ufro didia, vidre kompiuteruli damuSavebis gareSe. 

SemTxveviT SeirCa studentTa ori jgufi. sagamocdo qulebis Sesworebuli 

SerCeviTi dispersia studentebisaTvis, romelTa saleqcio kursi Serwymuli 

iyo kompiuteruli damuSavebis komponentTan aRmoCnda 103, xolo 

meore jgufisaTvis ki 73. TiToeuli SerCeva Sedgeba 20 – 20 studentisagan. 

0.05 mniSvnelovnebis doniT SeiZleba Tu ara davadasturoT pedagogis 

mtkicebuleba? 

4. momxmarebelTa interesebis damcvelis mtkicebiT A da B kompaniebis 

mier warmoebuli elementebis muSaobis xangrZlivobis dispersiebi 

ar gansxvavdeba erTmaneTisagan. SemTxveviT SerCeuli iqna TiToeuli tipis 

10 -10 elementi da aRmoCnda, rom Sesabamisad s '2 A 

24 da s '2 B 

40 . 0.1 

mniSvnelovnebis doniT SeiZleba Tu ara davadasturoT momxmarebelTa 

interesebis damcvelis mtkicebuleba? 

167

5. sagadasaxado inspeqtoris mtkicebiT or A da B qalaqSi dabegvrisagan 

Tavisufali Semosavlebis dispersiebi gansxvavebulia. qvemoT moyvanilia 

am qalaqebSi dabegvrisagan Tavisufali Semosavlebis SerCevebi 

(Semosavlebi gazomilia milion dolarebSi). 0.05 mniSvnelovnebis doniT 

SeiZleba Tu ara davadasturoT inspeqtoris mtkicebuleba? 

qalaqi A 

qalaqi B 

------------------------------------- --------------------------------------- 

113 25 44 31 82 295 12 20 

22 23 11 19 11 50 68 16 

14 23 19 5 5 12 81 4 

8 30 7 2 15 9 2 5 

6. saavadmyofos SemTxveviT SerCeul 20 derefanSi xmauris donis 

Sesworebuli SerCeviTi standartuli gadaxra aRmoCnda 4.1 decibali, xolo 

SemTxveviT SerCeul 24 oTaxSi xmauris donis Sesworebuli SerCevi- 

Ti standartuli gadaxra ki 7.5 decibali. 0.05 mniSvnelovnebis doniT 

SeiZleba Tu ara imis mtkiceba, rom standartuli gadaxrebi gansxvavebulia? 

7. ginekologis mtkicebiT axalSobili vaJebis simaRleTa cvalebadoba 

gansxvavdeba axalSobili gogonebis simaRleTa cvalebadobisagan. SemTxveviT 

SerCeuli 15 axalSobili vaJis simaRleTa Sesworebuli standartuli 

gadaxra aRmoCnda 1.3 diumi (1 diumi = 2.54 sm), xolo SemTxveviT 

SerCeuli 15 axalSobili gogonasaTvis ki 0.9 diumi. 0.1 mniSvnelovnebis 

doniT SeiZleba Tu ara davadasturoT ginekologis mtkicebuleba? 

8. mkvlevars ainteresebs aris Tu ara gansxvaveba dedaqalaqSi Semosuli 

avtomobilebisa da satranzito avtomobilebis ricxvebis cvalebadobas 

Soris. qvemoT moyvanili SerCevebis mixedviT, 0.05 mniSvnelovnebis 

doniT, SegviZlia Tu ara davaskvnaT, rom dispersiebi gansxvavebulia? 

ara satranzito 

satranzito 

------------------------------------------- --------------------------------------- 

3694560 3469431 2855109 2774251 3453745 499043 

3768285 4420553 719934 1068107 204369 253956 

1838271 1005469 3534092 456123 826710 

3358175 1112722 2016046 235752 133619 

9. mkvlevaris mtkicebiT arteriuli wnevis cvalebadoba Warbwonian 

adamianebSi ufro didia vidre normalurwonian adamianebSi. SemTxveviT 

SerCeul 28 Warbwonian adanmianSi arteriuli wnevis Sesworebuli standartuli 

gadaxra aRmoCnda 6.2 mm. vercx. sv., xolo 25 normalurwonian adamianSi 

ki 2.7 mm. vercx. sv.. 0.01 mniSvnelovnebis doniT SeamowmeT 

mkvlevaris mtkicebuleba. 

10. ganaTlebis saministros mtkicebiT pedagogebis swavlebis staJis 

ricxvis cvalebadoba umaRles saswavleblebSi ufro didia vidre saSualo 

skolebSi. umaRlesi saswavleblis SemTxveviT SerCeuli 26 pedagogis 

muSaobis staJis Sesworebuli standartuli gadaxra aRmoCnda 2.8 weli, 

xolo saSualo skolis 18 pedagogisaTvis ki 1.9 weli. 0.1 mniSvnelov- 

168

nebis doniT SeuZlia Tu ara mkvlevars daadasturos Tavisi mtkicebulebis 

marTebuleba? 

11. mkvlevars surs Seafasos ori popularuli A da B dietis Sedegad 

pacientTa mier dakarguli wonis dispersia. A dietis mimdevari Sem- 

TxveviT SerCeuli 10 adamianis mier erT TveSi dakarguli wonebis Sesworebuli 

standartuli gadaxra aRmoCnda 6.3 funti (1 funti = 453.6 gr), 

xolo B dietis mimdevari SemTxveviT SerCeuli 12 adamianis ki 4.8 funti. 

0.05 mniSvnelovnebis doniT, SegviZlia Tu ara davaskvnaT, rom A dietis 

SemTxvevaSi dakarguli wonis dispersia ufro didia vidre B dietis 

SemTxvevaSi? 

12. mkvlevaris mtkicebiT indianas StatSi Semavali sagrafoebis farTobebis 

dispersia naklebia vidre aiovas StatSi Semavali sagrafoebis 

farTobebis dispersia. qvemoT moyvanilia SemTxveviT SerCeuli sagrafoebis 

farTobebi (gazomili kvadratul kilometrebSi). 0.01 mniSvnelovnebis 

doniT, SegviZlia Tu ara davadasturoT mkvlevaris mtkicebuleba? 

indianas Stati 

aiovas Stati 

-------------------------------------- --------------------------------------- 

406 431 305 373 560 640 443 717 571 568 

393 430 215 148 384 580 569 568 577 434 

396 369 489 306 320 431 779 714 503 615 

485 408 293 509 407 416 381 731 501 402 


rom denverSi maRali Senobebis simaRleebis dispersia tolia detroitSi 

maRali Senobebis simaRleebis dispersiis, Tu saxlebis Sesabamisi SerCevebis 

futebSi gazomili (1 futi = 30.48 sm) simaRleebia: 

denveri 

detroiti 

------------------------------ ------------------------------ 

714 504 404 620 562 534 

698 438 544 472 448 436 

408 430 420 

14. qvemoT moyvanilia unciebSi gazomili (1 uncia = 28.3 gr) qalebisa 

da kacebis sportuli fexsacmlebis SerCevebis wonebi. gamoTvaleT Ti- 

Toeuli SerCevis Sesworebuli dispersia da SeamowmeT hipoTeza dispersiebis 

tolobis Sesaxeb 0.05 mniSvnelovnebis doniT. isargebleT P - 


qalebi 

kacebi 

---------------------------------------- ---------------------------------------- 

11.9 11.2 11.7 12.8 12.6 10.6 10.3 10.3 9.5 9.3 

12.3 13.8 10.8 14.5 14.7 9.6 9.8 11.2 8.8 9.5 

9.2 10.4 11.1 13.3 12.9 10.1 9.4 9.5 10.2 11.0 

169

15. qvemoT moyvanilia A da B tipis mtversasrutebis SerCevebis 

wonebi (gazomili funtebSi). 0.05 mniSvnelovnebis doniT SeamowmeT 

hipoTeza dispersiebis gansxvavebulobis Sesaxeb. 

A tipi 

B tipi 

------------------------------------ -------------------- 

21 16 23 13 18 24 12 

17 17 16 15 17 15 13 

15 17 16 20 20 11 

17 18 

pasuxebi: 

1. CV . . F15,22,0.01 

3.36 ; CV . . F24,13,0.01 

3.59 ; CV . . F45,29,0.05 

2.03 ; CV . . F20,16,0.1 

2.28 ; 

CV . . F 

2.98 . 2. 0.025 

10,10,0.05 

0.1; P -mniSvneloba = = 0.05; 0.005 

0.05; P -mniSvneloba > > 0.1; 0.05 

< 0.02. 3. H : , H : ; CV . . 2.23 ; f 1.41 ; ara. 

2 2 

2 2 

4. 

H : , 

2 2 

0 1 2 

2 2 

1 1 2 

0 1 2 

1 1 2 

H : ; CV . . 3.18 ; f 1.67 ; ki. 5. 

H : , 

2 2 

0 1 2 

H : ; 

2 2 

1 1 2 

CV . . 2.86 ; f 7.85 ; ki. 6. H0 : 1 2 

, H1 : 1 2 

; CV . . 2.51 ; f 3.346 ; ki. 7. 

2 2 

2 2 

2 2 

2 2 

H : , H : ; CV . . 2.53 ; f 2.09 ; ara. 8. H : , H : ; 

0 1 2 

1 1 2 

0 1 2 

1 1 2 

2 2 

2 2 

CV . . 3.53 ; f 1.88 ; ara. 9. H0 : 1 2 

, H1 : 1 2 

; CV . . 2.66 ; f 5.27 ; H 

1. 10. 

2 2 

2 2 

2 2 

2 2 

H : , H : ; CV . . 1.84 ; f 1.47 ; ara. 11. H : , H : ; 

0 1 2 

1 1 2 

CV . . 2.9 ; f 1.72 ; ara. 12. 

H : , 

2 2 

0 1 2 

2 2 

1 1 2 

2 2 

2 2 

H0 : 1 2 

, H1 : 1 2 

; CV . . 3.87 ; f 3.18 ; H 

0 

. 14. 

f 5.32 ; P -mniSvneloba < 0.01 < 0.05 ; H 

1. 15. 

CV . . 3.5 ; f 3.773 ; H 1 

. 

0 1 2 

1 1 2 

H : ; CV . . 3.15 ; f 1.45 ; ara. 13. 

H : , 

2 2 

0 1 2 

H : , 

2 2 

0 1 2 

H : ; 

2 2 

1 1 2 

H : ; 

2 2 

1 1 2 

170

Tavi XII 

statistikuri daskvnebi dawyvilebuli monacemebisaTvis 

statistikuri daskvnebi dawyvilebul monacemTa saSualoebis 

sxvaobebisaTvis (damokidebuli SerCevebi) 

monacemebi Sedgeba n damoukideblad SerCeuli wyvilisagan ( X1, Y 

1) 

, 

. . . , ( Xn, Y 

n) 

Di Xi Yi 

, i 1,..., n. D 1 

,..., D 

n 

warmoadgens damoukidebel normalurad 

ganawilebul SemTxveviT sidideTa mimdevrobas ucnobi parametrebiT 

aD 

a1 a2 

da 2 D 

(aq X 1 

,..., X 

n 

da Y 1 

,..., Y m 

-- damokidebuli SerCevebia). 

n 

1 

Dn 

: Di 

, '2 n 2 n 2 n 

1 1 1 

S 2 

n D 

( ) [ ( ) ] 

i 

Di Dn 

Di Di 

1 n 1 

i1 n 1 

; 

 

i1 n i1 

d 

n 

n 

1 1 

n : d i 

( x i 

y i) 

, '2 n 2 n 2 n 

1 1 1 

s 2 

n i1 n 

D ( di d n) [ di ( di) ] 

i1 

n 1 

i1 n 1 

. 

 

i1 n i1 

hipoTeza: H 0 

: aD 

0 


Dn 

aD 

kriteriumis statistika: T T( n 1) 

' 

S / n 

kriteriumis mniSvneloba T.V.: 

171 

D 

dn 

aD 

t , sadac dn 

-iT aRniSnulia 

s / n 

D n statistikis dakvirvebuli mniSvneloba. 



H1 : aD 

0 

t tn 

1, 

 

, 

H1 : aD 

0 

t tn 

1, 

 

, 

H1 : aD 

0 

t tn 

1, /2 

an t 

tn 

1, /2 

(sadac tn 1, 



1 FT 

( t), Tu H1 : a1 a2 

0; 

 

P - mniSvneloba: P FT 

( t), Tu H1 : a1 a2 

0; 

 

2 [1 FT 

(| t |)], Tu H1 : a1 a2 

0. 





0 



gvaqvs. 

(1 ) 

saimedoobis ndobis intervali dawyvilebuli monacemebis sa- 

SualoTa aD 

a1 a2 

sxvaobisaTvis: 

' 

D

s 

n 

s 

n 

' 

' 

D 

D 

d n tn 

1, /2 aD d n tn 

1, /2. 


magaliTi 1. farmacevtis mtkicebiT konkretuli vitamini zrdis aTletis 

mier simZimis awevis SesaZleblobas. SemTxveviT SearCies 8 aTleti 

da Seamowmes ra maqsimaluri simZimis aweva SeuZlia TiToeul maTgans 

vitaminis miRebamde da vitaminis erTi Tvis ganmavlobaSi miRebis Semdeg. 

igulisxmeT, rom sidide daaxloebiT normaluria da 0.05 mniSvnelovnebis 

doniT SeamowmeT vitaminis efeqturoba. 

aTletis # 1 2 3 4 5 6 7 8 

vit. miRebamde x i 

210 230 182 205 262 253 219 216 

vit. miRebis Semdeg y i 

219 236 179 204 270 250 222 216 

amoxsna. vitaminis efeqturoba niSnavs, rom vitaminis miRebamde aweuli 

simZime unda iyos arsebiTad (mniSvnelovnad) naklebi, vidre vitaminis 

miRebis Semdeg. Sesabamisad ZiriTadi da alternatiuli hipoTezebi iqneba: 

H 0 

: aD 

0, H 1 

: aD 

0. 

gamovTvaloT kritikuli mniSvneloba. aq Tavisuflebis xarisxi iqneba 

d. f . n181 7. amitom CV . . tn 

1, t7,0.05 

1.895 . 

gamovTvaloT kriteriumis mniSvneloba. a). SevavsoT Semdegi cxrili: 

aTletis vit. vit. miRebis 

2 2 

di xi yi 

di ( xi yi 

) 

# miRebamde, x i 

Semdeg, y i 

1 210 219 -9 81 

2 230 236 -6 36 

3 182 179 3 9 

4 205 204 1 1 

5 262 270 -8 64 

6 253 250 3 9 

7 219 222 -3 9 

8 216 216 0 0 

d 19 

2 

d 209 

 

i 

 

i 

b). vipovoT sxvaobaTa saSualo: 1 8 19 

d 8 d 2. 375 

8 1 8 

 

i 

; 

i 

g). vipovoT sxvaobaTa Sesworebuli SerCeviTi standartuli gadaxra: 

172

s 1 1 1 1 

[ d ( d ) ] [209 ( 19) ] 4.84 

n 

n 

' 2 2 2 

D i i 

n1 i1 n i1 

81 8 

; 

d). vipovoT kriteriumis mniSvneloba: 

dn 

aD 

2.375 0 

t 1.388 . 

' 

sD 

/ n 4.84/ 8 

vinaidan kriteriumis mniSvneloba ar ekuTvnis kritikul ares -- 

C. R. ( , t 

] ( , 1.895] (-1.388 > -1.895), amitom Cven ara gvaqvs nulovani 

9,0.05 

hipoTezis uaryofis safuZveli, anu ar arsebobs sakmarisi safuZveli, 

raTa davaskvnaT, rom vitamini aZlierebs aTletis SesaZleblobebs. 

magaliTi 2. dietologs surs gaarkvios Seicvleba Tu ara pacientis 

organizmSi qolesterinis done sakvebSi garkveuli mineralis damatebis 

Sdegad. SemTxveviT SearCies 6 pacienti da Seamowmes qolesterinis 

done mineralis miRebamde da mineralis miRebis Semdeg. igulisxmeT, rom 

sidide daaxloebiT normaluria. 0.1 mniSvnelovnebis doniT SegviZlia 

Tu ara davaskvnaT, rom qolesterinis done Seicvala? aageT 90%-iani 

ndobis intervali saSualoTa sxvaobisaTvis. 

pacientis # 1 2 3 4 5 6 

mineralis miRebamde, x i 

210 235 208 190 172 244 

min. miRebis Semdeg, y i 

190 170 210 188 173 228 


H0 : aD 

0, H 1 

: aD 

0. 

gamovTvaloT kritikuli mniSvneloba. aq Tavisuflebis xarisxi iqneba 

d. f . n1 61 5. kriteriumi ormxrivia. amitom kritikuli mniSvnelobebia: 

CV . . tn 

1, /2 

t5,0.05 

2.015 . 

gamovTvaloT kriteriumis mniSvneloba. a). SevavsoT Semdegi cxrili: 

pacientis min. min. miRebis 

2 2 

di xi yi 

di ( xi yi 

) 

# miRebamde, x i 

Semdeg, y i 

1 210 190 20 400 

2 235 170 65 4225 

3 208 210 -2 4 

4 190 188 2 4 

5 172 173 -1 1 

6 244 228 16 256 

d 100 

2 

d 4890 

g). vipovoT sxvaobaTa Sesworebuli SerCeviTi standartuli gadaxra: 

b). vipovoT sxvaobaTa saSualo: 

173 

d 

6 

 

i 

6 

1 100 

di 

6 i1 

6 

 

: 16.7 

; 

i

s 1 1 1 1 

[ d ( d ) ] [4890 (100) ] 25.4 

n 

n 

' 2 2 2 

D i i 

n1 i1 n i1 

61 6 

; 

d). vipovoT kriteriumis mniSvneloba: 

dn 

aD 

16.7 0 

t 1.61 . 

' 

sD 

/ n 25.4/ 6 

vinaidan kriteriumis mniSvneloba ar ekuTvnis kritikul ares -- 

C. R. ( , t ] [ t , ) ( , 2.015] [2.015, ) 

( 2.015 1.61 2.015 ), amitom 

5,0.05 5,0.05 

Cven ara gvaqvs nulovani hipoTezis uaryofis safuZveli, anu ar arsebobs 

sakmarisi safuZveli, raTa davaskvnaT, rom minerali cvlis qolesterinis 

dones. 

avagoT ndobis intervali. SevitanoT saWiro sidideebi Sesabamis 

formulaSi. gvaqvs: 

25.4 25.4 

16.7 2.015 a D 

16.7 2.015 

, 

6 6 

4.19 a D 

37.59 . 

rogorc vxedavT 0 Sedis ndobis intervalSi. Sesabamisad, gadwyvetileba 

emTxveva zemoT miRebuls: Cven ara gavaqvs safuZveli nulovani hipoTezis 

uaryofis. 

magaliTi 3. 0.05 mniSvnelovnebis doniT gaarkvieT Tu ra gavlenas 

axdens sqesi im kursdamTavrebulTaTvis SeTavazebuli xelfasis sidideze, 

romelTac gaaCniaT erTnairi ganaTleba, asaki da samuSao gamocdileba 

da aageT 90%-iani ndobis intervali saSualoTa sxvaobisaTvis, Tu 

maTTvis SeTavazebuli dRiuri xelfasebis monacemebia: 

wyvilis nomeri 1 2 3 4 5 6 7 8 9 10 

qali, X i 

22 17 21 19 26 23 21 31 25 18 

vaJi, Y i 

25 18 27 17 29 25 19 27 36 23 

sxvaobebi, D i 

-3 -1 -6 2 -3 -2 2 4 -11 -5 


H0 : aD 

0, H1 : aD 

0. gamovTvaloT sxvaobebis SerCeviTi saSualo da 

Sesworebuli SerCeviTi dispersia. gvaqvs: 

d 1 

10 ( 3 1 5) 2.3 

10 

, 1 

s '2 [( 3 2.3) 2 ( 5 2.3) 2 

D 

] 19.78 . 

9 

Sesabamisad, kriteriumis mniSvneloba iqneba: 

dn 

aD 

2.3 0 

T. V. t 1.635 . 

' 

sD 

/ n 19.78 / 10 

ramdenadac kriteriumi marcxena calmxrivia, amitom kritikuli mni- 

Svneloba iqneba: CV . . tn 

1, t9,0.05 

1.833 , xolo kritikuli area: 

CR . . ( , 1.833] . rogroc vxedavT, T. V. C. R. 

, amitom nulovani hipoTezis 

174

uaryofis safuZveli ara gvaqvs, anu ar arsebobs sakmarisi safuZveli 

raTa davaskvnaT, rom xelfasze gavlenas axdens sqesi. 

saZiebeli ndobis intervali iqneba: 

19.78 19.78 

( 2.3 t9,0.05, 2.3 t9,0.05 

) , 

10 10 

( 2.3 1.4064 1.833, 2.3 1.4064 1.833) , 

(-4.278, 0.278). 

amocanebi 

qvemoTmoyvanil amocanebSi igulisxmeba, rom yvela sidide ganawilebulia 

normalurad an daaxloebiT normalurad 

1. restoranTa qselis menejerma TanamSromlebs SesTavaza gamajansaRebeli 

programa, raTa Seemcirebina avadmyofobis gamo samuSaos gacdenaTa 

ricxvi. qvemoT naCvenebia SemTxveviT SerCeuli 10 muSakis mier Tvis 

ganmavlobaSi samuSaos gacdenaTa ricxvi gamajansaRebeli programis gavlamde 

da gavlis Semdeg. 0.05 mniSvnelovnebis doniT SegviZlia Tu 

ara davaskvnaT, rom programam gaamarTla? 

programis gavlamde 2 3 6 7 4 5 3 1 0 0 

programis gavlis Semdeg 1 4 3 8 3 3 1 0 1 0 

2. im mizniT, rom gaezardaT studentis mier kviraSi damoukidebeli 

muSaobis saaTebi, SemTxveviT SearCies 9 studenti da waukiTxes leqcia 

ganaTlebis mniSvnelobis Sesaxeb cxovrebaSi. qvemoT naCvenebia maTi 

kvireuli damoukidebeli muSaobis saaTebi leqciamde da leqciis mosmenis 

Semdeg. 0.1 mniSvnelovnebis doniT SegviZlia Tu ara davaskvnaT, 

rom leqciis mosmenis Semdeg gaizarda studentis damoukidebeli muSaobis 

saaTebi? 

leqciis mosmenamde 9 12 6 15 3 18 10 13 7 

leqciis mosmenis Semdeg 9 17 9 20 2 21 15 22 6 

3. profesors ainteresebs ra gavlenas axdens savarjiSoebis Sesaxeb 

filmi studentis savarjiSos mimarT damokidebulebis xarisxze. SemTxveviT 

SerCeuli 10 studentis gamokiTxvis Sedegebis mixedviT (ufro didi 

ricxvi gviCvenebs damokidebulebis ufro maRal xarisxs), 0.05 mni- 

Svnelovnebis doniT, gvaqvs Tu ara sakmarisi safuZveli davaskvnaT, rom 

filmis naxvis Semdeg damokidebuleba Seicvala? aageT 95%-iani ndobis 

intervali saSualoTa sxvaobisaTvis. 

filmis naxvamde 12 11 14 9 8 6 8 5 4 7 

filmis naxvis Semdeg 13 12 10 9 8 8 7 6 5 5 

175

4. mkvlevars ainteresebs axali dietis gavlena sisxlSi natriumis 

doneze. SemTxveviT SearCies 10 pacienti da gazomes sisxlSi natriumis 

done dietamde da dietis Semdeg. 0.05 mniSvnelovnebis doniT, gvaqvs 

Tu ara sakmarisi safuZveli davaskvnaT, rom dietis Semdeg sisxlSi natriumis 

done Semcirda? 

pacienti 1 2 3 4 5 6 7 8 9 10 

dietamde 146 138 152 163 136 147 148 141 143 142 

Semdeg 135 133 147 156 138 141 139 132 138 131 

5. ofisis menejers surs gaarkvios gazrdis Tu ara mdivnebis muSaobis 

siswrafes sabeWdi manqanebis Secvla kompiuterebiT. qvemoT naCvenebia 

10 mdivnis mier wuTSi dabeWdili sityvebis raodenoba sabeWd manqanaze 

da kompiuterze. 0.1 mniSvnelovnebis doniT, gvaqvs Tu ara sakmarisi 

safuZveli davaskvnaT, rom kompiuterze gadasvla zrdis dabeWdili 

sityvebis raodenobas? 

mdivani 1 2 3 4 5 6 7 8 9 10 

sabeWdi manqana 63 72 85 97 82 101 73 62 58 75 

kompiuteri 68 80 95 93 80 106 82 78 65 83 

6. mSobliuri enis pedagogs surs gaarkvios Seamcirebs Tu ara gramatikis 

axali programa im gramatikuli Secdomebis raodenobas, romelsac 

uSveben studentebi orgverdiani Txuzulebis dawerisas. qvemoTmoyvanili 

monacemebis mixedviT, 0.025 mniSvnelovnebis doniT, SegviZlia 

Tu ara davaskvnaT, rom Secdomebis raodenoba Semcirda? 

studenti 1 2 3 4 5 6 

Secdomebi programamde 12 9 0 5 4 3 

Secdomebi programis Semdeg 9 6 1 3 2 3 

7. fexsacmlis mwarmoebelis mtkicebiT is sportsmenebi, romlebsac 

acviaT misi firmis mier gamoSvebuli fexsacmeli darbian ufro swrafad 

vidre sxva sportsmenebi. qvemoTmoyvanili SemTxveviT SerCeuli 8 morbenalis 

siswarafis monacemebis mixedviT, 0.025 mniSvnelovnebis doniT, 

SegviZlia Tu ara davadasturoT mwarmoeblis mtkicebuleba? 

morbenali 1 2 3 4 5 6 7 8 

firmis fexsacmeliT 8.2 6.3 9.2 8.6 6.8 8.7 8 6.9 

sxva fexsacmeliT 7.1 6.8 9.8 8 5.8 8 7.4 8 

8. mkvlevars ainteresebs Seadaros tyupebis pulsTa ricxvi erTmaneTs. 

SerCeul iqna tyupebis rva wyvili da maTi pulsTa ricxvis monacemebi 

wuTSi moyvanilia qvemoT. 0.01 mniSvnelovnebis doniT aris Tu ara 

pulsTa ricxvs Soris gansxvaveba arsebiTi? aageT 99%-iani ndobis 

176

intervali saSualoTa sxvaobisaTvis. isargebleT P -mniSvnelobis meTodiT. 

tyupiscali I 87 92 78 83 88 90 84 93 

tyupiscali II 83 95 79 83 86 93 80 86 

9. uZravi qonebis agentis mtkicebiT dedaqalaqSi samoTaxiani binis 

fasebi Seicvala. qvemoT moyvanilia SemTxveviT SerCeuli 16 binis saeqsperto 

fasebi (gazomili aTasobiT larebSi) Sesabamisad 2003 da 2008 wleb- 

Si. 0.05 mniSvnelovnebis doniT SegviZlia Tu ara davaskvnaT, rom binebis 

saeqsperto fasebi Seicvala? isargebleT P -mniSvnelobis meTodiT. 

2003 2008 2003 2008 2003 2008 2003 2008 

----------------------------------------------------------------------------------------------------------- 

184 161 116 120 282 297 12 20 

414 382 49 52 25 40 37 38 

22 22 24 28 141 148 9 9 

99 109 50 50 45 56 17 19 

pasuxebi: 

1. H0 : aD 

0, H1 : aD 

0; CV . . t9,0.05 

1.833 ; t 1.56 ; ara -- H 

0 

. 2. H0 : aD 

0, 

H1 : aD 

0; CV . . t8,0.1 

1.397 ; t 2.8 ; ki -- H 

1. 3. H0 : aD 

0, H1 : aD 

0; 

CV . . t9,0.025 

2.262 ; t 0.176 ; ara -- H 

0 

; ( 1.18,1.38) . 4. H0 : aD 

0, H1 : aD 

0; 

CV . . t9,0.05 

1.833 ; t 5.435 ; ara -- H 

1. 5. H0 : aD 

0, H1 : aD 

0; 

CV . . t9,0.1 

1.383 ; t 3.4 ; ki -- H 

1. 6. H0 : aD 

0, H1 : aD 

0; CV . . t5,0.025 

2.571 ; 

t 2.24 ; ki -- H 

0 

. 7. H0 : aD 

0, H1 : aD 

0; CV . . t7,0.025 

2.365 ; t 0.765 ; ara -- 

H 

0 

. 8. H0 : aD 

0, H 1 

: aD 

0; df . . 7; t 0.978 ; < 0.2 

H 

0 

. 9. H 0 

: aD 

0, H 1 

: aD 

0; df . . 15 ; t 0.5 ; P -mniSvneloba > 0.5 > ; ara - 

H . 

- 0 

177

Tavi XIII 

oramokrefiani amocanebi bernulis sqemaSi 

HhipoTezaTa Semowmeba warmatebaTa albaTobebisaTvis bernulis 

cdaTa ori damoukidebeli mimdevrobisaTvis 

X1,..., X 

n 

da Y 1 

,..., Y m 

ori damoukidebeli SerCevaa bernulis kanoniT 

ganawilebuli populaciidan Sesabamisad warmatebis ucnobi p 1 

da p 2 

albaTobebiT; 

q11 

p1, q2 1 p2; S1 

Xi 

, S2 

Yi 

; P1 

, P2 

; 

n 

m 

S1 

S2 

i1 

i1 

n m 

s1 

1 n s2 

1 

p1 

x 

m n m 

i 

, p2 

yi 

n n i 

; P P1 

P2; n m 

p p1 

p2; 

1 m m i 1 nm nm 

nm nm 

Q1 P; q1 p. 

kriteriumi: 


H0 : p1 p2 

H0 : p1 p2 

H0 : p1 p2 

an H0 : p1 p2 

an H0 : p1 p2 

H : p p 

H : p p 

H : p p 

1 1 

 

2 

1 1 

 

2 

1 1 2 

hipoTeza: H 0 

: p 1 

p 2 


kriteriumis statistika: 

( P1 P2) ( p1 

p2) 

as 

Z 

N(0,1) 

PQ(1/ n1/ m) 

( p1 p2) ( p1 

p2) 


pq(1/ n1/ m) 



H1 : p1 p2 

z z , 

H1 : p1 p2 

z z , 

H1 : p1 p2 

z z /2 

an z 

z /2 

(sadac z aris N(0,1) 

-is zeda -kritikuli wertili anu kritikuli mniSvneloba 

C.V.). 

1 ( z), Tu H1 : a1 a2 

0; 

 

P - mniSvneloba: P ( z), Tu H1 : a1 a2 

0; 

= 

 

2 [1 (| z |)], Tu H1 : a1 a2 

0. 




178


0 



gvaqvs. 

(1 ) 

saimedoobis ndobis intervali p 1 

p 2 

sxvaobisaTvis: 

p1q1 p2q2 p1q1 p2q2 

(( p1 p2) z/2 , ( p1 p2) z/2 

) 

n m n m 

(sadac q 1 

1 p 1 

, q 2 

1 p 2 

). 

SezRudvebi: np1 , nq1 , mp2 , mq2 

5. 


magaliTi 1. mkvlevarma daadgina, rom 34 mcire klinikidan 12-Si, iseve 

rogorc 24 didi klinikidan 17-Si, dakavebuli sawolebis wili naklebia 

80%-ze. 0.05 mniSvnelovnebis doniT SeamowmeT hipoTeza imis Sesaxeb, 

rom proporciebi im mcire da did klinikebis romlebSic dakavebuli 

sawolebis wili naklebia 80%-ze erTidaigivea. aageT 95%-iani 

ndobis intervali proporciaTa sxvaobisaTvis. 


H0 : p1 p2 

, H1 : p1 p2 

. aRvniSnoT p 1 

(Sesabamisad, p 2 

) simboloTi proporcia 

im mcire (Sesabamisad, im didi) klinikebis romlebSic dakavebuli sawolebis 

wili naklebia 80%-ze. gvaqvs: 

s1 

12 s 

p1 

0.35 

2 

17 

da p2 

0.71 . 

n 34 

m 24 

n m 34 24 

amitom p p1 p2 

0.35 0.71 0.5 , q1 p 0.5 . 

nm nm 

34 24 34 24 

vipovoT kritikuli mniSvneloba. radaganac kriteriumi 

ormxrivia da 0.05 , kritikuli mniSvnelobebi iqneba: 

CV . . z 

/2 

z0.025 

1.96 . 


( p1 p2) ( p1 p2) (0.35 0.71) 0 

T. V. z 2.7 

. 

pq(1/ n1/ m) 

0.5 0.5 (1/34 1/ 24) 

vinaidan -2.7 < -1.96 (e.i. T. V. C. R. 

), amitom Cven unda ukuvagdoT 

nulovani hipoTeza, anu Cven gvaqvs sakmarisi safuZveli uarvyoT 

hipoTeza imis Sesaxeb, rom proporciebi ar gansxvavdeba. 

avagoT ndobis intervali. aq q11 p1 0.65 , q2 1 p2 

0.29 . 

amitom saZiebeli ndobis intervali iqneba: 

0.35 0.65 0.71 0.29 0.35 0.65 0.71 

0.29 

((0.35 0.71) 1.96 , (0.35 0.71) 1.96 ) 

34 24 34 24 

, 

179

( 0.36 0.242, 0.36 0.242) , 

( 0.602, 0.118) . 

vinaidan ndobis intervali moicavs 0-s, gadawyvetileba isev iqneba: 

ukuvagdoT nulovani hipoTeza. 

magaliTi 2. SemTxveviT SerCeul 50 adamians erTi Tvis ganmavloba- 

Si asmevdnen garkveul wamals, ramac gamoiwvia maT 60%-Si qolesterinis 

donis Semcireba. sxva SemTxveviT SerCeul 80 adamians erTi Tvis ganmavlobaSi 

asmevdnen sxva wamals, ramac gamoiwvia maT 25%-Si qolesterinis 

donis Semcireba. 0.01 mniSvnelovnebis doniT aris Tu ara proporciebs 

Soris mniSvnelovani gansxvaveba? amoxseniT rogorc Cveulebrivi, ise 

P -mniSvnelobis maTodiT. 


H0 : p1 p2 

, H 1 

: p 1 

p 2 

. 

gvaqvs: n 50 , m 80 , p 1 

60% 0.6 , p 2 

25% 0.25 . gamovTvaloT p : 

n m 50 50 

p p1 p2 

0.6 0.25 0.385 . 

nm nm 

50 80 50 80 

Sesabamisad, q1 p 0.615 . 

vipovoT kritikuli mniSvneloba. radaganac kriteriumi ormxrivia 

da 0.01 , kritikuli mniSvnelobebi iqneba: CV . . z 

/2 

z0.005 

2.58 . 


( p1 p2) ( p1 p2) (0.6 0.25) 0 

T. V. z 3.99 . 

pq(1/ n1/ m) 

0.385 0.615 (1/50 1/80) 

radganac 3.99 > 2.58 (e.i. T. V. C. R. 

), Cven unda ukuvagdoT nulovani 

hipoTeza, anu Cven gvaqvs sakmarisi safuZveli davaskvnaT, rom proporciebs 

Soris gansxvaveba mniSvnelovania. 

visargebloT P -mniSvnelobis meTodiT: normaluri ganawilebis 

cxrilidan vpoulobT, rom P -mniSvneloba iqneba: 

2 [1 (| z |)] 2 [1 (3.99)] 2 (1 0.999) 0.002 . 

vinaidan P -mniSvneloba = 0.002 < 0.01 , amitom gadawyvetileba isev 

iqneba: 0.01 mniSvnelovnebis doniT proporciebs Soris gansxvaveba 

mniSvnelovania. 

amocanebi 

1. muSaTa dasaxlebidan 150 SemTxveviT SerCeul adamians Soris 80-s 

awuxebs filtvebis avadmyofoba, xolo 100 SemTxveviT SerCeuli soflis 

mosaxlidan 30-s awuxebs filtvebis avadmyofoba. 0.05 mniSvnelovnebis 

doniT aris Tu ara gnsxvaveba am or dasaxlebaSi filtvebis avadmyofobis 

mqone adamianebis proporciebs Soris? 

2. gamokiTxvis Tanaxmad SemTxveviT SerCeuli 80 eqimidan 15 mwevelia, 

xolo 50 iuristidan ki sigarets eweva 5. 0.05 mniSvnelovnebis 

180

doniT aris Tu ara eqimebSi mwevelTa proporcia ufro maRali vidre iuristebSi? 

3. SemTxveviT SerCeuli 100 momxmareblidan 43 angariSsworebisaTvis 

iyenebs “masterqards”, xolo SemTxveviT SerCeuli sxva 100 momxmareblidan 

58 angariSsworebisaTvis iyenebs “vizaqards”. 0.05 mniSvnelovnebis 

doniT aris Tu ara gnsxvaveba im momxmarebelTa proporciebs Soris, 

romlebic angariSsworebisaTvis iyeneben sxvadasxva tipis sakredito 

baraTebs? 

4. TbilisSi SemTxveviT SerCeuli 73 fostalionidan erTi kviris 

ganmavlobaSi 10 dakbenil iqna mawanwala ZaRlis mier, xolo quTaisSi ki 

SemTxveviT SerCeuli 80 fostalionidan 16 dakbenil iqna ZaRlis mier. 

0.05 mniSvnelovnebis doniT aris Tu ara mniSvnelovani gansxvaveba 

proporciebs Soris? aageT 95%-iani ndobis intervali proporciaTa 


5. gamokiTxvam aCvena, rom gamokiTxuli mamakacebis 83% leqciasTan 

SedarebiT upiratesobas aniWebs kompiuterul swavlebas, xolo igive maCvenebeli 

qalebisaTvis Seadgens 75%-s. TiToeuli populaciisaTvis aRebuli 

iyo 100 – 100 adamianisagan Semdgari SerCeva. 0.05 mniSvnelovnebis 

doniT SeamowmeT hipoTeza imis Sesaxeb, rom mamakacebsa da qalebSi ar 

arsebobs gansxvaveba im adamianebis proporciebs Soris, romlebic leqciasTan 

SedarebiT upiratesobas aniWeben kompiuterul swavlebas. aageT 

95%-iani ndobis intervali proporciaTa sxvaobisaTvis. 

6. SemTxveviT SerCeuli 200 qirurgidan 15%-s miaCnia, rom janmrTelobis 

dacvaze kontrols unda axorcielebdes saxelmwifo, xolo SemTxveviT 

SerCeuli 200 Terapevtidan ase fiqrobs 21%. 0.1 mniSvnelovnebis 

doniT aris Tu ara gansxvaveba proporciebs Soris? aageT 90%-iani 


7. 80 gamokiTxuli amerikelidan 55% Tvlis, rom is mdidaria, xolo 

90 gamokiTxuli evropelidan 45% Tvlis, rom is mdidaria. 0.01 mni- 

Svnelovnebis doniT aris Tu ara gansxvaveba proporciebs Soris? aageT 


8. 200 gamokiTxuli mamakacidan 130 xmarobs usafrTxoebis Rveds, 

xolo 300 gamokiTxuli qalidan 63 xmarobs usafrTxoebis Rveds. 0.01 

mniSvnelovnebis doniT SeamowmeT hipoTeza imis Sesaxeb, rom mamakacebi 

ufro seriozulad ekidebian usafrTxoebas, vidre qalebi. isargebleT P - 


9. 80 gamokiTxuli Tbiliselidan 45-s aqvs kondicioneri, xolo 120 

gamokiTxuli quTaiselidan 63-s aqvs kondicioneri. 0.05 mniSvnelovnebis 

doniT aris Tu ara gansxvaveba proporciebs Soris? aageT 95%-iani 


10. 100 bavSvisgan Semdgar SerCevaSi 30% namyofia cirkSi. sxva 100 

bavSvisagan Semdgar SerCevaSi 24% namyofia cirkSi. 0.02 mniSvnelovnebis 

doniT aris Tu ara gansxvaveba proporciebs Soris? isargebleT P - 


181

11. 200 gamokiTxuli mozardidan 50-s sjera, rom omi Seuqcevadia, ma- 

Sin roca 300 asakovani adamianidan 93-s sjera, rom omi Seuqcevadia. 

0.01 mniSvnelovnebis doniT aris Tu ara gansxvaveba proporciebs Soris? 

aageT 99%-iani ndobis intervali proporciaTa sxvaobisaTvis. 

12. 50 pirvelkurseli studentidan 8-s aqvs sakuTari avtomanqana, 

xolo 75 meoTxekurselidan ki 20-s. 0.05 mniSvnelovnebis doniT Segvi- 

Zlia Tu ara, rom sakuTari avtomobilis mqone studentebis proporcia 

meoTxekurselebSi ufro maRalia? isargebleT P -mniSvnelobis meTodiT. 


igulisxmeT, rom sidideebi ganawilebulia normalurad an daaxloebiT 

normalurad 

13. SemTxveviT SerCeuli 100 – 100 qirurgisa da stomatologis saSualo 

wliuri Semosavali Seadgens Sesabamisad 54107-sa da 58417 lars. orive 

SerCevis Sesworebuli standartuli gadaxraa 81 lari. 0.1 mniSvnelovnebis 

doniT SeamowmeT hipoTeza imis Sesaxeb, rom am populaciaTa 

saSualoebi ar gansxvavdeba. aageT 90%-iani ndobis intervali saSualoTa 


14. gamokiTxuli martoxela da daqorwinebuli mZRolebis ori jgufis 

mier kviris ganmavlobaSi ganvlili manZilebis qvemoT moyvanili monacemebis 

mixedviT 0.01 mniSvnelovnebis doniT SeiZleba Tu ara davaskvnaT, 

rom martoxela mZRolebi saSualod ufro mets gadian kviraSi daqorwinebulebTan 

SedarebiT? 

martoxela mZRolebi 

daqorwinebuli mZRolebi 

------------------------------------------------- --------------------------------------------------- 

106 119 110 115 108 97 133 139 140 101 

110 97 117 114 117 115 113 104 120 108 

115 118 116 103 152 136 114 109 119 138 

121 122 138 98 147 119 117 113 116 147 

132 135 142 99 117 99 102 136 145 113 

154 107 86 133 115 113 106 108 115 96 

138 116 142 104 140 114 150 135 88 105 

15. ganaTlebis inspeqtors ainteresebs Seadaros moswavleebze daxarjuli 

Tanxebis dispersiebi qalaqad da soflad. Sesabamisi SemTxveviTi 

'2 

SerCevebidan miRebuli Sesworebuli SerCeviTi dispersiebia: s1 585, 

'2 

s2 261 (SerCevaTa moculobebia: n 18, m 16 ). 0.05 mniSvnelovnebis 

doniT aris Tu ara mniSvnelovani gansxvaveba populaciaTa dispersiebs 

Soris? 

16. mkvlevars surs Seadaros I da II ligis fexburTelebis simaRleebis 

dispersiebi. TiToeuli ligidan SemTxveviT SearCies 25 – 25 moTama- 

182

Se da maTi simaRleebis Sesworebuli SerCeviTi dispersiebi aRmoCnda Sesabamisad 

2.25 da 4.85. 0.1 mniSvnelovnebis doniT aris Tu ara mniSvnelovani 

gansxvaveba I da II ligis fexburTelebis simaRleebis dispersiebs 

Soris? 

17. klinikis intensiuri Terapiis SemTxveviT SerCeul 11 oTaxSi xmauris 

donis Sesworebuli standartuli gadaxra aRmoCnda 4.2 decibali, 

xolo SemTxveviT SerCeul arasamkurnalo daniSnulebis 24 oTaxSi -- 

13.2 decibali. 0.1 mniSvnelovnebis doniT aris Tu ara mniSvnelovani 

gansxvaveba klinikis samkurnalo da arasamkurnalo daniSnulebis oTaxebSi 

xmauris donis standartul gadaxrebs Soris? 

18. SemTxveviT SerCeuli 30 dRis ganmavlobaSi dedaqalaqis A skolaSi 

gakveTilebis gacdenaTa Sesworebuli standartuli gadaxra aRmoCnda 

4.9, xolo dedaqalaqis B skolaSi – 2.5. 0.01 mniSvnelovnebis doniT 

SegviZlia Tu ara davaskvnaT, rom standartuli gadaxrebi gansxvavebulia? 

19. mkvlevaris varaudiT, rom qarxnis muSakTa mier avadmyofobis gamo 

weliwadSi gacdenili dReebis raodenobis cvalebadoba ufro metia 

vidre klinikis muSakTaTvis. didi klinikis SemTxveviT SerCeuli 42 muSakzisaTvis 

Sesworebuli standartuli gadaxra aRmoCnda 2.1 dRe, xolo 

didi qarxnis SemTxveviT SerCeuli 65 muSakisaTvis – 3.2 dRe. SeamowmeT 

hipoTeza 0.1 mniSvnelovnebis doniT. 

20. 15 saxis sokos sufis saSualo Rirebuleba Seadgens 0.73 lars, 

xolo Sesworebuli SerCeviTi standartuli gadaxra ki 0.05 lars. SemTxveviT 

SerCeuli 24 saxeobis qaTmis sufisaTvis analogiuri maxasiaTeblebi 

Sesabamisad aris 0.91 lari da 0.03 lari. 0.01 mniSvnelovnebis doniT 

aris Tu ara fasebs Soris gansxvaveba mniSvnelovani? 

21. SemTxveviT SerCeul 25 dRian periodSi A da B saqonelze arsebuli 

fasebis monacemebis mixedviT, 0.1 mniSvnelovnebis doniT, SeiZleba 

Tu ara davaskvnaT, rom A saqoneli ufro Zviria, vidre B saqoneli? 

A saqoneli 

B saqoneli 

---------------------------------------- ----------------------------------------- 

78 82 68 67 68 70 74 73 60 77 

75 73 75 64 68 71 72 71 74 76 

62 73 77 78 79 71 80 65 70 83 

74 72 73 78 68 67 76 75 62 65 

73 79 82 71 66 66 65 77 66 64 

22. SemTxveviT SerCeuli 15 evropeli pedagogis saSualo wliuri 

Semosavali Seadgens 35270$-s, xolo Sesworebuli standartuli gadaxra 

tolia 3256$-is. analogiuri maxasiaTeblebi SemTxveviT SerCeuli 30 amerikeli 

pedagogisaTvis Sesabamisad tolia 29512$-isa da 1432$-is. aris Tu 

ara mniSvnelovani gansxvaveba Semosavlebs Soris 0.02 mniSvnelovnebis 

doniT? aageT 99%-iani ndobis intervali saSualoTa sxvaobisaTvis. 

183

23. dedaqalaqis SemTxveviT SerCeuli 16 ojaxis saSualo wliuri 

Semosavali aris 54356 lari, xolo Sesworebuli standartuli gadaxra 

ki 8256 lari. gareubnis SemTxveviT SerCeuli 12 ojaxis saSualo wliuri 

Semosavali Sesadgens 46512 lars, xolo Sesworebuli standartuli 

gadaxra tolia 1311 laris. 0.05 mniSvnelovnebis doniT SeiZleba Tu 

ara davaskvnaT, rom dedaqalaqis mcxovrebTa wliuri Semosavali metia 

vidre gareubanSi mcxovreblebis? isargebleT P -mniSvnelobis meTodiT? 

24. pedagogs surs gazardos 10 studentis sityvebis maragi ucxo enaSi. 

am mizniT igi maT kviraSi erTjer utarebs damatebiT seminarebs. damatebiTi 

seminarebis Catarebamde da seminarebis Catarebis Semdeg Catarebuli 

testirebis qvemoTmoyvanili Sedegebis mixedviT, 0.01 mniSvnelovnebis 

doniT, SeuZlia Tu ara pedagogs davaskvnas, rom seminarebma 

Sedegi gamoiRo? 

studentis nomeri 1 2 3 4 5 6 7 8 9 10 

seminarebis 83 76 92 64 82 68 70 71 72 63 

Catarebamde 

seminarebis Semdeg 88 82 100 72 81 75 79 68 81 70 

25. Sromis nayofierebis gazrdis mizniT avtonawilebis saamqros menejerma 

gadawyvita samuSao adgilas samuSao drois ganmavlobaSi CaerTo 

msubuqi musika. SemTxveviT SearCies rva muSa da daTvales maT mier 

gamoSvebuli detalebis raodenoba musikis CarTvamde da musikis CarTvidan 

erTi kviris Semdeg. qvemoT moyvanili Sesabamisi monacemebis mixedviT, 

0.05 mniSvnelovnebis doniT, SeuZlia Tu ara menejers davaskvnas, rom 

musikam gazarda Sromis nayofiereba? 

muSis nomeri 1 2 3 4 5 6 7 8 

musikis CarTvamde 6 8 10 9 5 12 9 7 

musikis Semdeg 10 12 9 12 8 13 8 10 

26. 365 dRidan londonSi 207 dRe nisliania, xolo stokholmSi ki 

166 dRe. 0.02 mniSvnelovnebis doniT, SegviZlia Tu ara davaskvnaT, 

rom nisliani dReebis proporcia am or qalqSi gansxvavebulia? aageT 


27. 50 gamokiTxuli dedaqalaqis mcxovrebidan 32 aqvs mikrotalRovani 

qura, xolo 60 gamokiTxuli gareubanSi mcxovrebidan 24 aqvs mikrotalRovani 

qura. 0.05 mniSvnelovnebis doniT, SegviZlia Tu ara vTqvaT, 

rom proporciebi erTidaigivea? aageT 95%-iani ndobis intervali proporciaTa 


184

pasuxebi: 

1. H0 : p1 p2 

, H1 : p1 p2 

; CV . . z0.025 

1.96 ; z 3.64 ; ki -- H 

1. 2. H0 : p1 p2 

, 

H1 : p1 p2 

; CV . . z0.05 

1.65 ; z 1.36 ; ki -- H 

1. 3. H0 : p1 p2 

, H1 : p1 p2 

; 

CV . . z0.025 

1.96 ; z 2.12 ; ki -- H 

1. 4. H0 : p1 p2 

, H1 : p1 p2 

; 

CV . . z0.025 

1.96 ; z 0.99 ; ara -- H 

0 

; (-0.181, 0.055). 5. H0 : p1 p2 

, H1 : p1 p2 

; 

CV . . z0.025 

1.96 ; z 1.39 ; H 

0 

; (-0.032, 0.192). 6. H0 : p1 p2 

, H1 : p1 p2 

; 

CV . . z0.05 

1.65 ; z 1.56 ; ara -- H 

0 

; (-0.123, 0.003). 7. H0 : p1 p2 

, H1 : p1 p2 

; 

CV . . z0.005 

2.58 ; z 1.302 ; ara -- H 

0 

; (-0.097, 0.297). 8. H0 : p1 p2 

, H 1 

: p 1 

p 2 

; 

z 9.9 ; P -mniSvneloba < 0.001 < 0.01 ; H 

1. 9. H0 : p1 p2 

, H1 : p1 p2 

; 

CV . . z0.025 

1.96 ; z 0.521 ; ara -- H 

0 

; (-0.103, 0.178). 10. H0 : p1 p2 

, H 1 

: p 1 

p 2 

; 

z 0.96 ; P -mniSvneloba 0.337 > 0.02 ; ara -- H 

0 

. 11. H0 : p1 p2 

, H1 : p1 p2 

; 

CV . . z0.005 

2.58 ; z 1.45 ; ara -- H 

0 

; (-0.165, 0.045). 12. H0 : p1 p2 

, H 1 

: p 1 

p 2 

; 

z 1.41; P -mniSvneloba = 0.0793 > ; ara -- H 

0 

. 13. H0 : 1 2 

, H1 : 1 2 

; 

CV . . z0.05 

1.65 ; z 3.76 ; H 

1; (-62, -24.2). 14. H0 : 1 2 

, H1 : 1 2 

; 

2 2 

2 2 

CV . . z0.01 

2.33; z 0.59 ; H 

0 

. 15. H : , H : ; CV . . F17,15,0.025 

2.86 ; 

T. V. f 2.24 ; ara -- H 

0 

. 16. 

0 1 2 

H : , 

2 2 

0 1 2 

1 1 2 

H : ; CV . . F24,24,0.05 

1.98 ; 

2 2 

1 1 2 

T. V. f 2.16 ; ki -- H 

1. 17. H0 : 1 2 

, H1 : 1 2 

; CV . . F23,10,0.05 

2.77 ; 

TV . . f 10.365 ; ki -- H 

1. 18. H0 : 1 2 

, H1 : 1 2 

; CV . . F29,29,0.005 

2.76 ; 

TV . . f 3.84 ; ki -- H 

1. 19. 

T. V. f 2.32 ; H 

1. 20. I). 

H : , 

2 2 

0 1 2 

H : , 

2 2 

0 1 2 

H : ; CV . . F64,41,0.1 

1.47 ; 

2 2 

1 1 2 

H : ; CV . . F14,23,0.005 

3.47 ; 

2 2 

1 1 2 

TV . . f 2.78 ; H 

0 

; II). H0 : 1 2 

, H1 : 1 2 

; CV . . t37,0.005 

2.74 ; t 14.09 ; 

2 2 

2 2 

H 

1.21. I). H0 : 1 2 

, H1 : 1 2 

; CV . . 1.98 ; TV . . f 1.11 ; H 

0 

; II). H0 : 1 2 

, 

2 2 

2 2 

H1 : 1 2 

; CV . . 1.28 ; t 1.31; H 

1. 22. I). H0 : 1 2 

, H1 : 1 2 

; CV . . 2.87 ; 

TV . . f 5.17 ; H 1 

; II). H0 : 1 2 

, H 1 

: 1 

2 

; CV . . 2.624 ; df . . 14 ; t 6.54 ; H 1 

; 

2 2 

2 2 

III). (3447.8, 8068.2). 23. I). H0 : 1 2 

, H1 : 1 2 

; TV . . f 39.7 ; P -mniSvneloba 

< 0.05; H 1 

; II). H0 : 1 2 

, H 1 

: 1 

2 

; df . . 11; t 3.74 ; P -mniSvneloba < 0.005; 

H 

1. 24. H0 : aD 

0, H1 : aD 

0; CV . . t9,0.01 

2.821 ; T. V. t 

4.17 ; ki -- H 

1. 25. 

H0 : aD 

0, H 1 

: aD 

0; CV . . t7,0.05 

1.895 ; T. V. t 

2.73 ; ki -- H 1 

. 26. H0 : p1 p2 

, 

H1 : p1 p2 

; CV . . z0.01 

2.33 ; z 3.03 ; ki -- H 

1; (0.027, 0.198). 27. H0 : p1 p2 

, 

H1 : p1 p2 

; CV . . z0.025 

1.96 ; z 2.51 ; ara -- H 1 

; (0.058, 0.422). 

185

Tavi XIV 

Tanxmobis kriteriumebi 

xi-kvadrat Tanxmobis kriteriumis formula: 

2 

2 ( O 

E) 

, 

E 

sadac O -- dakvirvebuli sixSirea (observed frequency), E -- mosalodneli 

sixSire (expected frequency), xolo Tavisuflebis xarisxi tolia kategoria- 

Ta (klasTa) raodenobas ( n) gamoklebuli 1 ( d. f . n 1). 

SezRudvebi: 1. monacemebi miRebulia SemTxveviTi SerCevidan; 2. TiToeuli 

kategoriis mosalodneli sixSire unda iyos meti an toli 5-is 

(Tu ukanaskneli piroba ar sruldeba, unda moxdes kategoriis gaerTianeba 

sxva kategoriasTan ise, rom gaerTianebuli kategoriis mosalodneli 

sixSire gaxdes meti an toli 5-is), roca kategoriaTa raodenoba 2-ze 

metia (Tu gvaqvs mxolod 2 kategoria, maSin TiToeulis sixSire meti an 

toli unda iyos 10-is). 

2 

2 ( oi 

ei) 

kriteriumis mniSvneloba T.V. 

. 

i ei 

kriteriumi marjvena calmxrivia. 

2 

kritikuli mniSvneloba CV . . n 

1, . 

2 

kritikuli are C.R. ( H0 

-is uaryofis are) = [ n 

1, , ) 

2 

P - mniSvneloba = P{ ( n1) T. V.} 

gadawyvetileba: Tu kriteriumis mniSvneloba metia an toli kritikul 

mniSvnelobaze (T.V. C.V.), maSin nulovan hipoTezas ukuvagdebT, winaaRmdeg 



0 




magaliTi 1. maRaziis menejers ainteresebs aniWebs Tu ara momxmarebeli 

upiratesobas limonaTis xuTi gansxvavebuli gemodan romelimes. 100 

SemTxveviT SerCeuli momxmareblidan 32-ma aarCia alubalis, 28-m marwyvis, 

16-ma forToxalis, 14-ma feixos da 10-ma yurZnis gemos mqone limonaTi. 

0.05 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi safuZveli ukuvagdoT 

hipoTeza imis Sesaxeb, rom momxmarebeli ar aniWebs upiratesobas 

arc erT gemos? 

amoxsna. im SemTxvevaSi, Tu momxmarebeli ar aniWebs upiratesobas 

arc erT gemos, maSin unda velodoT, rom yvela gemos sixSire tolia, 

anu Cvens SemTxvevaSi mosalodneli sixSireebia: 100/5 = 20. Sesabamisad, 

Cven gvaqvs sixSireTa ganawilebis Semdegi cxrili: 

186

sixSire alubali marwyvi forToxali feixo yurZeni 

dakvirvebuli, o i 

32 28 16 14 10 

mosalodneli, e i 

20 20 20 20 20 

nabiji 1. CamovayaliboT ZiriTadi da alternatiuli hipoTezebi: 

H 

0 

: momxmareblebisaTvis mniSvneloba ara aqvs gemos. 

H 

1 

: momxmareblebisaTvis mniSvneloba aqvs gemos. 

nabiji 2. vipovoT kritikuli mniSvneloba. aq Tavisuflebis xarisxia 

5 – 1 = 4, 0.05 . Sesabamisad, xi-kvadrat ganawilebis cxrilidan 

2 2 

vpoulobT, rom CV . . n 

1, 4,0.05 

9.488 .. 

nabiji 3. gamovTvaloT kriteriumis mniSvneloba: 

2 2 2 2 

2 ( oi 

ei) (32 20) (28 20) (16 20) 

... 

 

e 20 20 20 

i 

i 

2 2 

(14 20) (10 20) 

18 . 

20 20 

nabiji 4. gadawyvetilebis miReba: vinaidan 18 > 9.488, amitom nulovani 

hipoTeza unda ukuvagdoT, anu Cven gvaqvs sakmarisi safuZveli raTa CavTvaloT, 

rom momxmareblebisaTvis mniSvneloba aqvs limonaTis gemos. 

magaliTi 2. statistikis departamentis monacemebiT, im adamianebidan 

vinc erTxel dakarga samuSao, 38% Soulobs axal samuSaos, 32% xdeba 

naxevraddasaqmebuli, 23% muSaobs kontraqtis gareSe da 7% iwyebs sakuTar 

bizness. imis gasarkvevad, es procentebi aris Tu ara mdgradi, gamoki- 

Txul iqna A kompaniidan daTxovnili 300 adamiani da aRmoCnda, rom 122 

iSova axali samuSao, 85 naxevraddasaqmebulia, 76 muSaobs kontraqtis gareSe 

da 17 adamianma daiwyo sakuTari biznesi. 0.1 mniSvnelovnebis doniT 

SeamowmeT hipoTeza procentebis mdgradobis Sesaxeb. isargebleT rogorc 

Cveulebrivi, ise P -mniSvnelobis meTodiT.misargebleT 

amoxsna. 


H 

0 

: meored dasaqmebul adamianTa ganawileba Semdegia: 38% muSaobs 

axal samsaxurSi, 32% -- naxevraddasaqmebulia, 23% muSaobs kontraqtis 

gareSe da 7% daiwyo sakuTari biznesi. 

H 

1 

: ganawileba ar aris iseTi rogorc nulovan hipoTezaSia. 

nabiji 2. vipovoT kritikuli mniSvneloba. am SemTxvevaSi Tavisuflebis 

xarisxi df . . 41 3, 0.1. amitom, xi-kvadrat ganawilebis cxrilidan 

vpoulobT, rom CV . . n 

1, 3,0.1 

6.251 .. 

2 2 

nabiji 3. gamovTvaloT kriteriumis mniSvneloba. mosalodnel sixSireTa 

mniSvnelobebi gamoiTvleba Semdegnairad: 

e1 300 0.38 114 , e 2 

300 0.32 96 , e 3 

300 0.23 69 , e 4 

300 0.07 21 . 

Sesabamisad, 

2 2 2 2 2 

2 ( oi 

ei) (122 114) (85 96) (76 69) (17 21) 

... 3.2939 . 

e 114 86 69 21 

i 

i 

187

nabiji 4. gadawyvetilebis miReba: radganac 3.2939 < 6.251, amitom ZiriTadi 

hipoTezis uaryofis safuZveli ara gvaqvs, anu procentebi ar aris 

mniSvnelovnad gansxvavebuli imisagan, rac Camoyalibebulia nulovan hipoTezaSi 

(procentebi mdgradia). 

visargebloT axla P -mniSvnelobis meTodiT: xi-kvadrat ganawilebis 

cxrilidan vpoulobT, rom P - mniSvneloba = P{ (3) 3.293} > 0.1 

2 

(radganac 2 3,0.1 

6.251 ). Sesabamisad, gadawyvetileba iqneba igive: ZiriTadi 

hipoTezis uaryofis safuZveli ara gvaqvs. 

magaliTi 3. universitetis ekologiuri klubis menejeris mtkicebiT 

klubis 10%-s Seadgens pirvelkurselebi, 20% -- meorekurselia, 40% -- mesamekurseli 

da 30% ki meoTxekurseli. mimdinare wels klubSi iricxeba 

14 pirvelkurseli, 19 meorekurseli, 51 mesamekurseli da 16 meoTxekurseli. 

0.1 mniSvnelovnebis doniT SeamowmeT menejeris hipoTeza. 

amoxsna. 


H 

0 

: ekologiuri klubis Semadgenlobis 10%, 20%, 40% da 30% 

Sesabamisad aris I, II, III da IV kurseli. 

H : ganawileba ar aris iseTi rogorc nulovan hipoTezaSia. 

1 

2 2 

nabiji 2. vipovoT kritikuli mniSvneloba: CV . . n 

1, 3,0.1 

6.251 .. 

nabiji 3. gamovTvaloT kriteriumis mniSvneloba. mosalodnel sixSireTa 

mniSvnelobebia: 

e1 100 0.1 10 , e 2 

100 0.2 20 , e 3 

100 0.4 40 , e 4 

100 0.3 30 . 

amitom 

2 2 2 2 2 

2 ( oi 

ei) (14 10) (19 20) (51 40) (16 30) 

... 11.208 

. 

i ei 

10 20 40 30 

nabiji 4. gadawyvetilebis miReba: radganac 11.208 > 6.251, amitom ZiriTad 

hipoTezas ukuvagdebT, anu Cven gvaqvs sakmarisi safuZveli uarvyoT 

menejeris mtkicebuleba klubis Semadgenlobis Sesaxeb. 

normalurobis Semowmebis xi-kvadrat kriteriumi 

magaliTi 5. xi-kvadrat kriteriumis gamoyenebiT SeamowmeT sidide, 

romlis sixSiruli ganawileba moyvanilia qvemoT, 0.05 mniSvnelovnebis 

doniT, aris Tu ara normalurad ganawilebuli. 

klasis sazRvrebi 

sixSire 

89.5—104.5 24 

104.5—119.5 62 

119.5—134.5 72 

134.5—149.5 26 

149.5—164.5 12 

164.5—179.5 4 

188

200 

amoxsna. 


H 

0 

: sidide ganawilebulia normalurad. 

H 

1 

: sidide ar aris ganawilebuli normalurad. 

nabiji 2. vipovoT sididis saSualo da Sesworebuli standartuli 

gadaxra. visargebloT Semdegi cxriliT: 

2 

sazRvrebi sixSire, f m 

Suawertilebi, x m 

fm 

xm 

fm 

xm 

------------------------------------------------------------------------------------------------------------------------ 

89.5—104.5 24 97 2328 225816 

104.5—119.5 62 112 6944 777728 

119.5—134.5 72 127 9144 1161288 

134.5—149.5 26 142 3692 524264 

149.5—164.5 12 157 1884 295788 

164.5—179.5 4 172 688 118336 

 

200 24680 3103220 

Sesabamisad gvaqvs: x 24680/200 123.4 da 

1 1 

s f x ( f x ) / n 3103220 24680 / 200 17.03 

n 

n 

' 2 2 2 

m m m m 

n1 m1 m1 

199 

nabiji 3. vipovoT klasis sazRvrebSi sididis moxvedris albaTobebi 

misi normalurad ganawilebulobis daSvebis SemTxvevaSi. amisaTvis, 

movaxdinoT sididis, da Sesabamisad klasis sazRvrebis standartiza- 

x 

x 

cia, anu TiToeuli sazRvari gadaviyvanoT e. w. z sazRvarSi z 

' 

s 

formulis mixedviT. maSin sazRvrebi 104.5, 119.5, 134.5, 149.5, 164.5 Seicvleba 

Sesabamisad sazRvrebiT: -1.11, -0.23, 0.65, 1.53, 2.41 da normaluri ganawilebis 

cxrilis gamoyenebiT gveqneba: 

P{ 104.5} P{ z 1.11} 0.1335 ; 

P{104.5 

119.5} P{ 1.11 z 0.23} 0.2755 ; 

P{119.5 

134.5} P{ 0.23 z 0.65} 0.3332 ; 

P{134.5 

149.5} P{0.65 z 1.53} 0.1948 ; 

P{149.5 

164.5} P{1.53 z 2.41} 0.055 ; P{ 164.5} P{ z 2.41} 0.008 . 

nabiji 4. vipovoT mosalodnel sixSireTa mniSvnelobebi, risTvisac 

zemoTmiRebuli albaTobebi gavamravloT 200-ze. mosalodnel sixSireTa 

mniSvnelobebia: 

e1 200 0.1335 26.7 , e 2 

200 0.2755 55.1 , e 3 

200 0.3332 66.64 , 

e4 200 0.1948 38.96 ; e 5 

200 0.055 11; e 6 

200 0.008 1.6 . 

SeniSvna: vinaidan ukanaskneli kategoriis mosalodneli sixSire 

naklebia 5-ze, is unda gaerTiandes wina kategoriasTan da sabolood 

gveqneba sixSireTa ganawilebis Semdegi cxrili: 

o 24 62 72 26 16 

i 

189

e 26.7 55.1 66.64 38.96 12.6 

i 

(aq 16 12 4 da 12.6 11 1.6 ) 


2 2 2 2 

2 ( oi 

ei) (24 26.7) (62 55.1) (72 66.64) 

 

 

e 26.7 55.1 66.64 

i 

i 

2 2 

(26 38.96) (16 12.6) 

6.797 

38.96 12.6 

nabiji 6. vipovoT kritikuli mniSvneloba. vinaidan kategoriaTa raodenoba 

gaxda 5 amitom Tavisuflebis xarisxi iqneba df . . 51 

4. Sesabamisad, 

CV . . n 

1, 4,0.05 

9.488 . 

2 2 

nabiji 7. gadawyvetilebis miReba: radganac 6.797 < 9.488, amitom nulovani 

hipoTezis uaryofis safuZveli ara gvaqvs, anu sididis ganawileba 

SeiZleba CaiTvalos, rom daaxloebiT normaluria. 

amocanebi 

1. gadaudebeli samedicino samsaxuris xelmZRvanels surs daadginos 

aris Tu ara Tanabrad ganawilebuli kviris ganmavlobaSi gamoZaxeba- 

Ta ricxvi. qvemoT moyvanilia SemTxveviT arCeuli kviris ganmavlobaSi 

samsaxurSi Semosul gamoZaxebaTa raodenobebi. 0.05 mniSvnelovnebis 

doniT gvaqvs Tu ara sakmarisi safuZveli uarvyoT hipoTeza imis Sesaxeb, 

rom kviris ganmavlobaSi gamoZaxebaTa ricxvi ganawilebulia Tanabrad? 

dRe orS. samS. oTxS. xuTS. paras. Sab. kvira 

gamoZ. raod. 28 32 15 14 38 43 19 

2. bavSvTa sawvimari labadis mwarmoebels surs gaarkvios aniWebs 

Tu ara momxmarebeli upiratesobas labadis romelime fers. SemTxveviT 

SearCies 50 gayiduli sawvimari labada da CainiSnes maTi feri. monacemebi 

moyvanilia qvemoT. 0.1 mniSvnelovnebis doniT eniWeba Tu ara upiratesoba 

romelime fers? 

feri yviTeli wiTeli mwvane lurji 

gayidv. raod. 17 13 8 12 

3. baris menejers surs gaarkvios aniWebs Tu ara momxmarebeli upiratesobas 

SarbaTSi romelime xilis gemos. qvemoTmoyvanili monacemebis 

mixedviT, 0.01 mniSvnelovnebis doniT, SegviZlia Tu ara davaskvnaT, 

rom momxmarebelTa mier SekveTil SarbaTSi xilis gemo ganawilebulia 

Tanabrad? 

gemo limoni forToxali marwyvi feixo 

Sekv. raod. 12 24 19 9 

4. bankis menejers ainteresebs drois romel monakveTs aniWeben bankis 

klientebi upiratesobas bankSi misasvlelad. SearCies 6 saaTi da qve- 

190

moT moyvanilia TiToeuli saaTis ganmavlobaSi bankSi mosuli klientebis 

raodenobebi. 0.05 mniSvnelovnebis doniT Cans Tu ara, rom bankis 

klientebi drois romelime monakveTs aniWeben upiratesobas? 

dro 10:00 11:00 12:00 13:00 14:00 15:00 

klient. raod. 26 33 42 36 24 19 

5. aSS-is wiTeli jvris sazogadoebis monacemebiT amerikelTa 42%-s 

aqvs O jgufis sisxli, 44%-s – A jgufis sisxli, 10%-s – B jgufis sisxli 

da 4%-s – AB jgufis sisxli. garkveuli sagrafos samedicino 

eqspertis mosazrebiT mis sagrafoSi sisxlis jgufis ganawileba igivea 

rac zogadnacionaluri. am sagrafodan SemTxveviT SerCeuli 200 

adamianis sisxlis jgufis monacemebis mixedviT, 0.1 mniSvnelovnebis 

doniT, SeamowmeT eqspertis mosazreba. 

jgufi A O B AB 

sixSire 58 65 55 22 

6. ekonomikisa da biznesis fakultetis dekanatis mtkicebiT studentTa 

saboloo Sefasebebi garkveul disciplinaSi ganawilebulia Semdegnairad: 

40% -- A, 30% -- B, 20% -- C, 5% -- D da 5% -- F. gasuli semestris bolos 

am disciplinaSi studentTa mier miRebuli Sefasebebis mixedviT, 

0.05 mniSvnelovnebis doniT, gansxvavdeba Tu ara am disciplinaSi saboloo 

Sefasebebi mosalodnelisagan?F 

Sefaseba A B C D E 

raodenoba 45 52 39 8 6 

7. kvlevis Tanaxmad momxmarebelTa 53% arCevs naRdi fulis gadaxdas 

salaroSi, 30% sargeblobs CekebiT, 16% iyenebs sakredito baraTebs, 

da 1%-saTvis ara aqvs mniSvneloba ra formiT gadaixdis. didi maRaziis 

mepatronem SemTxveviT SearCia 800 myidveli da SeekiTxa maT Tu ra formiT 

arCevdnen gadaxdas. aRmoCnda, rom maT Soris 400 ixdis naRdi fuliT, 

210 ixdis CekebiT, 170 sargeblobs sakrediti baraTebiT da 20-saTvis mniSvneloba 

ara aqvs gadaxdis formas. 0.01 mniSvnelovnebis doniT SeamowmeT 

hipoTeza imis Sesaxeb, rom am maRaziis momxmareblebis gadaxdis formebTan 

damokidebuleba emTxveva kvlevis Sedegebs. 

8. iaraRis maRaziis menejers surs daadginos Tu romel TveSi amjobineben 

momxmareblebi sanadiro Tofis yidvas. qvemoTmoyvanili gasuli 

wlis gayidvebis maCveneblebis mixedviT, 0.05 mniSvnelovnebis doniT, 

SeamowmeT hipoTeza imis Sesaxeb, rom arcerT Tves ara aqvs upiratesoba 

sanadiro Tofis yidvisas. 

Tve seqtemberi oqtomberi noemberi dekemberi 

gayidv. raod. 18 23 28 15 

9. programuli uzrunvelyofis departamentis menejers mosazrebiT 

misi momxmareblebis 50% yidulobs teqstis dasamuSavebel programebs, 

25% -- eleqtronuli gaTvlis programebs, da 25% -- monacemTa bazebis 

191

programebs. SemTxveviT SerCeul myidvelTa qvemoTmoyvanili monacemebis 

mixedviT, 0.05 mniSvnelovnebis doniT, sworia Tu ara menejeris mosazreba? 


programa teqstis damuS. eleqtr. gaTv. monac. bazebi 

gayidv. raod. 38 23 19 

10. sam monetas agdeben 72-jer. mosul gerbTa raodenobis qvemoTmoyvanili 

monacemebis mixedviT, 0.05 mniSvnelovnebis doniT, SeamowmeT 

hipoTeza imis Sesaxeb, rom monetebi wesieria. 

gerbTa raod. 0 1 2 3 

sixSire 3 10 17 42 

pasuxebi: 

2 

2 

1. H 

0 

: Tanabaria, H 1 

: araTanabaria; CV . . 6,0.05 

12.59 ; TV . . 

28.887 ; H 

1. 

2 

2 

2. H 

0 

: ar eniWeba, H 

1 

: eniWeba; CV . . 3,0.1 

6.251 ; TV . . 

3.28 ; H 

0 

. 3. H 

0 

: 

2 

ar eniWeba (Tanabaria), H 

1 

: eniWeba (Tanabaria); CV . . 3,0.01 

11.345 ; 

2 

TV . . 

8.625 ; ki -- H 

0 

. 4. H 

0 

: ar aniWeben, H 

1 

: aniWeben; 

2 

2 

CV . . 5,0.05 

11.071 ; TV . . 

12.067 ; ki -- H 

1. 5. H 

0 

: igivea, H 

1 

: 

2 

2 

gansxvavebulia; CV . . 3,0.1 

6.251 ; TV . . 

100.275 ; H 

1. 6. H 

0 

: igivea, H 

1 

: 

2 

2 

gansxvavebulia; CV . . 4,0.05 

9.488 ; TV . . 

7.872 ; ara -- H 

0 

. 7. H 

0 

: emTxveva, 

2 

2 

H 

1 

: ar emTxveva; CV . . 3,0.01 

11.345 ; TV . . 

36.8897 ; H 

1. 8. H 

0 

: ara aqvs 

2 

2 

upiratesoba, H 1 

: aqvs upiratesoba; CV . . 3,0.05 

7.815 ; TV . . 

4.67 ; H 

0 

. 9. 

2 

H 

0 

: sworia, H 1 

: arasworia; TV . . 

0.6 ; P -mniSvneloba > 0.1; ki -- H 

0 

. 10. 

2 

2 

H 

0 

: wesieria, H 1 

: arawesieria; CV . . 3,0.05 

7.815 ; TV . . 

139.4 ; H 1 

. 

192

Tavi XV 

damoukideblobis Semowmebis xi-kvadrat kriteriumi 

193 

i, j i, 

j 

mowmdeba erTi populaciis ori sxvadasxva A da B niSnis erTmane- 

TTan damokidebulebis sakiTxi. A niSani iyofa R kategoriad, xolo B 

niSani -- C kategoriad. 

nulovani hipoTeza: H 0 

: A da B niSnebi damoukidebelia 

alternativa: H 1 

: A da B niSnebi damokidebelia 


kriteriumis statistika: xi-kvadrati 

2 

( o 

2 

i, j 

ei , j) 


, sadac o 

e 

ij , 

-- dakvirvebuli 

sixSireebia, xolo e ij , 

ki mosalodneli sixSireebi; 

oi , j 

o 

i 

j i, 

j 

ei, 

j . 

o 

 

i, 

j 

i, 

j 

SezRudva: yvela eij 

, 

5 (winaaRmdeg SemTxvevaSi vaxdenT dajgufebas), 

roca cxrili 2X2-ze didia (2X2 cxrilis SemTxvevaSi ki yvela 

e i , j 

10). 


2 

kritikuli mniSvneloba CV . . k, 

, sadac Tavisuflebis xarisxi 

k ( R1)( C 1) 

2 


-is uaryofis are) = [ n 

1, , ) 

P - mniSvneloba = P{ 

2 ( k) 

T. 

V.} 





0 




magaliTi 1. 200 gamokiTxuli eqTanidan (Sesabamisad, eqimidan) 100 

(Sesabamisad, 50) upiratesobas aniWebs axal proceduras, 80 (Sesabamisad, 

120) – Zvel proceduras, xolo 20-saTvis (Sesabamisad, 30-saTvis) mniSvneloba 

ara aqvs romel proceduras gamoiyenebs. 0.05 mniSvnelovnebis doniT 

aris Tu ara eqTanisa da eqimis SexedulebebSi gansxvaveba? 

amoxsna. 


H 

0 

: proceduraze Sexeduleba damoukidebelia profesiisagan. 

H : proceduraze Sexeduleba damokidebelia profesiaze. 

1

nabiji 2. SevadginoT niSanTa SeuRlebis R C cxrili: 

amjobinebs amjobinebs 

jgufi axal proced. Zvel proced. erTidaigivea 

------------------------------------------------------------------------------------------------------------------ 

eqTnebi 100 80 20 

eqimebi 50 120 30 

ij , 

am SemTxvevaSi gvaqvs 2 striqoni da 3 sveti ( R2, C 3), anu gvaqvs 2 

3 

SeuRlebis cxrili. am cxrilis i -uri striqonisa da j -uri svetis 

gadakveTaze mdgomi elementi ( ( i, j ) blokSi mdgomi elementi) aRvniSnoT 

o -Ti. magaliTad, o 1,2 

80 . Sesabamisad, niSanTa SeuRlebis cxrili sqematurad 

ase gamoisaxeba: 

sveti 1 sveti 2 sveti 3 

---------------------------------------------------------------------------------------------------------- 

striqoni 1 

striqoni 2 

o 

1,1 

o 

1,2 

o 

1,3 

o 

2,1 

o 

2,2 

o 

2,3 

nabiji 3. vipovoT kritikuli mniSvneloba. aq Tavisuflebis xarisxi: 

d. f . ( R1)( C 1) (21) (31) 2. Sesabamisad, xi-kvadrat ganawilebis cxrilidan 

vpoulobT, rom: CV . . 2,0.05 

5.991 . 

2 

nabiji 4. gamovTvaloT striqonebSi da svetebSi mdgomi sidiideebisa 

da yvela sididis jamebi. 



----------------------------------------------------------------------------------------------------------------------- 

eqTnebi 100 80 20 

200 

eqimebi 50 120 30 

200 

150 200 

50 

400 

nabiji 5. TiToeuli ( i, j ) blokisaTvis gamovTvaloT mosalodnel 

sixSireTa mniSvnelobebi e ij , 

Semdegi formulis mixedviT: 

striqonisjami ×svetisjami 

mosalodneli mniSvneloba e ij , 

= 

. 

mTliani jami 

magaliTad, (1,1) blokisaTvis mosalodneli mniSvneloba e 1,1 

iqneba: 

(I striqonisjami) × ( I svetisjami) 200 150 e1,1 

75 . 

mTlianijami 

400 

analogiurad, danarCeni mosalodneli mniSvnelobebi iqneba: 

e1,2 100 , e 1,3 

25, e 2,1 

75 , e 2,2 

100 , e 2,3 

25 . 

TvalsaCinoebisaTvis mosalodneli mniSvnelobebi SeiZleba CavweroT 

frCxilebSi niSanTa SeuRlebis cxrilis Sesabamis blokSi: 

194



----------------------------------------------------------------------------------------------------------------------- 

eqTnebi 100 (75) 80 (100) 20 (25) 

200 

eqimebi 50 (75) 120 (100) 30 (25) 

200 

150 200 

50 

400 


2 2 2 2 

( o 

2 

i, j 

ei , j) (100 75) (80 100) (20 25) 

 

 

e 75 100 25 

ij , ij , 

2 2 2 

(50 75) (120 100) (30 25) 

26.67 . 

75 100 25 

nabiji 7. gadawyvetilebis miReba: vinaidan 26.67 > 5.991, nulovani hipoTeza 

unda ukuvagdoT, anu Cven gvaqvs sakmarisi safuZveli raTa davaskvnaT, 

rom profesiaze Sexeduleba damokidebulia profesiaze – e. i. eqTanisa 

da eqimis Sexeduleba proceduraze gansxvavebulia erTmaneTisagan. 

magaliTi 2. sociologs ainteresebs gaarkvios aris Tu ara kavSiri 

adamianis sacxovrebel adgilsa da mis mier universitetSi gatarebul 

wlebs Soris. SemTxveviT SerCeuli 88 adamianisaTvis niSanTa SeuRlebis 

cxrils aqvs Semdegi saxe: 

sacxovr. universit. bakalavris magistris 

adgili II kurseli xarisxi xarisxi jami 

qalaqi 15 12 8 35 

qalaqgareT 8 15 9 32 

sofeli 6 8 7 21 

jami 29 35 24 88 

0.05 mniSvnelovnebis doniT SeuZlia Tu ara sociologs daaskvnas, 

rom adamianis sacxovrebeli adgili dakavSirebulia mis mier 

universitetSi gatarebuli wlebis raodenobasTan? 

amoxsna. 


H 

0 

: adamianis sacxovrebeli adgili damoukidebelia universitetSi 

gatarebuli wlebis raodenobisagan. 

H 

1 

: adamianis sacxovrebeli adgili damokidebelia universitetSi 

gatarebul wlebis raodenobaze. 

nabiji 2. vipovoT kritikuli mniSvneloba. radganac Tavisuflebis 

xarisxia d. f . ( R1)( C 1) (31) (31) 4 da 0.05 , amitom kritikuli mni- 

2 

Svneloba iqneba: CV . . 4,0.05 

9.488 . 

nabiji 3. gamovTvaloT mosalodneli mniSvnelobebi. gvaqvs: 

35 

29 3535 35 

24 

e1,1 

11.53 , e 

88 

1,2 

13.92 , e 

88 

1,3 

9.55 , 

88 

195

32 

29 3235 32 

24 

e2,1 

10.55 , e 

88 

2,2 

12.73 , e 

88 

2,3 

8.73 , 

88 

21 

29 2135 21 

24 

e3,1 

6.92 , e 

88 

3,2 

8.35 , e 

88 

3,3 

5.73 . 

88 


2 2 2 2 

( o 

2 

i, j 

ei , j) (15 11.53) (12 13.92) (8 9.55) 

 

 

e 11.53 13.92 9.55 

ij , ij , 

2 2 2 

(8 10.55) (15 12.73) (9 8.73) 

 

10.55 12.73 8.73 

2 2 2 

(66.96) (8 8.35) (7 5.73) 

3.01 . 

6.96 8.35 5.73 

nabiji 5. gadawyvetilebis miReba: radganac 3.01 < 9.488, amitom sociologs 

ara aqvs safuZveli nulovani hipoTezis uaryofis, anu sociologs 

ara aqvs sakmarisi safuZveli raTa daaskvnas, rom adamianis sacxovrebeli 

adgili damokidebulia mis mier universitetSi gatarebuli wlebis 

raodenobaze. 

magaliTi 3. mkvlevars ainteresebs aris Tu ara kavSiri adamianis 

sqessa da mis mier miRebuli alkoholis raodenobas Soris. SemTxveviT 

SerCeuli 68 adamianisaTvis miRebuli monacemebis mixedviT, 0.1 mniSvnelovnebis 

doniT, SeuZlia Tu ara mkvlevars daaskvnas, rom miRebuli 

alkoholis raodenoba dakavSirebulia sqesTan? 

alkoholis moxmareba 

sqesi dabali saSualo maRali jami 

------------------------------------------------------------------------------------------------------ 

mamrob. 10 9 8 27 

mdedrob. 13 16 12 41 

jami 23 25 20 68 

amoxsna. 


H 

0 

: miRebuli alkoholis raodenoba damoukidebelia adamianis sqesisagan. 

H 

1 

: alkoholis raodenoba damokidebelia adamianis sqesze. 

nabiji 2. vipovoT kritikuli mniSvneloba. aq Tavisuflebis xarisxia 

d. f . ( R1)( C 1) (21) (31) 2 da 0.1. Sesabamisad, kritikuli mniSvneloba 

iqneba: CV . . 2,0.1 

4.605 . 

2 


27 

23 27 

25 27 

20 

e1,1 

9.13 , e 

68 

1,2 

9.93 , e 

68 

1,3 

7.94 , 

68 

41 

23 41 

25 41 

20 

e2,1 

13.87 , e 

68 

2,2 

15.07 , e 

68 

2,3 

12.06 . 

68 

Sesabamisad, dasrulebul SeuRlebis cxrils eqneba Semdegi saxe: 

196

alkoholis moxmareba 

sqesi dabali saSualo maRali jami 

------------------------------------------------------------------------------------------------------ 

mamrob. 10 (9.13) 9 (9.93) 8 (7.94) 27 

mdedrob. 13 (13.87) 16 (15.07) 12 (12.06) 41 

jami 23 24 20 68 


2 2 2 2 

( o 

2 

i, j 

ei , j) (10 9.13) (9 9.93) (8 7.94) 

 

 

e 9.13 9.93 7.94 

ij , ij , 

2 2 2 

(13 13.87) (16 15.07) (12 12.06) 

0.283 

13.87 15.07 12.06 

nabiji 5. gadawyvetilebis miReba: ramdenadac 0.283 < 4.605, amitom nulovani 

hipoTezis uaryofis safuZveli ara gvaqvs, anu ar arsebobs sakmarisi 

safuZveli vamtkicoT, rom alkoholis moxmareba damokidebulia sqesze. 

amocanebi 

1. swavloben aris Tu ara kavSiri sportsa da sisxlis wnevas Soris. 

SemTxveviT SerCeuli 210 adamianis qvemoTmoyvanili monacemebis mixedviT, 

0.05 mniSvnelovnebis doniT, SeamowmeT hipoTeza imis Sesaxeb, rom 

sporti da wneva ar arian damokidebuli. 

sisxlis wneva 

----------------------------------------------------- 

dabali saSualo maRali 

-------------------------------------------------------------------------------------------------------- 

sportsmeni 34 57 21 

ara sportsmeni 15 63 20 

2. mkvlevars ainteresebs aris Tu ara damokidebuli moxmarebuli 

yavis raodenoba adamianis asakze. 152 SemTxveviT SerCeuli adamianis qvemoTmoyvanili 

monacemebis mixedviT, 0.01 mniSvnelovnebis doniT, aris 

Tu ara damokidebuli yavis moxmareba asakze? 

yavis moxmareba 

----------------------------------------------------- 

asaki dabali saSualo maRali 

----------------------------------------------------------------------------------------------- 

21—30 18 16 12 

31—40 9 15 27 

41—50 5 12 10 

51 

13 9 6 

3. mkvlevars ainteresebs aris Tu ara damokidebuli erTmaneTze myidvelis 

asaki da SeZenili avtomobilis fasi. SemTxveviT SerCeuli 222 avtomflobelis 

qvemoTmoyvanili monacemebis mixedviT, 0.05 mniSvnelov- 

197

nebis doniT, aris Tu ara avtomobilis fasi damoukidebeli avtomflobelis 

asakisagan? 

gayidvis fasi 

----------------------------------------------------- 

asaki araumetes 20000 l-dan -- 30000 l-dan -- 

20000 l -- 30000 l-mde -- 40000 l-mde 

----------------------------------------------------------------------------------------------------------- 

21—30 16 25 3 

31—40 44 23 15 

41—50 31 15 18 

51 

9 11 12 

4. mkvlevars ainteresebs damokidebulia Tu ara informaciis miRebis 

saSualebebi adamianis ganaTlebis doneze. gamokiTxuli 400 bakalavrisa 

da magistris qvemoTmoyvanili monacemebis mixedviT, 0.05 mniSvnelovnebis 

doniT, SeamowmeT hipoTeza imis Sesaxeb, rom informaciis miRebis 

gzebi damoukidebelia ganaTlebis donisagan. 

televizia gazeTebi sxva saSualebebi 

--------------------------------------------------------------------------------------------------- 

bakalavri 159 90 51 

magistri 27 42 31 

5. universitetis reqtors ainteresebs axdens Tu ara gavlenas leqtoris 

samecniero xarisxi studentis Sexedulebaze am leqtoris muSaobis 

xarisxis Sesaxeb. gamokiTxuli studentebis qvemoTmoyvanili monacemebis 

mixedviT, 0.1 mniSvnelovnebis doniT, SeuZlia Tu ara reqtors daaskvnas, 

rom studentis Sexeduleba leqtoris muSaobis efeqturobis Sesaxeb 

damokidebulia leqtoris samecniero xarisxze? 

xarisxi 

-------------------------------------------------------------------- 

Sexeduleba bakalavri magistri doqtori 

------------------------------------------------------------------------------------------------------ 

umaRlesi 14 9 4 

saSualo 16 5 7 

dabali 3 12 16 

6. mkvlevars ainteresebs dakavSirebulia Tu ara studentis ojaxuri 

mdgomareoba mis SefasebasTan statistikaSi. 142 SemTxveviT SerCeuli 

studentis qvemoTmoyvanili monacemebis mixedviT, 0.05 mniSvnelovnebis 

doniT, damokidebulia Tu ara studentis Sefaseba statistikaSi mis ojaxur 

mdgomareobaze? 

Sefaseba 

------------------------------------------------------------ 

ojaxuri mdgomareoba A B C D an E 

----------------------------------------------------------------------------------------------- 

martoxela 27 32 16 10 

daojaxebuli 14 19 16 8 

198

7. ikvleven kavSirs filmis tipsa da mayureblis asaks Soris. qvemo- 

Tmoyvanili monacemebis mixedviT, 0.1 mniSvnelovnebis doniT, aris Tu 

ara kavSiri filmis tipsa da mayureblis asaks Soris? 

asaki dokumenturi komedia misteria 

----------------------------------------------------------------------------------------------------- 

12—20 14 9 8 

21—29 15 14 9 

30—38 9 21 39 

39—47 7 22 17 

48 

6 38 12 

8. ikvleven damokidebulebas avtomobilis tipsa (wamyvani 2 an 4 

TvaliT) da myidvelis sqess Soris. SemTxveviT SerCeuli 90 myidvelis 

qvemoTmoyvanili monacemebis mixedviT, 0.05 mniSvnelovnebis doniT, 

SeamowmeT hipoTeza imis Sesaxeb, rom myidvelis arCevani damoukidebelia 

misi sqesisagan. 

sqesi 2 wamyvani Tvali 4 wamyvani Tvali 

------------------------------------------------------------------------------------------------ 

mamrobiTi 23 43 

mdedrobiTi 18 6 

9. ikvlevdnen fexburTis TamaSis dros mayureblebis mier arCeul 

sakvebis saxeobas. qvemoTmoyvanili monacemebis mixedviT, 0.1 mniSvnelovnebis 

doniT, aris Tu ara mayureblis mier arCeuli sakvebi damoukidebeli 

mayureblis sqesisagan? 

sqesi “hoT dogi” araxisi Rvezeli 

------------------------------------------------------------------------------------------- 

mamrobiTi 12 21 19 

mdedrobiTi 13 8 25 

10. axali preparatis gamokvlevis mizniT SerCeul iqna pacientTa 

ori jgufi, romelTagan erTs aZlevdnen axal preparats, xolo meores ki 

placebos. qvemoTmoyvanili kvlevis Sedegebis mixedviT, 0.1 mniSvnelovnebis 

doniT, SeiZleba Tu ara davaskvnaT, rom preparati efeqturia? isargebleT 


Sedegi 

efeqturi araefeqturi 

----------------------------------------------------------------------- 

preparati 32 9 

placebo 12 18 

11. wignis gamomcemels ainteresebs aris Tu ara gansxvaveba mamakacebisa 

da qalebis mier wasakiTxad arCeul wignebis tipebs Soris. qvemoTmoyvanili 

monacemebis mixedviT, 0.05 mniSvnelovnebis doniT, SeamowmeT 

199

hipoTeza imis Sesaxeb, rom arCeuli wignis tipi damoukidebelia mkiTxvelis 

sqesisagan. isargebleT P -mniSvnelobis meTodiT. 

sqesi mistika romani deteqtivi 

------------------------------------------------------------------------------------------- 



pasuxebi: 

2 

2 

1. H 

0 

: damoukid., H 

1: damokid.; CV . . 2,0.05 

5.991 ; TV . . 

6.789 ; H 

1. 2. H 

0 

: 

2 

2 

damoukid., H 1 

: damokid.; CV . . 6,0.01 

16.812 ; TV . . 

15.824 ; ara -- H 0 

. 3. H 0 

: 

2 

2 

damoukid., H 

1: damokid.; CV . . 6,0.05 

12.592 ; TV . . 

24.004 ; ara -- H 

1. 4. 

2 

2 

H 

0 


1: damokid.; CV . . 2,0.05 

5.991 ; TV . . 

21.347 ; H 

1. 5. H 

0 

: 

2 

2 

damoukid., H 

1: damokid.; CV . . 4,0.1 

7.779 ; TV . . 

19.507 ; ki -- H 

1. 6. H 

0 

: 

2 

2 

damoukid., H 

1: damokid.; CV . . 3,0.05 

7.815 ; TV . . 

2.218 ; ara -- H 

0 

. 7. H 

0 

: 

2 

2 


: damokid.; CV . . 8,0.1 

13.362 ; TV . . 

46.733 ; ara -- H 0 

. 8. H 0 

: 

2 

2 

damoukid., H 

1: damokid.; CV . . 1,0.05 

3.841 ; TV . . 

11.441 ; H 

1. 9. H 

0 

: 

2 

2 

damoukid., H 

1: damokid.; CV . . 2,0.1 

4.605 ; TV . . 

6.342 ; ara -- H 

1. 10. H 

0 

: 

2 

araefeqt., H 

1: efeqt.; TV . . 

10.643 ; P -mniSvneloba < 0.005; H 

1. 11. H 

0 

: 

2 


: damokid.; TV . . 

19.43 ; P -mniSvneloba < 0.005; H 1 

. 

200

Tavi XVI 

erTgvarovnebis Semowmebis xi-kvadrat kriteriumi 

mowmdeba populaciaTa erTgvarovnebis hipoTeza, romelic eqvivalenturia 

hipoTezisa imis Sesaxeb, rom ama Tu im niSnis proporciebi sxvadasxva 

populaciebSi erTidaigivea. 


: p 1 

p 2 

p R 

. 


: erTi proporcia mainc gansxvavdeba danarCenebisagan. 

201 

i, j i, 

j 


kriteriumis statistika: xi-kvadrati. 

2 

( o 

2 

i, j 

ei , j) 


, sadac o 

e 

ij , 

-- dakvirvebuli 

sixSireebia, xolo e ij , 

ki mosalodneli sixSireebi: 

oi , j 

o 

i 

j i, 

j 

ei, 

j . 

o 

 

i, 

j 

i, 

j 

SezRudva: yvela e 

, 

5 (winaaRmdeg SemTxvevaSi xdeba dajgufeba), 

ij 

roca cxrili 2X2-ze didia (2X2 cxrilis SemTxvevaSi ki yvela e i , j 

10). 


2 

kritikuli mniSvneloba CV . . ( R 1)( C 1), , sadac R warmoadgens SerCevaTa 

raodenobas, xolo C ki klasebis raodenobaa. 

rodesac SeuRlebis cxrilis Tavisuflebis xarisxi erTis tolia 

2 

(| o 

2 

i, j 

ei , j 

| 0.5) 

( df . . 1), gamoiyeneba kriteriumis mniSvneloba: TV . . 

. 

e 


-is uaryofis are) = 

[ 

, ) . 

2 

n 1, 

i, j 

i, 

j 

P - mniSvneloba = P{ 2 ( k) 

T. 

V.} 

, sadac k ( R1)( C 1) . 





0 




magaliTi 1. mkvlevarma SemTxveviT SearCia 150 damamaTavrebeli kursis 

studenti sami umaRlesi saswaleblidan (instituridan) da TiToeuls 

dausva SekiTxva: “dadis Tu ara igi institutSi Tavisi an mSoblebis manqaniT”. 

gamokiTxvis Sedegebi moyvanilia cxrilis saxiT. 0.05 mniSvnelovnebis 

doniT SeamowmeT hipoTeza imis Sesaxeb, rom im studentebis

proporcia, romlebic institutSi dadian sakuTari an mSoblebis 

manqanebiT erTidaigivea samive institutisaTvis. 

instituti I instituti II instituti III jami 

------------------------------------------------------------------------------------------------------------------- 

ki 18 22 16 56 

ara 32 28 34 94 

jami 50 50 50 150 

amoxsna. 


H 

0 

: p 1 

p 2 

p 3 

(proporciebi erTidaigivea). 

H 

1 

: sul cota erTi proporcia mainc gansxvavdeba danarCenebisagan. 

nabiji 2. vipovoT kritikuli mniSvneloba. Tavisuflebis xarisxi 

tolia d. f . ( R1)( C 1) (21) (31) 2. amitom xi-kvadrat ganawilebis 

cxrilidan vpoulobT, rom 0.05 -saTvis kritikuli mniSvneloba iqneba: 

2 

CV . . 2,0.05 

5.991 . 

nabiji 3. gamovTvaloT mosalodneli mniSvnelobebi. visargebloT 

formuliT: 

striqonisjami ×svetisjami 

mosalodneli mniSvneloba e ij , 

= 

. 

mTliani jami 

5650 5650 5650 e1,1 

18.67 , e 

150 

1,2 

18.67 , e 

150 

1,3 

18.67 , 

150 

9450 9450 9450 e2,1 

31.33 , e 

150 

2,2 

31.33 , e 

150 

2,3 

31.33 . 

150 

dasrulebuli SeuRlebis cxrili iqneba: 

instituti I instituti II instituti III jami 

------------------------------------------------------------------------------------------------------------------- 

ki 18 (18.67) 22 (18.67) 16 (18.67) 56 

ara 32 (31.33) 28 (31.33) 34 (31.33) 94 

jami 50 50 50 150 


2 2 2 2 

( o 

2 

i, j 

ei , j) (18 18.67) (22 18.67) (16 18.67) 

 

 

e 18.67 18.67 18.67 

ij , ij , 

2 2 2 

(32 31.33) (28 31.33) (34 31.33) 

1.596 . 

31.33 31.33 31.33 


hipoTezis uaryofis safuZveli ara gvaqvs, anu ar arsebobs sakmarisi 

safuZveli vamtkicoT, rom proporciebi gansxvavebulia. 

magaliTi 2. sami qarxana uSvebs erTidaigive detals. am qarxnebidan 

aRebulia Sesabamisad 250, 200 da 150 detali, romelTagan arastandartuli 

aRmoCnda Sesabamisad 10, 9 da 11 detali. SevamowmoT hipoTeza aRniSnul 

SerCevaTa erTgvarovnebis Sesaxeb 0.1 mniSvnelovnebis doniT. 

amoxsna. 

202


H 

0 

: SerCevebi erTgvarovania. 

H 

1 

: ori SerCeva mainc gansxvavebulia erTmaneTisagan. 

nabiji 2. vipovoT kritikuli mniSvneloba: 

2 2 2 

CV . . ( R1)( C1), (3 1)(2 1),0.1 2,0.1 

4.605 . 


250 570 250 30 200 570 e1,1 

297.5 , e 

600 

1,2 

12.5 , e 

600 

2,1 

190 , 

600 

200 30 150 570 150 30 e2,2 

10 , e 

600 

3,1 

142.5 , e 

600 

3,2 

7.5 . 

600 

amrigad, dasrulebuli SeuRlebis cxrili iqneba: 

standartuli arastandartuli jami 

---------------------------------------------------------------------------------------------------- 

I qarxana 240 (297.5) 10 (12.5) 250 

II qarxana 191 (190) 9 (10) 200 

III qarxana 139 (142.5) 11 (7.5) 150 

jami 570 30 600 


2 2 2 2 

( o 

2 

i, j 

ei , j) (240 237.5) (10 12.5) (191 190) 

 

 

e 237.5 12.5 190 

i, j i, 

j 

2 2 2 

(9 10) (139 142.5) (11 7.5) 

2.36 . 

10 142.5 7.5 


hipoTezis uaryofis safuZveli ara gvaqvs, anu 0.1 mniSvnelovnebis 

doniT SerCevebi erTgvarovania. 

amocanebi 

1. kvlevis Tanaxmad 6-dan 17-wlamde mozardebis 64%-s ar SeuZlia 

gadalaxos sabazo Sesabamisobis testi. mkvlevars ainteresebs aris Tu 

ara am kategoriis moswavleebis proporcia erTidaigive sxvadasxva skolebSi. 

testireba Cautarda SemTxveviT SerCeul 120--120 moswavles oTxi 

sxvadasxva skolidan. 0.05 mniSvnelovnebis doniT SeamowmeT hipoTeza 

imis Sesaxeb, rom im moswavleebis proporcia, romlebmac gadalaxes Sesabamisobis 

testi, erTidaigivea. 

I skola II skola III skola IV skola 

--------------------------------------------------------------------------------------------------------------- 

gadalaxa 49 38 46 34 

ver gadalaxa 71 82 74 86 

jami 120 120 120 120 

2. sareklamo kompaniam gadawyvita sami sxvadasxva maRaziis 92—92 

momxmarebels SesTavazos monawileoba miiRos bazris kvlevis gamokiTxva- 

Si. wina kvlevis mixedviT, 38%-ma uari ganacxada aseT gamokiTxvaSi monawileobis 

miRebaze. 0.01 mniSvnelovnebis doniT SeamowmeT hipoTeza im- 

203

is Sesaxeb, rom im momxmareblebis proporcia, romlebic apireben monawileoba 

miiRon gamokiTxvaSi, erTidaigivea. 

I maRazia II maRazia III maRazia 

---------------------------------------------------------------------------------------------------- 

miiRebs 

52 45 36 

monawileobas 

ar miiRebs 40 47 56 

monawileobas 

jami 92 92 92 

3. sadazRvevo kompanias ainteresebs icvleba Tu ara asakis mixedviT 

proporcia im mZRolebis, vinc dalevis Semdeg jdeba saWesTan. gamokiTxul 

iqna 4 asakobrivi jgufis 86--86 mZRoli. 0.05 mniSvnelovnebis doniT 

SeamowmeT hipoTeza imis Sesaxeb, rom proporcia im mZRolebis, vinc 

kiTxavas: “jdebiT Tu ara nasvami saWesTan” -- upasuxa “diax”, erTidaigivea 

yvela asakobrivi jgufisaTvis. 

asaki 21—29 30—39 40—49 50-ze meti wlis 

-------------------------------------------------------------------------------------------------- 

diax 32 28 26 21 

ara 54 58 60 65 

jami 86 86 96 86 

4. gamokvlevis mixedviT 8-dan 17 wlamde mozarebis 59% urCevnia, 

rom mama dasaqmebuli iyos biznesSi, vidre iyos partiuli muSaki. SerCeul 

iqna 60—60 moswavle dawyebiTi, saSualo da maRali klasebidan, raTa 

daedginaT maTi damokidebuleba mamis samsaxuris mimarT. 0.1 mniSvnelovnebis 

doniT SeamowmeT hipoTeza imis Sesaxeb, rom proporciebi im moswavleebis, 

romelTac urCevniaT, rom mama iyos biznesmeni, tolia. 

dawyebiTi saSualo maRali 

------------------------------------------------------------------------------------------------- 

biznesis sferoSi 29 38 51 

partiul samuSaoze 31 22 9 

jami 60 60 60 

5. gamokiTxul iqna 100—100 adamiani qveynis oTxive kuTxeSi. maT 

dausves kiTxva: “aqvT Tu ara mudmivi samuSao”. 0.1 mniSvnelovnebis 

doniT SeamowmeT hipoTeza imis Sesaxeb, rom proporciebi im adamianebis, 

romelTac aqvT mudmivi samuSao, tolia qveynis oTxive kuTxeSi. 

aRmosavleTi dasavleTi CrdiloeTi samxreTi 

------------------------------------------------------------------------------------------------------ 

diax 43 39 22 28 

ara 57 61 78 72 

jami 100 100 100 100 

6. amerikeli mamebis saSualod 79% sakuTari Svilis gaCenisas imyofeba 

samSobiaro ganyofilebaSi. gamokiTxul iqna 300 amerikeli mama. maT 

dausves kiTxva: “imyofebodnen Tu ara samSobiaro ganyofilebaSi meuRlis 

mSobiarobisas”. 0.05 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi 

safuZveli ukuvagdoT hipoTeza im mamebis proporciebis tolobis Ses- 

204

axeb, romlebic meuRlis mSobiarobisas imyofebodnen samSobiaro ganyofilebaSi? 

I samSobiaro II samSobiaro III samSobiaro IV samSobiaro 

------------------------------------------------------------------------------------------------------------------- 

ki 66 60 57 56 

ara 9 15 18 19 

jami 75 75 75 75 

7. msoflio jandacvis organizaciis monacemebiT zooparkebSi bavSvebis 

mier miRebuli tramvebis 55% modis maimunis nakbenze. gamokvleul 

iqna 4 sxvadasxva zooparkSi bavSvTa tramvebis 30—30 SemTxveva. 0.05 

mniSvnelovnebis doniT SeamowmeT hipoTeza proporciebis tolobis Sesaxeb. 


tramva zooparki A zooparki B zooparki C zooparki D 

----------------------------------------------------------------------------------------------------------------------- 

maimunis kbena 15 18 13 16 

sxva tramva 15 12 17 14 

jami 30 30 30 30 

8. sasursaTo maRaziebis qselis mepatrones surs daadginos misi ma- 

Raziebis momxmareblebi maRaziaSi wasvlamde winaswar adgenen Tu ara sa- 

Wiro produqtebis sias. am mizniT gamokiTxul iqna 288 momxmarebeli. 

0.1 mniSvnelovnebis doniT SeamowmeT hipoTeza imis Sesaxeb, rom proporciebi 

im momxmareblebis, romlebic maRaziaSi wasvlamde adgenen SesaZeni 

produqtebis sias, tolia samive maRaziisaTvis. 

maRazia A maRazia B maRazia C 

------------------------------------------------------------------------------------------------------ 

adgens sias 77 74 68 

ar adgens sias 19 22 28 

jami 96 96 96 

pasuxebi: 

H : p p p p , H 1 

: erTi mainc gansx.; 

1. 

0 1 2 3 4 

2 

CV . . 

7.815 ; TV . . 

5.317 ; 

2 

2 

H 

0 

. 2. H 0 

: p 1 

p 2 

p 3 

, H 1 

: erTi mainc gansx.; CV . . 2,0.01 

9.210 ; TV . . 

8.046 ; 

2 

H 

0 

. 3. H0 : p1 p2 p3 p4 

, H1 : erTi mainc gansx.; CV . . 3,0.05 

7.815 ; 

2 

2 

TV . . 

3.4 ; H 

0 

. 4. H0 : p1 p2 p3 

, H1 : erTi mainc gansx.; CV . . 2,0.1 

4.605 ; 

2 

TV . . 

18.06 ; H 

1. 5. H0 : p1 p2 p3 p4 

, H1 : erTi mainc gansx.; 

2 

2 

CV . . 3,0.1 

6.251 ; TV . . 

12.755 ; H 

1. 6. H0 : p1 p2 p3 p4 

, H1 : erTi mainc 

2 

2 

gansx.; CV . . 3,0.05 

7.815 ; TV . . 

5; ara -- H 

0 

. 7. H0 : p1 p2 p3 p4 

, 

2 

H1 : erTi mainc gansx.; TV . . 

1.734 ; P -mniSvneloba > 0.1; H 

0 

. 8. 

2 

2 

H0 : p1 p2 p3 

, H 1 

: erTi mainc gansx.; CV . . 2,0.1 

4.605 ; TV . . 

2.401 ; H 0 

. 

2 

3,0.05 

205

fatalur avariaTa 

ricxvi 

Tavi XVII 

dawyvilebuli monacemebi, korelacia 

ama Tu im or maxasiaTebels (an or maCvenebels) Soris kavSiris Seswavlis 

mizniT am maxasiaTeblebis dakvirvebuli mniSvnelobebis simravles 

warmoadgenen wyvilebis simravlis saxiT: ( x1, y 

1), 

( x2, y 

2) 

, . . . , ( xn, y 

n) 

. 

am monacemebis mixedviT ageben e. w. gabnevis diagramas Semdegi wesiT: sibrtyeze, 

koordinatTa marTkuTxa sistemaSi, moniSnaven wertilebs, romel- 

Ta koordinatebia ( xi, y 

i) 

, i 1,2,..., n. 

magaliTi 1. mkvlevars ainteresebs daadginos arsebobs Tu ara kavSiri 

velosipediT wvimian amindSi momxdar msubuq avariaTa ricxvsa da sikvdiliT 

(fataluri SedegiT) damTavrebul avariaTa ricxvs Soris. qvemoT 

moyvanilia 10 wlis ganmavlobaSi wvimian amindSi velosipediT momxdari 

avariebis monacemebi: 

msub. avar. 376 650 884 1162 1513 1650 2236 3002 4028 4010 

ricxvi, x 

fat. avar. 5 20 20 28 26 34 35 56 68 55 

ricxvi, y 

avagoT gabnevis diagrama am monacemebisaTvis. 

amoxsna. 

nabiji 1. sibrtyeze avagoT marTkuTxa koordinatTa sistema da movniSnoT 

x da y RerZebi. 

nabiji 2. avagoT wertilTa wyvilebi sakoordinato sibrtyeze ise 

rogorc es qvemoTaa naCvenebi: 

gabnevis diagrama 

80 

70 

60 

50 

40 

30 

20 

10 

0 

0 1000 2000 3000 4000 5000 

msubuq avariaTa ricxvi 

magaliTi 2. avagoT gabnevis diagrama monacemebisaTvis, romelic miiReba 

6 SemTxveviT SerCeuli pirovnebis asaksa da arteriuli wnevas Soris 

damokidebulebis Seswavlisas. monacemebis cxrilia: 

206

w n eva 

persona asaki, x wneva, y 

A 43 128 

B 48 120 

C 56 135 

D 61 143 

E 67 141 

F 70 152 

amoxsna. 

nabiji 1. davxazoT sakoordinato sibrtye da movniSnoT x da y 

RerZebi. 

nabiji 2. movniSnoT TiToeuli wertili grafikze rogorc es qvemo- 

Taa naCvenebi: 


160 

150 

140 

130 

120 

110 

30 40 50 60 70 80 

x da y sidideebs Soris SesaZlebelia arsebobdes ramodenime gansxvavebuli 

tipis damokidebuleba: 

1. dadebiTi wrfivi kavSiri arsebobs (gvaqvs) im SemTxvevaSi, rodesac 

wertilebi daaxloebiT koncentrirebulia aRmavali swori xazis irgvliv 

da erTdroulad orive x da y sididis mniSvnelobebi zrdadia. aseTi 

kavSiri gvaqvs maSin, rodesac x sididis zrdasTan erTad izrdeba y 

sididec. 

asaki 

dadebi Ti wrfi vi kavSi ri 

400 

300 

200 

100 

0 

0 50 100 150 

2. uaryofiTi wrfivi kavSiri arsebobs im SemTxvevaSi, rodesac wertilebi 

daaxloebiT koncentrirebulia daRmavali swori xazis irgvliv. 

aseTi kavSiri gvaqvs maSin, rodesac x sididis zrdasTan erTad klebulobs 

y sidide da piriqiT. 

207

uaryofi Ti wrfi vi kavSi ri 

400 

300 

200 

100 

0 

0 50 100 150 

3. arawrfivi kavSiri arsebobs im SemTxvevaSi, rodesac wertilebi 

daaxloebiT koncentrirebulia arawrfivi (mrudi) xazis (wiris) irgvliv. 

es damokidebuleba aRiwereba Sesabamisi wiris bunebiT. 

arawrfi vi kavSi ri 

400 

300 

200 

100 

0 

0 50 100 150 

4. ar aris kavSiri im SemTxvevaSi, rodesac wertilebi uwesrigodaa 

mimofantuli, anu ar Cans rom isini raime wiris irgvlivaa koncentrirebuli. 

ar ari s kavSi ri 

600 

400 

200 

0 

0 50 100 150 

zemoT moyvanil magaliTSi x (msubuq avariaTa ricxvi) da y (fatalur 

avariaTa ricxvi) sidideebs Soris adgili aqvs wrfiv dadebiT kavSirs. 

wirs, romelic aRwers kavSirs (damokidebulebas) x da y sidideebis 

mniSvnelobebs Soris misadagebis wiri ewodeba. am wiris povnis sakiTxebs 

swavlobs maTematikuri statistikis nawili, romelsac regresiuli analizi 

ewodeba. wiris mosaZebnad iyeneben e. w. umcires kvadratTa meTods. 

Tu sidideebs Soris gvaqvs wrfivi kavSiri y() 

x ax b, maSin amboben 

rom x da y sidideebs Soris wrfivi regresiuli kavSiria da y() 

x ax b 

wrfes uwodeben regresiis wrfis gantolebas (xolo am formuliT gamo- 

Tvlil y ' i 

y( xi ) axi 

b mniSvnelobas prognozi ewodeba). umcires kvadratTa 

meTodi gvaZlevs, rom am SemTxvevaSi: 

208

n n n 

 

n( x y ) ( x ) ( y ) 

a 

i i i i 

i1 i1 i1 

n 

n 

2 2 

n( xi 

) ( xi) 

i1 i1 

da 

b 

n n n n 

2 

( yi ) ( xi ) ( xi ) ( xi yi 

) 

i1 i1 i1 i1 

n 

n 

2 2 

n( xi 

) ( xi) 

i1 i1 

Sefasebis standartuli Secdoma ewodeba sidides: 

n 

( ' )) 2 

y 

1 i 

y 

i 

i 

Sef : 

, 

s 

n2 

igi warmoadgens damokidebuli cvladis dakvirvebuli mniSvnelobebis 

standartul gadaxras saprognozo mniSvnelobebisagan. 

yx () mniSvnelobis mniSvnelovnebis donis mqone saprognozo intervali 

(ndobis intervalis analogi) moicema TanafardobiT: 

 

1 n( x x) 1 n( x x) 

y( x) t s 1 , y( x) t s 1 

n 

n 

( ) 

n2, /2 Sef 

n n n2, /2 

Sef 

n n 

2 2 2 2 

n( xi ) ( xi ) n( xi ) ( xi 

) 

i1 i1 i1 i1 

magaliTi 3. menejerma Seagrova monacemebi da daadgina, rom gasamravlebeli 

manqanis asaksa da TveSi teqnikuri momsaxurebis xarjebs Soris 

mniSvnelovani kavSiria. regresiis wrfis gantolebaa y( x) 8.13x 

55.57 . 

ipoveT Sefasebis standartuli Secdoma, Tu monacemebia: 

manqana A B C D E F 

asaki, x 1 2 3 4 5 6 

xarji, y 62 78 70 90 93 103 

amoxsna. 

nabiji 1. gavakeToT Semdegi cxrili: 

' 

' 

' 2 

x 

i 

y 

i y 

i 

yi 

yi 

( yi 

yi) 

1 62 

2 78 

3 70 

4 90 

4 93 

6 103 

nabiji 2. visargebloT regresiis wrfis gantolebiT da gamovTvaloT 

saprognozo mniSvnelobebi TiToeuli x i 

-saTvis: y ' i 

: y( xi ) 8.13 xi 

55.57 . 

gvaqvs: y ' 1 

63.7 , y ' 2 

71.83 , y ' 3 

79.96 , y 

' ' 

4 

y5 88.09 da y ' 6 

104.35 . SevitanoT 

es monacemebi cxrilis Sesabamis svetSi. 

nabiji 3. TiToeul y i 

-s gamovakloT Sesabamisi y ' i 

da Sedegebi SevitanoT 

Sesabamis svetSi: y1 y1 1.7 , y2y2 6.17 , y3 y3 9.96 , y4y4 1.91 , 

' 

' 

' 

' 

' 

y5y5 1.35 . 

nabiji 4. miRebuli sxvavobebi aviyvanoT kvadratSi da SevitanoT 

Sesabamis svetSi. 

nabiji 5. SevkriboT bolo svetis monacemebi: 

. 

209

' 

' 

' 2 

x 

i 

y 

i y 

i 

yi 

yi 

( yi 

yi) 

1 62 63.7 -1.7 2.89 

2 78 71.83 6.17 38.0689 

3 70 79.96 -9.96 99.2016 

4 90 88.09 1.91 3.6481 

4 93 88.09 4.91 24.1081 

6 103 104.35 -1.35 1.8225 

jami 169.7392 

nabiji 6. gamovTvaloT Sefasebis standartuli Secdoma: 

n 

( ' )) 2 

y 

1 i 

y 

i 

i 169.7392 

Sef : 6.51. 

s 

 

n2 62 

Sefasebis standartuli Secdoma SeiZleba agreTve gamoiTvalos 

fomuliT: 

n 2 n n 

i 1 i i 1 i i 1 i i 

: 

y b y a x y 

 

Sef 

. 

s 

 

n2 

magaliTi 4. wina magaliTSi: a). gamoTvaleT Sefasebis standartuli 

Secdoma ukanaskneli formulis mixedviT da b). aageT 95%-iani saprognozo 

intervali 3 wlis gasamravlebeli manqanisaTvis TveSi gasawevi teqnikuri 

momsaxurebis xarjisaTvis. 

amoxsna. a). 

nabiji 1. SevavsoT qvemoTmoyvanili cxrili: 

2 

x 

i 

y 

i 

xy 

i i y 

i 

1 62 62 3844 

2 78 156 6084 

3 70 210 4900 

4 90 360 8100 

4 93 372 8649 

6 103 618 10609 

jami 496 1778 42186 

nabiji 2. vipovoT Sefasebis standartuli Secdoma: 

 

n y 2 

b n y a n 

x y 

i i i i i 

i i 

 

1 1 1 

42186 55.57 496 8.13 1778 

sSef 

: 6.48 

n2 62 

(gasagebia, rom 6.48 axlosaa wina magaliTSi miRebul 6.51-Tan). 

b). 

2 20 

nabiji 1. vipovoT: xi 

=20, x i =82, x 3.3 . 

6 

nabiji 2. xarjis prognozi, roca x 3: y(3) 8.13 355.57 79.96 . 

nabiji 3. gamovTvaloT Sefasebis standartuli Secdoma. a). punqtis 

mixedviT: s 6.48 

Sef . 

210

nabiji 4. SevitanoT saWiro monacemebi saprognozo intervalis TanafardobaSi 

(aq (1 ) 95 100 , 0.05 ; df . . 62 4 ; tn 

2, /2 

2.776 ): 

2 2 

1 6 (33.3) 1 6 (33.3) 

(79.96 2.776 6.48 1 , 79.96 2.776 6.48 1 ) , 

2 2 

6 682 20 6 682 20 

(79.96 2.776 6.48 1.08, 79.96 2.776 6.48 1.08) , 

(60.53, 99.39). 

or sidides Soris damokidebulebis xarisxis gasazomad iyeneben e. 

w. korelaciis koeficients. arsebobs sxvadasxva saxis korelaciis koeficienti. 

Cven aq moviyvanT e. w. pirsonis momentTa namravlis korelaciis 

koeficients (Pearson product moment correlation coefficient -- PPMC ), romelic 

dakavSirebulia cnobili statistikosis karl pirsonis saxelTan. 

korelaciis koeficienti, gamoTvlili monacemTa SerCevis mixedviT, 

warmoadgens or sidides Soris wrfivi damokidebulebis xarisxisa da mimarTulebis 

sazoms. SerCeviTi korelaciis koeficienti aRiniSneba r aso- 

Ti, xolo populaciis mixedviT gamoTvlili korelaciis koeficienti aRiniSneba 

berZnuli (ro) asoTi. 

or x da y sidides Soris kovariaciis koeficienti (cov( xy) , ) ganimarteba 

Semdegi TanafardobiT: cov( x , y n 

n 

1 1 

) ( x i 

x )( y y i 

) x y xy i i 

, 

n 

n 

sadac 

1 n xi 

n i 1 

x , 

y 

1 n yi 

n i 1 

211 

i1 i1 

. SerCeviTi korelaciis koeficienti moicema 

n 

1 

( xi 

x)( yi 

y) 

cov( xy , ) 

 

n i1 

TanafardobiT: r 

ss 

n 

n 

x y 1 2 1 

2 

( xi 

x) ( yi 

y) 

n 

n 

 

n 

 

i1 i1 

( x x)( y y) 

i i n 

i1 

i i 

n 

n 

2 2 

( xi 

x) ( yi 

y) 

i1 i1 

1 x x y y 

( )( ) . 

n s s 

i1 

x y 

korelaciis koeficientis Tvisebebi: 

1. 1 r 1. r -is dadebiTi mniSvneloba miuTiTebs sidideebs Soris 

pozitiuri damokidebulebis arsebobaze, xolo uaryofiTi mniSvneloba ki 

– uaryofiTze. 

2. | r | 1 maSin da mxolod maSin, roca x da y sidideebs Soris arsebobs 

wrfivi kavSiri y i 

ax i 

b . amasTanave, Tu r 1, maSin a 0 da Tu 

r 1, maSin a 0 . 

3. korelaciis koeficienti ucvleli rCeba sidideebis wrfivi gardaqmnisas. 

4. korelaciis koeficienti zomavs sidideebs Soris wrfivi damokidebulebis 

siZlieres. sxva, arawrfivi, Tundac deterministuli kavSiri 

korelaciaSi ar aisaxeba.

B trestis 

monacemebi 

5. rac ufro koncentrirebulia monacemebi raime wrfis midamoSi, 

miT ufro didia korelaciis koeficienti. 

6. or sidides Soris kavSiri arsebobs, Tu adgili aqvs Tanafardobas: 

| r| 2/ n, sadac n SerCevis moculobaa. 

7. Tu mocemulia monacemTa ori simravle x 1 

,..., x 

n 

da y 1 

,..., y 

n 

, maSin 

monacemTa axali z i 

x i 

y i 

( i 

1,..., n) simravlisaTvis SerCeviTi dispersia 

2 2 2 

2 2 2 

ase gamoiTvleba: s s s 2cov( x, y) 

(Tu cov( xy , ) 0, maSin s s s ). 

analogiurad, 

z x y 

s s s 2cov ( x, y) 

, sadac 

'2 '2 '2 ' 

z x y 

1 

n 

. 

n 

' 

cov ( x, y) ( xi 

x)( yi 

y) cov( x, y) 

n1 

i1 

n1 

z x y 

magaliTi 5. mocemulia ori trestis wliuri saprocento Semosavlis 

monacemebi bolo 10 wlis ganmavlobaSi: 

tresti A 8.3 -6.2 20.9 -2.7 33.6 42.8 24.4 5.2 3.1 30.5 

tresti B 12.1 -2.8 6.4 12.2 27.8 25.3 18.2 10.7 -1.3 11.4 

avagoT gabnevis diagrama da gamovTvaloT korelaciis koeficienti. 

amoxsna. A trestis monacemebi gadavzomoT x RerZze, B trestisa ki 

y RerZze. 


30 

25 

20 

15 

10 

5 

0 

-20 -5 0 20 40 60 

A trestis monacemebi 

am SemTxvevaSi x 16 , y 12, s 17.65 da s 10.51 . amitom gvaqvs: 

x 

n 

1 

cov( x, y) 

x y xy 

 

n i 1 

1 

[8.3 12.1 ( 6.2) ( 2.8) 30.5 11.4] 16 12 71.48 , 

10 

Sesabamisad, 

cov( xy , ) 71.48 

r 0.39 . 

ss 17.65 10.51 

x 

y 

i 

i 

y 

212

gayidvis moculoba 

korelaciis koeficientis es mniSvneloba gvaewmunebs, rom arsebobs 

sustad gamoxatuli pozitiuri wrfivi kavSiri A da B trestebis amonagebs 

Soris, rasac gabnevis diagramac adasturebs. 

magaliTi 6. axali produqciis fasis dasadgenad kompaniam SearCia 

10 qalaqi, romelTac arsebiTad erTi da igive yidvis potenciali gaaCnia 

da gamoitana produqcia bazarze yovel maTganSi sxvadasxva fasad. kompaniis 

mier miRebuli Sedegebi moyvanilia cxrilSi: 

qalaqis nomeri 1 2 3 4 5 6 7 8 9 10 

fasi, x 15. 15. 16. 16. 17. 17. 18. 18. 19. 19. 

0 5 0 5 0 5 0 5 0 5 

gayidvis moculoba, y 15 14 16 9 12 10 8 9 6 5 

davadginoT gayidvis moculobis fasze damokidebulebis saxe. 

amoxsna. avagoT gabnevis diagrama 


20 

15 

10 

5 

0 

12 14 16 18 20 22 

rogorc diagrama gviCvenebs, Cvens winaSea uaryofiTi wrfivi kavSiris 

tendencia: monacemebi Tavmoyrili arian uaryofiTi daxrilobis mqone 

wrfis midamoSi. radganac korelaciis koeficienti zomavs or sidides 

Soris arsebuli wrfivi kavSiris siZlieres, unda movelodeT, rom korelaciis 

koeficienti sakmaod axlos iqneba -1-TTan. gamovTvaloT korelaciis 

koeficienti. gvaqvs: 

x 17.25 , sx 

1.44 , y 10.4 , sy 

3.56 , cov( xy , ) 4.75 da 

cov( xy , ) 4.75 

r 0.93 . 

ss 

x y 

5.13 

korelaciis koeficientis es mniSvneloba adasturebs gabnevis diagramis 

daxmarebiT miRebul daskvnas. wrfes, romlis midamoSiac Tavmoyrilia 

dakvirvebuli wyvilebi aqvs saxe: y ax b 

, sadac 

cov( xy , ) 4.75 

a 2.29 da b yax 

10.4 2.29 17.25 49.90 

2 2 

s 1.44 

, 

x 

fasi 

213

anu y 2.29 x 49.9 . 

es gantoleba kompanias saSualebas aZlevs axali produqciis fasis 

mixedviT gaakeTos prognozi gayidvis SesaZlo moculobis Sesaxeb. 

amrigad, korelaciis koeficientis mniSvneloba moTavsebulia -1-sa 

da 1-s Soris. Tu or sidides Soris adgili aqvs mkacrad dadebiT wrfiv 

damokidebulebas, maSin r -is mniSvneloba axlosaa +1-Tan. Tu or sidides 

Soris adgili aqvs mkacrad uaryofiT wrfiv damokidebulebas, maSin r -is 

mniSvneloba axlosaa -1-Tan. rodesac or sidides Soris ara gvaqvs wrfivi 

damokidebuleba, an am sidideebs Soris susti kavSiria, maSin r -is mni- 

Svneloba axlosaa 0-Tan. qvemoT moyvanilia korelaciis xarisxis sityvieri 

daxasiaTebisaTvis miRebuli terminologia da Sesabamisi diagramebi: 

srulyofili 

uaryofiTi 

korelacia 

Zlieri 

uaryofiTi 

korelacia 

saSualo 

uaryofiTi 

korelacia 

saSualo 

dadebiTi 

korelacia 

ar aris 

korelacia 

susti susti Zlieri 

uaryofiTi dadebiTi dadebiTi 

korelacia korelacia korelacia 

-1 -0.5 0 0.5 1 

uaryofiTi korelacia dadebiTi korelacia 

Y Y 

srulyofili 

dadebiTi 

korelacia 

r=–1 r=1 

srulyofili uaryofiTi 

korelacia 

Y 

X X 

. . 

. . 

. . 

. 

. 

. 

. 

. . 

. 

. . 

. . . 

ar aris korelacia 

srulyofili dadebiTi 

korelacia 

X 

Y 

. 

. 

. 

. 

. 

. 

. 

. 

. . 

. . 

. . 

. 

X 

Y 

. 

. . 

. . 

. . . 

. 

. 

. . 

. 

. 

X 

214

Zlieri uaryofiTi Zlieri dadebiTi 

korelacia 

korelacia 

korelaciis koeficientis pirdapiri gamoTvlisaTvis iyeneben formulas: 

r 

n n n 

 

n( x y ) ( x ) ( y ) 

i i i i 

i1 i1 i1 

n n n n 

2 2 2 2 

[ n( xi ) ( xi ) ] [ n( yi ) ( yi 

) ] 

i1 i1 i1 i1 

magaliTi 7. gamovTvaloT korelaciis koeficientis mniSvneloba 6 

SemTxveviT SerCeuli pirovnebis asakisa da arteriuli wnevis monacemebis 

mixedviT: 

persona A B C D E F 

asaki 43 48 56 61 67 70 

wneva 128 120 135 143 141 152 

amoxsna. 

nabiji 1. gavakeToT cxrili ise rogors es qvemoTaa naCvenebi: 

persona asaki, x wneva, y xy 2 

x 

2 

y 

A 43 128 

B 48 120 

C 56 135 

D 61 143 

E 67 141 

F 70 152 

nabiji 2. gamovTvaloT xy , x 2 da y 2 gamosaxulebaTa mniSvnelobebi 

da SevitanoT cxrilis Sesabamis svetebSi. SevkriboT svetebSi moTavebli 

mniSvnelobebi. dasrulebul cxrils eqneba Semdegi saxe: 

persona asaki, x wneva, y xy 2 

x 

2 

y 

A 43 128 5.504 1.849 16.384 

B 48 120 5.760 2.304 14.400 

C 56 135 7.560 3.136 18.225 

D 61 143 8.723 3.721 20.449 

E 67 141 9.447 4.489 19.881 

F 70 152 10.640 4.900 23.104 

x 345 y 819 xy 47.634 

2 

x 20.399 

2 

y 112.443 

nabiji 3. miRebuli mniSvnelobebi SevitanoT korelaciis koeficientis 

formulaSi da gamovTvaloT r : 

r 

647.634 345 819 

0.897 

. 

2 2 

(620.399 345 ) (6112.443 819 ) 

. 

215

daskvna. korelaciis koeficientis miRebuli mniSvneloba gveubneba, 

rom adamianis asaksa da arteriul wnevas Soris arsebobs mkacrad dadebiTi 

wrfivi damokidebuleba. 

daskvnebi korelaciis koeficientis Sesaxeb 

SerCeviTi korelaciis koeficienti SeiZleba aviRoT populaciis 

korelaciis koeficientis Sefasebad, Tu x da y SemTxveviTi sidideebi 

wrfivadaa dakavSirebuli da maTi erToblivi ganawileba normaluria. 

hipoTeza: H0 : 0 (sidideebs Soris ar aris korelacia populacia- 

Si) 


n2 

kriteriumis statistika: T r T( n2) 

2 

1r 

n2 

kriteriumis mniSvneloba T.V.: t 

r 

2 

1r 



H1 : 0 

t tn 

2, 

, 

H1 : 0 

t tn 

2, 

, 

H1 : 0 

t tn2, /2 

an t 

tn 

2, /2 

(sadac tn 2, 



P{ T t | H0}, Tu H1: 0; 

 

P - mniSvneloba: P P{ T t | H0}, Tu H1: 0; = 

 

P{| T | | t || H0}, Tu H1: 0. 

1 FT 

( t), Tu H1 

: 0; 

 

= FT 

( t), Tu H1 

: 0; 

 

2 [1 FT 

(| t |)], Tu H1 

: 0. 





0 



magaliTi 8. 0.05 mniSvnelovnebis doniT SeamowmeT hipoTeza wina 

magaliTSi gamoTvlili korelaciis koeficientis mniSvnelovnebis Sesaxeb. 

amoxsna (standartuli meTodi). 


H0 : 0, H 1 

: 0. 

216

nabiji 2. vipovoT kritikuli mniSvneloba. aq n 6 , e. i. Tavisuflebis 

xarisxia df . . 62 4 da radganac 0.05 , amitom stiudentis ganawilebis 

cxrilidan vpoulobT, rom CV . . tn 

2, /2 

t4,0.025 

2.776 . 


n2 62 

t r 0.897 4.059 . 

2 2 

1r 

10.897 

nabiji 4. gadawyvetilebis miReba: vinaidan kriteriumis mniSvneloba 

vardeba kritikul areSi, amitom nulovan hipoTezas ukuvagdebT, anu asaksa 

da wnevas Soris aris mniSvnelovani wrfivi kavSiri. 

amoxsna ( P -mniSvnelobis meTodi). 


H0 : 0, H 1 

: 0. 

nabiji 2. vipovoT Tavisuflebis xarisxi. vinaidan n 6 , amitom Tavisuflebis 

xarisxia df . . 62 4 . 


n2 62 

t r 0.897 4.059 . 

2 2 

1r 

10.897 

nabiji 4. stiudentis ganawilebis cxrilSi Tavisuflebis 4-is toli 

xarisxisaTvis movZebnoT ori mniSvneloba, romelTa Soris vardeba 

kriteriumis mniSvneloba 4.059. aseTi mniSvnelobebia 3.747 da 4.604. maTi 

Sesabamisis -ebis moZebniT vpoulobT, rom Sesabamisad, 0.01 

< 0.02. 

nabiji 5. gadawyvetilebis miReba: vinaidan P -mniSvneloba < , amitom 

nulovan hipoTezas ukuvagdebT. 

magaliTi 9. dawereT regresiis wrfis gantoleba magaliT 5-Si da 

gaakeTeT 50 wlis asakis adamianis arteriuli wnevis prognozi. 

6 

 

i1 

x 

2 

i 

amoxsna. gvaqvs: n 6 , 

6 

xi 

345 , 

i1 

217 

6 

yi 

819 , 

i1 

6 

xy 

i i 

47.634 , 

20.399 . gamovTvaloT regresiis wrfis koeficientebi: 

n n n 

 

n( x y ) ( x ) ( y ) 

a 

0.964 

, 

i i i i 

i1 i1 i1 

n 

n 

2 2 

n( xi 

) ( xi) 

i1 i1 

b 

i1 

n n n n 

2 

( yi ) ( xi ) ( xi ) ( xi yi 

) 

i1 i1 i1 i1 

n 

n 

2 2 

n( xi 

) ( xi) 

i1 i1 

81.048 

. 

amitom regresiis wrfis gantoleba iqneba: 

y0.964 x 81.048 . 

gavakeToT prognozi: 

y 0.964 50 81.048 129.248 129 . 

maSasadame, 50 wlis adamianis savaraudo arteriuli wneva iqneba 129. 

SeniSvna: im SemTxvevaSi, roca korelaciis koeficienti mniSvnelovnad 

ar gansxvavdeba nulisagan, y -is saukeTeso prognozia y -is mniSvnelobebis 

saSualo.

218 

amocanebi 

1. gamoTvaleT korelaciis koeficienti da gaakeTeT Sesabamisi daskvnebi 

x da y sidideebs Soris kavSiris Sesaxeb: 

x 99 150 130 105 219 167 180 

y 1200 1000 1050 1150 700 850 900 

2. gamoTvaleT korelaciis koeficienti da gaakeTeT Sesabamisi daskvnebi 

x da y sidideebs Soris kavSiris Sesaxeb: 

x 40 31 50 53 36 54 36 44 

y 32 24 47 50 30 55 33 41 

3. sadazRvevo kompania swavlobs kavSirs sicocxlis dazRvevaze gamoyofil 

saxsrebsa ( y ) da ojaxis Semosavlebs ( x ) Soris. am mizniT Seiswavles 

6 klientis Sesabamisi maCveneblebi: 

x 45 20 40 40 30 55 

y 70 50 60 50 55 105 

gamoTvaleT korelaciis koeficienti. 

qvemoTmoyvanil amocanebSi 0.05 mniSvnelovnebis doniT SeamowmeT 

hipoTeza korelaciis koeficientis mniSvnelovnebis Sesaxeb. 

4. gayidvebis menejers ainteresebs aris Tu ara kavSiri kviris ganmavlobaSi 

garkveul produqciaze radioTi gakeTebul sareklamo gancxadebaTa 

raodenobasa da am produqciis gayidvis Sedegad amoRebul Tanxas 

Soris (Tanxa izomeba aTasobiT larebSi): 

gacxadebaTa raodenoba, x 2 5 8 8 10 12 

amoRebuli Tanxa, y 2 4 7 6 9 10 

5. mkvlevars ainteresebs aris Tu ara kavSiri adamianis asaksa da 

mis mier kviris ganmavlobaSi gamajansaRebel varjiSze daxarjul saaTebs 

Soris: 

asaki, x 18 26 32 38 52 59 

saaTebi, y 10 5 2 3 1.5 1 

6. kvlevis mizania daadginos damokidebuleba adamianis Tviur Semosavalsa 

da mis mier Tvis ganmavlobaSi saxlis gareT sadilobis raodenobas 

Soris: 

Semosavali, x 500 1200 1500 945 850 400 540

sadilebi, y 8 12 16 10 9 3 7 

7. maRaziis menejers ainteresebs aris Tu ara kavSiri gamyidvelis 

asaksa da mis mier wlis ganmavlobaSi avadmyofobis gamo gacdenil dReebis 

raodenobas Soris: 

asaki, x 18 26 39 48 53 58 

dReebi, y 16 12 9 5 6 2 

8. pedagogs ainteresebs ramdenad Zlieri kavSiria studentis testirebis 

qulasa da mis saSualo Sefasebas Soris: 

testis qula, x 98 105 100 100 106 95 116 112 

saSualo qula, y 2.1 2.4 3.2 2.7 2.2 2.3 3.8 3.4 

9. sadazRvevo kompanias ainteresebs ramdenad Zlieri kavSiria adamianis 

kvireuli muSaobis saaTebis raodenobasa da mis mier amave period- 

Si miRebuli tramvebis raodenobas Soris: 

saaTebi, x 40 32 36 44 41 

tramvebi, y 1 0 3 8 5 

qvemoTmoyvanil amocanebSi 0.05 mniSvnelovnebis doniT SeamowmeT 

hipoTeza korelaciis koeficientis mniSvnelovnebis Sesaxeb da dawereT 

regresiis wrfis gantoleba. 

10. gamokiTxvis Sedegad miRebuli qvemoTmoyvanili monacemebis mixedviT 

gaakeTeT prognozi 30 wliani mowevis staJis mqone adamianis avad 

gaxdomaTa raodenobis: 

mowevis staJi, x 22 14 31 36 9 41 19 

avad gaxdoma, y 20 14 54 63 17 71 23 

11. mkvlevars ainteresebs rogor aris damokidebuli adamianis asaki 

mis mier kviraSi fizikur varjiSze daxarjuli saaTebis raodenobasTan: 

asaki, x 34 22 48 56 62 

saaTebi, y 5.5 7 3.5 3 1 

12. pedagogs ainteresebs damokidebulebis siZliere studentebis mier 

“statistika I”-sa da “statistika II”-Si miRebul saboloo Sefasebebs 

Soris: 

statistika I, x 87 92 68 72 95 78 83 98 

statistika II, y 83 88 70 74 90 74 83 99 

13. pedagods ainteresebs rogoraa damokidebuli studentis mier 

leqciebis gacdenaTa ricxvi mis mier miRebul saboloo Sefasebaze: 

219

gacdena, x 10 12 2 0 8 5 

Sefaseba, y 70 65 96 94 75 82 

qvemoTmoyvanil amocanebSi isargebleT P -mniSvnelobis meTodiT 

14. medikoss ainteresebs rogori kavSiria mamis wonasa (funtebSi) 

da axalSobili vaJis wonas Soris: 

mamis wona, x 176 160 187 210 196 142 205 215 

vaJis wona, y 6.6 8.2 9.2 7.1 8.8 9.3 7.4 8.6 

15. avtomobilebis dilers ainteresebs aris Tu ara kavSiri gamyidvelis 

muSaobis staJsa da mis mier TveSi gayiduli avtomobilebis raodenobas 

Soris: 

staJi, x 3 9 2 5 1 

raodenoba, y 5 14 12 21 8 


qvemoTmoyvanil 4 amocanaSi aageT gabnevis diagrama, ipoveT korelaciis 

koeficienti, SeamowmeT korelaciis koeficientis mniSvnelovneba 

0.05 -saTvis, dawereT regresiis wrfis gantoleba, gaakeTeT prognozi. 

16. swavloben damokidebulebas mamis asaksa da Svilebis raodenobas 

Soris. gaakeTeT 35 wlis mamis Svilebis raodenobis prognozi, Tu: 

asaki, x 19 21 27 20 35 31 32 38 

raodenoba, y 1 2 3 0 1 4 3 4 

17. mkvlevars ainteresebs kavSiri mZRolis asaksa da weliwadSi momxdar 

avto-sagzao SemTxvevaTa raodenobas Soris. gaakeTeT 64 wlis 

mZRolis avto-sagzao SemTxvevaTa raodenobis prognozi, Tu: 

asaki, x 63 65 60 62 66 67 59 

raodenoba, y 2 3 1 0 3 1 4 

18. medikoss ainteresebs icodes damokidebuleba bavSvis asaksa da 

kbilebze kariesis raodenobas Soris. gaakeTeT kariesis raodenobis prognozi 

11 wlis bavSvisaTvis, Tu: 

asaki, x 6 8 9 10 12 14 

raodenoba, y 2 1 3 4 6 5 

220

19. ikvleven yoveldRiurad miRebuli cximis raodenobis damokidebulebas 

pacientis qolesterinis doneze. gaakeTeT qolesterinis donis 

prognozi 8.5 grami cximis miRebisas, Tu: 

cximis raodenoba, x 6.8 5.5 8.2 10 8.6 9.1 8.6 10.4 

qolester. done, y 183 201 193 283 222 250 190 218 

20. me-18 amocanaSi ipoveT Sefasebis standartuli Secdoma. 

21. me-19 amocanaSi ipoveT Sefasebis standartuli Secdoma. 

22. me-18 amocanaSi aageT 90%-iani saprognozo intervali 7 wlis bavSvis 

kariesTa raodenobisaTvis. 

23. me-19 amocanaSi aageT 95%-iani saprognozo intervali im pacientis 

qolesterinis donisaTvis, vinc moixmars 10 gram cxims. 

pasuxebi: 

1. r 0.98 , Zlieri uaryofiTi korelacia. 2. r 0.98 , Zlieri dadebiTi korelacia. 

3. r 0.62 . 4. H0 : 0, H1 : 0; r 0.988 ; CV . . tn 

2, /2 

t4,0.025 

 

2.776 ; H 

1. 5. H0 : 0, H1 : 0; r 0.832 ; H 

1. 6. H0 : 0, H1 : 0; 

r 0.952 ; H 

1. 7. H0 : 0, H1 : 0; r 0.979 ; H 

1. 8. H0 : 0, H1 : 0; 

r 0.716 ; H 

1. 9. H0 : 0, H1 : 0; r 0.814 ; H 

0 

. 10. H0 : 0, H1 : 0; 

r 0.956 ; H 

1; y1.969 x 10.944 ; y(30) 48 . 11. H0 : 0, H1 : 0; r 0.984 ; 

H 

1; y 0.14 x 10.199 . 12. H0 : 0, H 1 

: 0; r 0.963 ; H 1 

; y0.86 x 10.251 . 13. 

H0 : 0, H1 : 0; r 0.981 ; H 

1; y 2.668 x 96.784 . 14. H0 : 0, H1 : 0; 

r 0.306 ; 0.2 

0 

; r aramniSvnelovania. 15. H0 : 0, 

H1 : 0; r 0.491 ; 0.2 

0 

; r aramniSvnelovania. 16. 

H0 : 0, H1 : 0; r 0.857 ; H 

1; y0.19 x 2.818 ; y(35) 3.84 4 . 17. H0 : 0, 

H1 : 0; r 0.078 ; H 

0 

; korelacia aramniSvnelovania. 18. H0 : 0, 

H1 : 0; r 0.842 ; H 

1; y0.551 x 1.918 ; y(11) 4.14 4 . 19. H0 : 0, H1 : 0; 

r 0.602 ; H 

0 

; korelacia aramniSvnelovania. 20. 1.129. 21. 29.5. 22. (0, 5). 23. 

217.5 – dakvirvebul mniSvnelobaTa saSualo. 

221

Tavi XVIII 

regresia da korelacia 

1. martivi wrfivi regresia. 

zogadad amocana ase aRiwereba: mocemulia ( xY 

i, i) 

, i 1,2,..., n wyvilebi, 

romelTa Soris Semdegi damokidebulebaa 

Y B B x , i 1,2,..., n, 

i 0 1 i i 

2 

sadac B 0 

da B 1 

ucnobi mudmivebia, i N(0, ) ki normalurad ganawilebuli 

SemTxveviTi sidideebia saSualoTi nuli da ucnobi 2 dispersiiT. 

y B0 B1x 

wrfes regresiis wrfe ewodeba. B 0 

da B 1 

koeficientebs – 

regresiis koeficientebi, xolo i 

SemTxveviT sidides – SemTxveviTi gadaxra 

ewodeba. i 

-- SemTxveviTi sididea, romelic ganapirobebs y B0 

B1x 

wrfidan gabnevas. x cvlads uwodeben amxsnel (damoukidebel) cvlads an 

prediqtors, xolo Y -s ki – mopasuxe (damokidebul) cvlads. 

B 

1 

koeficients SeiZleba mivceT Semdegi interpretacia: amxsneli x 

cvladis erTi erTeuliT cvlileba iwvevs damokidebuli Y cvladis 

cvlilebas saSualoD B 1 

sididiT. B 0 

koeficienti warmoadgens regresiis 

wrfis y RerZTan gadakveTis wertils. 

E(Y|X=x) 

y=B 0 +B 1 x 

 

tg = B 1 

B 0 

x 0 x 0 +1 x 

saWiroa ( x1, Y1 ), ( x2, Y2 

),...,( xn, Y 

n) 

damoukidebeli dakvirvebebis safuZvelze 

aigos B 0 

da B 1 

koeficientebisa da 2 dispersiis Sefasebebi. 

2. umcires kvadratTa meTodi. 

222

B 

0 

da B 1 

koeficientebis b 0 

da b 1 

Sefasebebs eZeben e. w. umcires 

kvadratTa meTodiT: 

b 

n 

2 

( Yi b0 b1x 

i) min 

i1 

n n n 

 

. 

 

n xY x Y 

i i i i 

n 

i1 i1 i1 

SSXY 

1 

2 

: , 0 i 1 i 1 

n n SS 

2 

X 

n i1 n i1 

nx xi 

 

i1 i1 

n 

n 

2 2 2 

x ( i ) i 

i1 i1 

, 

SS x x x nx 

 

 

223 

b n 

1 Y b 1 

x Y b x 

n 

, 

SS ( x x)( y y) 

x y nxy 

XY 

. 

i i i i 

i1 i1 

 

Y b0 b1x 

wrfes ewodeba regresiis wrfis Sefaseba, an umcires 

kvadratTa meTodiT agebuli wrfe. 

SerCeviTi korelaciis ricxviTi mniSvneloba: 

r 

n 

XY 

. 

SS 

SS 

X 

SS 

Sefasebuli regresiis wrfis gantoleba Semdegnairad gadaiwereba: 

y Y b1( x x) 

. es wrfe gadis ( xY) , wertilze da ( Y i Y i ) 0 . 

modeli romelic Y B0 B1x formuliTaa mocemuli gulisxmobs, 

rom gabnevis diagramaze wertilebis gabneva ganpirobebulia Y cvladis 

mniSvnelobaTa gabneviT regresiis mrudidan, xolo x cvladis mniSvnelobebi 

ganixileba zust mniSvnelobebad. Tu regresiis wrfis gantolebaSi 

x -is nacvlad CavsvamT dakvirvebul x 1 

,..., x 

n 

mniSvnelobebs, maSin Sesabamis 

 

Y1 b0 b1 x1,..., 

Y n b0 b1x 

n mniSvnelobebs ewodeba Y cvladis prognozireb- 

 

 

uli mniSvnelobebi, xolo e1 Y1 Y1,..., en 

Yn Y n sidideebs naSTebi ewodeba. 

sidides 

SS 

SSE : e ( Y Y ) Y ( b b x ) SS 

n n n 

2 

2 2 

2 

XY 

i i i 

i 0 1 i 

Y 

i1 i1 i1 

SSX 

ewodeba kvadratebis narCeni jami (naSTTa kvadratebis jami, sum of square 

for eror). 

SST 

n n n 

2 2 2 

( Yi Y) ( Yi Y i) ( Y iY) 

i1 i1 i1 

SST : : SSE SSR 

n 

i1 

. 

(total sum of squares) sidides srul variacias uwodeben, sidide 

n 

2 

i aris gabnevis is nawili, romelis axsnilia regresiis 

i1 

SSR : ( Y Y ) 

 

Y

wrfiT (regression sum of squares). SSE aris sruli gabnevis is nawili, romelic 

ar aixsneba regresiis saSualebiT. ~ 2 SSR 

SST 

( n 1) 

; ~ 

2 (1) 

; 

2 2 

 

 

SSE 

~ 

2 ( n 2) . SST SSE SSR . 

2 

 

n1 n2 1 

Y 

Y i 

i 

Y 

Y 

Y i – Y 

i 

 

Y i Y 

Y i –Y 

x x i x 

determinaciis koeficienti. regresiis xarisxis dasaxasiaTeblad 

SemohyavT e. w. SerCeviTi determinaciis koeficienti: 

R 

2 

 

n 

 

i1 

n 

 

i1 

 

( Y i Y 

) 

( Y Y 

) 

i 

2 

2 

SSR 

 

, 

SSR SSE 

1 

SST SST SST 

2 

R warmoadgens regresiiT axsnil gabnevis wils srul gabnevaSi. 

e. i. 

2 

mtkicdeba, rom R -- axals arafers iZleva, vinaidan 

SerCeviTi korelaciis koeficientia. 

R 

r , sadac r 

2 2 

Tu r 2 mcirea, e.i. regresiiT axsnilia xruli gabnevis mcire wili, 

maSin mosaZebnia sxva alternatiuli modeli (magaliTad, arawrfivi an 

mravlobiTi regresiis modeli), romelic ufro efeqturad axsnida Y 

cvladis dakvirvebuli mniSvnelobebis srul gabnevas Tavisi saSualo 

mniSvnelobebis mimarT. 

n 

n n 

2 2 2 2 2 

( Y iY ) R ( Yi Y ) r ( Yi 

Y ) 

i1 i1 i1 

, 

n 

n 

2 2 2 

( Yi 

Y i) (1 r ) ( Yi 

Y 

) 

i1 i1 

. 

1. statistikuri daskvnebi regresiis koeficientebis Sesaxeb. 

mtkicdeba, rom b 0 

da b 1 

Sesabamisad B 0 

da B 1 

parametrebis Caunacvlebeli 

Sefasebebia ( Eb0 B0 , Eb1 B1 

) da 

224

n 

2 

xi 

2 2 i 1 

b 

: Db 

0 0 

 

n 

n 

2 

2 

nxi 

xi 

 

i1 i1 

 

 

 

 

, b1 

: Db 

 

2 2 

1 

n 

n 

2 

2 

nxi 

xi 

 

i1 i1 

n 

 

 

ucnobi 2 -is Caunacvlebeli Sefaseba moicema TanafardobiT 

n 

n 

2 2 2 

i 

i i 

i1 i1 

. 

S ( e )/( n2) ( ( Y Y ) )/( n 2) SSE /( n 2) 

sidides S SSE /( n 2) -- standartuli Secdoma ewodeba. 

b 

0 

da b 1 

statistikebis dispersiebis Sefasebebia Sesabamisad 

2 

2 2 

xi 

Sb 

S 

 

0 

, 

2 2 1 

Sb 

S . 

n SS 

1 

SSX 

T 

b 

j 

X 

bj 

Bj 

t( n 2) , j 0,1. aqedan gamomdinare igeba ndobis interva- 

S 

b 

j 

B j 

-saTv- 

li B j 

-saTvis. mniSvnelovnebis donis mqone ndobis intervali 

is Semdegi saxisaa: ( bj sb tn2, /2, bj sb tn2, /2) 

. 

ganvixiloT ZiriTadi 

j 

0 

0 

: 

J j 

j 

H B B hipoTeza 

0 

1 

: 

J j 

winaaRmdeg. am SemTxvevaSi kriteriumis statistika iqneba 

0 

T ( b B )/ S t( n 2) . 

Tu 

b j j b 

j 

j 

H B B alternativis 

T 

b j 

statistikis dakvirvebuli mniSvnelobisaTvis 

b j 

t da mniSvnelovnebis 

donisaTvis sruldeba piroba : tb 

tnk 

1, /2 

, maSin uarvyofT 

H B B hipoTezas (Sesabamisad, vRebulobT 

0 

0 

: 

J j 

j 

0 

1 

: 

J j 

H B B hipoTezas), winaaRmdeg 

SemTxvevaSi H 0 

-is uaryofis safuZveli ar gagvaCnia. 

SevniSnavT, rom Tu B 1 

0, maSin regresiis wrfe abscisTa RerZis paraleluria, 

e. i. x -is cvlilebiT Y ar icvleba. Tu Cven gvinda, mniSvnelovnebis 

doniT, davrwmundeT, rom ar arsebobs raime (dadebiTi an uaryofiTi) 

wrfivi kavSiri x da Y sidideebs Soris, nulovani hipoTeza 

H0: B1 0 unda Semowmdes ormxrivi H 1 

: B 1 

0 alternativis winaaRmdeg. Tu 

gvsurs davrwmundeT dadebiTi (Sesabamisad, uaryofiTi) wrfivi kavSiris 

arsebobaSi, maSin vixilavT marjvena calmxriv H 1 

: B 1 

0 (Sesabamisad, marcxena 

calmxriv H 1 

: B 1 

0) alternativas. yvela SemTxvevaSi hipoTezis Semowmeba 

xdeba T : T b / S t( n 2) statistikis gamoyenebiT. gadawyvetile- 

1 b 1 

b 

1 1 

bis miRebis wesi ase yalibdeba: 

225

saxlis gayidvis 

fasi 

a). ormxrivi alternativis H 1 

: B 1 

0 SemTxvevaSi H0: B1 0 hipoTezas 

uarvyofT, Tu | t 1 

| tn 

2, /2 

; 

b). marjvena calmxrivi alternativis H 1 

: B 1 

0 SemTxvevaSi H0: B1 

0 

hipoTezas uarvyofT, Tu t 1 

t n 2, 

; 

g). marcxena calmxrivi alternativis H 1 

: B 1 

0 SemTxvevaSi H0: B1 

0 

hipoTezas uarvyofT, Tu t 1 

t n 2, 

, sadac t 1 

aris T 1 

statistikis dakvirvebuli 

mniSvneloba, xolo tn 2, 

-- stiudentis tn ( 2) ganawilebis zeda - 

kritikuli wertili. 

magaliTi 1. uZravi qonebis kompaniis agnts surs ganWvritos saxlis 

gasayidi fasi. misi azriT saxlis fasi yvelaze mWidro kavSirSia mis 

frTobTan. man SemTxveviT SearCia adre gayiduli 15 saxlis monacemebi, 

romlebic moyvanilia cxrilis saxiT (sadac x -- saxlis farTobia ( 10 

kv. m.), xolo Y -- saxlis gayidvis fasi (1000$)). agentis mizania saxlis 

fasis (damoukidebeli fasi) prognozireba misi farTobis (prediqtoris) 

mixedviT. 

x 

i 

Y 

i 

x 

i 

Y 

i 

x 

i 

Y 

i 

20.0 89.5 14.3 82.5 22.0 112.6 

14.8 79.9 27.5 126.3 19.0 120.8 

20.5 83.1 16.5 79.3 12.3 78.5 

12.5 56.9 24.3 119.9 14.0 74.3 

18.0 66.6 20.2 87.6 16.7 74.8 

amoxsna. 

nabiji 1. SevadginoT gabnevis diagrama 


150 

100 

50 

0 

0 10 20 30 

saxlis farTobi 

aqedan Cans, rom Y cvladis X -isagan damokidebulebaSi ikveTeba 

wrfivobis tendencia da SesaZlebelia, rom martivma wrfivma regresiam 

kargad aRweros igi. amitom maTematikur modelad SegviZlia avirCioT 

Y B B x . 

i 0 1 i i 

nabiji 2. gamovTvaloT sidideebi: 

226

n 

xi 

172.6 

, 

i1 

sidideebi: 

n 

Yi 

1332.6 

, 

i1 

n 

xi 

, 

i1 

n 

Yi 

, 

i1 

n 

2 

xi 

, 

i1 

n 

2 

xi 

5222.24 , 

i1 

n 

2 

Yi 

, 

i1 

n 

xY 

i i. 

i1 

n 

2 

Yi 

124618.42 

, 

i1 

n 

xY 

i i 

25257.97 . 

nabiji 3. gamovTvaloT b 0 

-isa da b 1 

-is gamosaTvlelad saWiro 

n 

1 272.6 

x xi 

18.17 , 

15 

n i 1 

x 

n 

2 

2 

i 

268.19 , 

i1 

SS x nx 

 

n 

xY i i 

i1 

n i 1 

i1 

n 

1 1332.6 

Y Yi 

88.84 . 

15 

Y 

n 

2 

2 

i 

6230.24 , 

i1 

SS Y nY 

 

SS xY nxY 

1040.18 

. 

nabiji 4. gamovTvaloT b 0 

da b 1 

Sefasebebi: 

SSxY 

1040.18 

b1 

3.8786 , 

SS 268.19 

b0 Y b1x 

88.84 3.8786 18.17 18.354 . 

nabiji 5. davweroT regresiis wrfis Sefaseba: 

 

Y 18.354 3.8786 x . 

miRebuli monacemebi gvaZlevs Semdeg grafikul suraTs : 

x 

150 

100 

y = 3.8786x + 18.354 

50 

0 

0 10 20 30 

Y Linear (Y) 

daskvna. b 1 

-is miRebuli mniSvneloba b 1 

3.8786 , niSnavs, rom saxlis 

farTobis yoveli erTi erTeuliT gazrda iwvevs saxlis fasis saSualod 

3.8786 1000$ 3878.6$ 

-iT gazrdas. 

nabiji 6. gamovTvaloT ESS . gvaqvs: 

2 2 

SSxY 

(1040.18) 

SSE SSY 

6230.24 21995.88 , 

SS 

268.19 

saidanac miiReba ucnobi 2 dispersiis Sefaseba 

x 

227

2 SSE 21995.88 

S 168.91 . 

n2 13 

2 

2 

nabiji 7. gamovTvaloT S 

b 0 

da S 

b 1 

: 

2 

S 2 S 2 xi 

5222.24 

b 

168.91 219.26 

0 

n 

SS 15268.19 

, 

X 

Sb 

S 

1 

SSX 

2 2 1 1 

168.91 0.6298 , 

268.19 

Sesabamisad, Sb 

1 

0.7936 . 

nabiji 8. gamovTvaloT determinaciis koeficienti: 

2 SSE 

R 1 0.648 . 

SST 

determinaciis koeficientis gamoTvlili mniSvneloba gviCvenebs, 

rom martivi wrfivi regresiiT aixsneba saxlis gasayidi fasis variaciis 

mxolod 64.8%, danarCeni 35.2% auxsneli rCeba. saWiroa SeirCes sxva 

modeli (arawrfivi regresia, an mravlobiTi regresia). 

magaliTi 2. magaliT 1-Si: 

a). avagoT 0.05 mniSvnelovnebis donis mqone ndobis intervali 

B -saTvis; 

1 

b). SevamowmoT H 0 

: B 1 

0 hipoTeza H 1 

: B 1 

0 alternativis winaaRmdeg. 

b1 

B1 

amoxsna. a). cnobilia, rom T1 : t( n 2) . stiudentis zeda kriti- 

S 

kuli wertilebis cxrilSi vpoulobT tn 

2, /2 

t13,0.025 2.16 . amitom saZiebeli 

ndobis intervali iqneba: 

3.88 2.16 0.7936 B 3.88 2.16 0.7936 , 

anu 2.1658 B1 

5.5942 . 

b). T 1 

statistikis gamoTvlili mniSvneloba t 1 

iqneba: 

b1 

B1 

3.88 0 t1 4.8891 t13,0.025 

2.16 . 

S 0.7936 

b 

1 

amitom H 0 

: B 1 

0 hipoTezas uarvyofT (vinaidan | t 1 

| tn 

2, /2 

). 

amrigad, 0.05 mniSvnelovnebis doniT sarwmunoa mtkiceba, rom 

saxlis fassa da mis farTobs Soris arsebobs wrfivi kavSiri. 

4. mravlobiTi regresia. 

ganvixiloT wrfivi mravlobiTi regresiis modeli 

Y B0 B1 X1 BkX k 

, 

sadac Y -- damokidebuli cvladia, X1,..., X 

k 

-- damoukidebeli cvladebia 

(maT amxsnel cvladebs, prediqtorebs, an regresorebs uwodeben), -- Se- 

228 

b 

1 

1

SfoTebaa, romelsac xSirad WeSmarit gadaxras an Secdomas uwodeben. 

2 

SemTxveviTi sididea, romlisTvisac E 0, xolo D 

0. igulisxmeba, 

rom damoukidebeli X 1 

,..., X 

k 

cvladebis mniSvnelobebi zustadaa cnobili 

da gadaxrebi miewereba mxolod Y cvlads. 

cxadia, rom Y SemTxveviTi sididis maTematikuri lodini pirobaSi, 

rom X 1 

x 1 

,..., Xk xk 

moicema formuliT: 

E( Y | X1 x1 ,..., Xk xk ) B0 B1 x1 

Bk xk 

, 

xolo dispersia imave pirobebSi iqneba: 

2 

D( Y | X1 x1,..., Xk 

xk) 

. 

k cvladis y B0 B1 x1 Bkx 

k 

funqcias WeSmariti regresiis funqcia 

ewodeba, xolo B 0 

, B 1 

,..., B 

k 

sidideebs ki – regresiis koeficientebi. 

sabolood, mravlobiTi regresiis amocana Semdegnairad aRiwereba: 

mocemulia n moculobis SerCeva ( Yi , x1 

i,..., xki 

), i 1,2,..., n, 

n k , sadac 

Yi B0 B1 x1 i 

Bk xki i, i 1,2,..., n, (3) 

xolo i 

SemTxveviTi sididea, romelic ganapirobebs Y i 

cvladis gadaxras 

WeSmariti regresiis funqciis B0 B1 x1i Bk xki 

mniSvnelobidan; 

2 

i N(0, ) , E( ij) 0 , i j. 

iseve rogorc martivi regresiis SemTxvevaSi, pirvel etapze xdeba 

regresiis Teoriuli B 0 

, B 1 

,..., B 

k 

koeficientebisa da ucnobi 2 dispersiis 

Sefaseba. regresiis koeficientebis Sefaseba isev warmoebs umcires kvadratTa 

meTodis gamoyenebiT. mas Semdeg rac miRebulia Teoriuli regres- 

 

iis koeficientebis b 0 

, b 1 

,..., b k 

Sefasebebi, Y b0 b1 x1 bkx 

k funqcia gansazRvravs 

regresiis funqciis Sefasebas. sidides 

 

ei Yi Yi 

Yi b0 b1 x1 

i 

bk xki 

naSTi ewodeba. 

iseve rogorc martivi regresiis modelSi, ucnobi 

2 Di 

dispersiis 

Sefaseba eyrdnoba naSTebis (Sesworebebis) kvadratebis jams (sum of 

square errors (residuals) – SSE): 

n n 

2 2 

i i i i 

i1 i1 

. 

SSE : e ( Y Y 

) 

vinaidan am gamosaxulebaSi Sedis ( k 1) Sefasebuli parametri, misi 

Tavisuflebis xarisxia n( k 1) da amitom dispersiis Sefasebad iReben 

sidides: 

2 SSE 

S MSE 

. 

n( k1) 

2 

radganac, daSvebis Tanaxmad i N(0, ) , amitom Y i 

normalurebia da 

rogorc maTi wrfivi kombinacia, aseve normaluri iqneba b j 

Sefasebebic. 

mtkicdeba, rom: T : ( b B )/ S t( n( k 1)) . 

j j j b j 

229

determinaciis koeficienti. davweroT dispersiuli Tanafardoba 

n 

 

i1 

SemoviRoT aRniSvnebi: 

maSin gvaqvs: 

n 

2 

( 

i 

) , 

i1 

SST Y Y 

n1 nk1 

k 

n 

 

i1 

 

i1 

 

Y i 

( Y Y 

) ( Y ) ( Y 

) . 

i 

2 

i 

 

2 

Y i 

n 

n 

2 

( 

i 

i) 

da 

i1 

SSE Y Y 

2 

n 

2 

( i 

) . 

i1 

SSR Y Y 

SST SSE SSR , sadac SSR -- aris sruli gabnevis regresiiT 

axsnili nawili, SSE -- sruli gabnevis is nawili, romelic ar aixsneba 

regresiiT. TiToeul wevrs qveviT miwerili aqvs Tavisuflebis xarisxi. 

e. w. mravlobiTi determinaciis koficienti warmoadgens sruli variaciis 

im nawils, romelic axsnilia mravlobiTi regresiis modeliT: 

2 SSE SSR 

R 1 ( ) 

SST 

SST 

. 

garda amisa, SemoaqvT e. w. Tavisuflebis xarisxTan SeTanxmebuli (adjusted) 

determinaciis koeficienti: 

2 SSE /( n k 

1) 

adjusted R 1 

, 

SST /( n 1) 

rac aixsneba im garemoebiT, rom Tu damoukidebel cvladTa raodenoba k 

Sedarebadia n-Tan, R 2 -is mniSvneloba SeiZleba ararealurad didi gamovides 

da migviyvanos mcdar daskvnamde. im SemTxvevaSi, roca n mniSvnelovnad 

didia k -sTan SedarebiT, maSin es koeficientebi daaxloebiT tolia, 

2 2 

sazogadog ki adjusted R R . 

mtkicdeba, rom R 2 statistika e. w. SerCeviTi mravlobiTi korelaciis 

koeficientis kvadratis tolia R 

2 r 

2 , romelic warmoadgens SerCevi- 

Ti korelaciis koeficients Y da Y 

SemTxveviT sidideebs Soris: 

r 

n 

 

i1 

( Y Y)( Y Y) 

i 

n 

n 

2 2 

( Yi 

Y) ( Yi 

Y) 

i1 i1 

modelis vargisianobis Semowmeba. modelis vargisianobis Semowmeba 

mravlobiTi regresiis SemTxvevaSi mdgomareobs H0 : B1 B2 

Bk 

hipoTezis 

SemowmebaSi H 1 

: erTi mainc Bi 

0, i 1,2,..., k alternativis winaaRmdeg. 

gasagebia, rom Tu H 

0 

hipoTeza dawunebuli ar iqneba, arc erTi damoukidebeli 

cvladi (regresori) ar aris wrfivad dakavSirebuli Y -Tan 

da amitom modeli uvargisia prognozirebis miznebisaTvis. maSin roca, 

Tu erTi mainc Bi 

0, modeli raime TvalsazrisiT mainc gamosadegia. 

mtkicdeba, rom nulovani hipoTezis samarTianobisas 

SSR / k MSR 

F F( k, n ( k 1)) 

SSE /( n( k 1)) 

MSE 

. 

 

i 

. 

230

F -statistika SeiZleba Caiweros mravlobiTi determinaciis koeficientis 

saSualebiT: 

2 2 

R / k n( k 1) 

R 

F , 

2 2 

(1 R )/( n( k 1)) k 1 

R 

e. i. F -statistika [ n( k 1)]/ k Tanamamravlis sizustiT emTxveva statistikas: 

R 

2 /(1 R 

2 ) , romelic warmoadgens axsnili variaciis fardobas auxsnelTan 

da Tu es fardoba sakmarisad didia, bunebrivia uarvyoT H 

0 

hipoTeza 

H 

1 

alternativis sasargeblod. Sesabamisad, kritikul are iqneba -- 

F F krit saxis. 

gadawyvetilebis miRebis wesi: Tu F statistikis dakvirvebuli mni- 

SvnelobisaTvis f da mniSvnelovnebis donisaTvis sruldeba piroba 

2 

n( k 1) 

R 

f F 

2 k, nk1, 

 

, 

k 1 

R 

maSin uarvyofT H 

0 

hipoTezas, winaaRmdeg SemTxvevaSi H0 

-is uaryofis safuZveli 

ar gagvaCnia. 

5. statistikuri daskvnebi mravlobiTi modelis parametrebis Sesaxeb. 

Tu H 0 

hipoTeza dawunebulia, maSin vamowmebT hipoTezas calkeuli 

regresiis koeficientis nulTan tolobis Sesaxeb. amrigad, yoveli j -saTvis 

( j 1,2,..., k) mowmdeba H 0 

: B 0 hipoTeza H 1 

: B 0 alternativis winaa- 

Rmdeg. am SemTxvevaSi testis statistikaa T b / S 

j 

b j b 

j 

j 

j 

, romelsac nulovani 

hipoTezis samarTlianobisas aqvs stiudentis ganawileba Tavisuflebis 

xarisxiT n( k 1) : T b / S t( n( k 1)) . 

b j b 

j 

gadawyvetilebis miRebis wesi: Tu 

j 

T 

b j 

statistikis dakvirvebuli mni- 

SvnelobisaTvis t 

b j 

da mniSvnelovnebis donisaTvis sruldeba piroba 

| tb 

| tnk 

1, /2, 

j 

maSin H 

0 

hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi H0 

-is uaryofis safuZveli 

ar gagvaCnia. 

mniSvnelovnebis donis ndobis intervali B 

j 

koeficientisaTvis 

Semdegia: bj sb t j nk1, /2 Bj bj sb t 

j nk1, /2, sadac tn k 1, /2 

-- Tavisuflebis 

n( k 1) xarisxis mqone stiudentis ganawilebis zeda /2-kritikuli wertilia. 

davubrundeT uZravi qonebis agentis amocanas, romelsac Cven vixilavdiT 

martivi wrfivi regresiis konteqstSi. rogorc vnaxeT, saxlis fasis 

cvalebadobis axsna mxolod farTobis cvalebadobis xarjze ar iZle- 

231

2 

va damakmayofilebel Sedegs (determinaciis koeficienti iyo R 0.648 , 

anu axsnilia cvalebadobis mxolod 64.8%). agentma gadawyvita ganexila 

mravlobiTi wrfivi regresiis modeli: 

Y B0 B1 X1 B2 X2 B3 X3 

, 

sadac Y -- saxlis gayidvis fasia ( 1000$ 

-Si), X 1 

-- saxlis farTobia ( 

10 kv. m.-Si), X 

2 

-- saxlis eqsploataciis xangrZlivobaa (wlebSi), xolo 

X 

3 

-- saxlis kuTvnili miwis nakveTis farTobia ( 100 kv. m.-Si). Segrovebuli 

monacemebi Cawerilia Excel-is davTris furcelSi (qveprograma: Tools/ 

Data Analysis/Regression): 

1 

A B C D E 

saxlis 

nomeri Y X 1 X 2 X 3 

2 1 89.5 20 5 4.1 

3 2 79.9 14.8 10 6.8 

4 3 83.1 20.5 8 6.3 

5 4 56.9 12.5 7 5.1 

6 5 66.6 18 8 4.2 

7 6 82.5 14.3 12 8.6 

8 7 126.3 27.5 1 4.9 

9 8 79.3 16.5 10 6.2 

10 9 119.9 24.3 2 7.5 

11 10 87.6 20.2 8 5.1 

12 11 112.6 22 7 6.3 

13 12 120.8 19 11 12.9 

14 13 78.5 12.3 16 9.6 

15 14 74.3 14 12 5.7 

16 15 74.8 16.7 13 4.8 

determinaciis koeficienti. mocemuli monacemebi SeviyvanoT kompiuterSi 

Semdegi wesiT: sawyisi (Semavali) monacemebis Y cvladis areSi 

CavweroT damokidebuli cvladis ricxviTi mniSvnelobebis misamarTi 

$ C $2: $ C $17 , X cvladis areSi -- $ D$2:$ F $17 : 

qveprograma Regression gvaZlevs Semdeg amonabeWds: 

SUMMARY OUTPUT 

Regression Statistics 

Multiple R / SerCeviTi korelaciis 0.957 

koeficienti 

232

R Square / determinaciis koeficienti 0.916 

Adjusted R Square / 

0.893 

SeTanxmebuli determinaciis 

koeficienti 

Standard Error / standartuli Secdoma 6.894 

Observations / dakvirvebebi 15.000 

rogorc vxedavT, am SemTxvevaSi determinaciis koeficienti (R-square) 

gamovida 2 2 

R 0.916 anu 91.6%, xolo adjusted R 0.893 anu 89.3%. gavixsen- 

2 

oT, rom martivi wrfivi regresiis SemTxvevaSi R 0.648 . amrigad, ori 

damatebiTi cvladis Semoyvanam sagrZnoblad gazarda determinaciis 

koeficienti, saxlis fasis variaciis 91.6% aixsna sami damoukidebeli 

cvladiT da auxsneli darCa mxolod 8.4%. 

modelis vargisianobis Semowmeba. kompiutruli amonabeWdis meore 

bloki (ANOVA) warmoadgens dispersiuli analizisaTvis saWiro sidideebis 

ricxviT mniSvnelobebs: 

ANOVA 

dispersiuli 

analizi 

df Tavisuflebis 

xarisxi 

SS 

kvadr. jami 

MS 

saS. kvadr. gadaxra 

SSR 

Regression 3 SSR=5707.43 MSR= =1902.479 

k 

Residual 11 SSE=522.79 MSE= 

Total 14 SST=6230.23 

MSE =47.527 

n k 1 

Significance F 

F 

mniSvnelovn- 

F statistika 

ebis done 

MSR 

f 

MSE 

40.029 

0.000 

aq bolo svetSi miTiTebulia P{ F f | H0} 

-is ricxviTi mniSvneloba, 

anu mniSvnelovnebis is minimaluri done, romliTac arsebuli monacemebis 

safuZvelze xdeba H 0 

hipoTezis uaryofa. 

Tu aviRebT magaliTad, 0.05 , maSin Fk , nk1, F3,11,0.05 8.76 da vinaid- 

40.029 f F 8.76 , amitom ZiriTad hipoTezas ukuvagdebT 0.05 

an 

3,11,0.05 

mniSvnelovnebis doniT da vaskvniT, rom erTi mainc regresiis koeficienti 

gansxvavebulia nulisagan. 

statistikuri daskvnebi mravlobiTi modelis parametrebis Sesaxeb. 

amonabeWdis mesame blokidan viRebT informacias ucnobi regresiis 

koeficientebis Sefasebis Sesaxeb: 

Coefficients 

koeficienti 

Standard Error 

standartuli 

Secdoma 

t Stat P-value Lower 

t testi P mniSv 95% 

neloba 

Upper 

95% 

Intercept (b 0 ) -16.058 19.071 -0.842 0.418 -58.033 25.917 

233

x1 (b 1 ) 4.146 0.751 5.520 0.000 2.493 5.800 

x2 (b 2 ) -0.236 0.881 -0.268 0.794 -2.176 1.703 

x3 (b 3 ) 4.831 0.901 5.361 0.000 2.848 6.814 

b j 

b j b 

aq pirvel svetSi mocemulia b 0 

, b 1 

, b 2 

, b 3 

Sefasebebi; meoreSi – maTi 

standartuli gadaxrebi, s ; mesameSi -- T b / S statistikis dakvirvebuli 

mniSvnelobebi, t b / s ; meoTxe svetSi -- 

b j b 

j 

j 

j 

j 

T 

b j 

statistikebisaTvis p - 

H B 

mniSvneloebebi Sesabamisi H0 : Bj 

0 hipoTezis Sesamowmeblad 

1 

: 

j 

0 

alternativis winaaRmdeg, xolo bolo or svetSi ki miTiTebulia 95%-iani 

ndobis intervalebi 

B -saTvis. 

j 

1 2 3 

0 

b0 16.058 mniSvneloba miuTiTebs saSualo gasayid fass, roca 

X X X da misi interpretacia azrs moklebulia; b 1 

4.146 mniSvneloba 

miuTiTebs imaze, rom farTobis yoveli damatebiTi erTeulisaTvis 

saxlis gayidvis fasi izrdeba saSualod 4146 dolariT; b 2 

0.236 mniSvneloba 

miuTiTebs imaze, rom saxlis eqsploataciis yoveli damatebiTi 

wlisaTvis fasi iklebs 236 dolariT; b 3 

4.831 mniSvneloba ki miuTiTebs 

imaze, rom miwis nakveTis yoveli damatebiTi 100 kv. metrisaTvis saxlis 

fasi izrdeba 4831 dolariT. 

testis statistikis Tb 

2 

b2 / Sb 

2 

dakvirvebuli mniSvnelobaa t 2 

0.268 , 

xolo p -mniSvnelobaa 0.794, rac imis maCvenebelia, rom nulovani hipoTezis 

uaryofa xdeba mniSvnelovnebis mxolod iseTi doniT, romelic aranaklebia 

0.794-ze. amitom mniSvnelovnebis 0.05 doniT (0.05

is Sedegebi Cawerilia Semdegi saxiT 

x x x 

 

, ,..., 

1 2 

n 

 

y 

 

1 

y 

 

2 

y 

 

n ,sadac i 

x da 

aris i -uri obieqtis damaxasiaTebeli niSnebi. am niSnebs SeiZleba arc 

hqondeT zusti ricxobrivi mniSvnelobebi, magram SesaZlebeli iyos maTi 

ise dalageba, rom TiToeuli niSnis mixedviT yovel obieqts mieces 

rigiTi nomeri -- rangi. 

magaliTi 1. qvemoT moyvanilia erTi da igive saxis produqciis ran- 

Jirebis Sedegebi Catarebuli ori A da B eqspertis mier. I svetSi mocemulia 

Sesamowmebeli produqciis pirobiTi nomeri, xolo II da III svetSi 

A da B eqspertebis Sefasebebi. magaliTad, A eqsperti saukeTesod miiCnevs 

produqciis #3 nimuSs, maSin roca B eqsperti mas meoTxe adgils miawers. 

ramodenime nimuSis erTnairi Sefasebisas, TiToeuls miewereba Sesabamisi 

rangebis saSualo ariTmetikuli. magaliTad, A eqsperti erTnairad 

afasebs produqciis #5 da #7 nimuSebs, isini iyofen II-III adgils, amitom 

orives miewereba (23)/ 2 2.5 

-is toli rangi. IV svetSi A eqspertis 

rangebi dalagebulia zrdis mixedviT, xolo momdevno V svetSi mas miwerili 

aqvs igive nimuSisaTvis B eqspertis mier mikuTvnebuli rangi. 

produqciis 

pirobiTi 

nomeri 

A eqspertis B eqspertis 

Sefaseba Sefaseba 

R 

A 

R 

B 

R 

A 

R 

B 

( R R ) 

1 4 8 1 4 -3 9 

2 9 7 2.5 1 1.5 2.25 

3 1 4 2.5 5 -2.5 6.25 

4 7 6 4 8 -4 16 

5 2.5 5 5 3 2 4 

6 8 10 6 2 4 16 

7 2.5 1 7 6 1 1 

8 6 2 8 10 -2 4 

9 5 3 9 7 2 4 

10 10 9 10 9 1 1 

 

63.5 

imisaTvis, rom SeviswavloT A da B eqspertebis kriteriumebs Soris 

urTierTdamokidebuleba, gamoviyenoT spirmenis e. w. rangobrivi korelaciis 

koeficienti, romelic Semdegi formuliT gamoiTvleba: 

n 

6 

1 

R 

2 

2 

A 

RB 

. 

nn ( 1) 

i1 

Tu eqspertebis mier gamoyenebul kriteriumebs Soris sruli Tanxvedraa, 

n 

RARB 

0 

da 1; Tu uaryofiTi kavSiria, maSin 1 da Tu 

maSin 2 

i1 

kavSiri ar arsebobs, maSin 0 . 

ganxilul magaliTSi 6 

1 63.5 1 0.385 0.615 

2 

10 (10 1) 

. 

A 

B 

2 

y i 

235

spirmenis rangobrivi korelaciis koeficienti gamoiyeneba Semdegi 

hipoTezebis Sesamowmeblad: 

H 

0 

: niSnebi damoukidebelia, e. i 0 , 

H niSnebs Soris dadebiTi kavSiria, e. i. 0 . 

1 

: 

cnobilia, rom didi moculobis SemTxvevaSi 

t n t n 

2 

2 / 1 ( 2) . 

amitom kriteriumi ase yalibdeba: Tu t statistikis dakvirvebuli t S 

mni- 

SvnelobisaTvis da mocemuli mniSvnelovnebis donisaTvis sruldeba 

piroba: tS tn 

2, 

, maSin H 

0 

hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi amis 

safuZveli ara gvaqvs. gasagebia, rom ormxrivi alternativis SemTxveva- 

Si -- H 1 

: 0, kritikuli are iqneba: U 1 

{| t | tn 

2, /2 

}. 

magaliTi 2. dRisa da saRamos swavlebis ekonomist studentebs eki- 

Txebodnen Tu rogor afaseben isini sxvadasxva profesiis prestiJulobas 

8 baliani sistemiT. ipoveT spirmenis rangobrivi korelaciis koeficienti, 

Tu gamokiTxvis monacemebia: 

profesia dRis saRamos 

buRalteri 6 3 

programisti 7 2 

bankis menejeri 2 6 

administratori 5 4 

statistikosi 1 7 

marketingis menejeri 4 8 

finansist-analitikosi 3 5 

warmoebis menejeri 8 1 

amoxsna. am SemTxvevaSi gvaqvs profesiebisagan Sedgenili erToblioba, 

xolo niSnebis rolSi ganixileba dRisa da saRamos swavlebis studentTa 

mier am profesiebis Sefasebebi. gavakeToT Semdegi cxrili: 

profesia dRis saRamos 

R 

A 

R 

buRalteri 6 3 3 9 

programisti 7 2 5 25 

bankis menejeri 2 6 –4 16 

administratori 5 4 1 1 

statistikosi 1 7 –6 36 

marketingis menejeri 4 8 –4 16 

finansist-analitikosi 3 5 –2 4 

warmoebis menejeri 8 1 7 49 

 

156 

B 

( R R ) 

6S 

6156 

1 2 

1 2 

0.857 

nn ( 1) 8 (8 1) 

A 

B 

. 

2 

236

amocanebi 

1. A da B eqspertebma Seafases 7 saxis erTi da igive produqcia: 

A 19 17 15 14 13 10 7 

B 16 18 10 14 15 11 5 

gamoTvaleT spirmenis rangobrivi korelaciis koeficienti. 

2. ipoveT SST , Tu SSR 36 da SSE 4. 

3. ipoveT determinaciis koeficienti, Tu SSR 30 da SSE 10 

4. rogoraa damokidebuli firmebis mier reklamaze TveSi daxarjuli 

Tanxa ( x -- aTasobiT larebSi) gayiduli saqonlis Rirebulebaze ( y - 

- aTasobiT larebSi). SeafaseT gayiduli saqonlis Rirebuleba, romelic 

gamowveulia reklamaze daxarjuli 4500 lariT, Tu: 

firma A B C D E 

x 2 4 5 7 3 

y 10 40 30 50 20 

5. rac ufro dabalia saprocento ganakveTi ( x ), miT ufro dabalia 

sesxebis daubruneblobis, anu defoltebis ( y ) ganakveTi. aageT 95%-iani 

saprognozo intervali defoltebis ganakveTisaTvis, rodesac saprocento 

ganakveTi 8%-ia, Tu SemTxveviT SerCeuli 9 banki monacemebia: 

x 7 6.6 6 8.5 8 7.5 6.5 7 8 

y 38 40 35 46 48 39 36 37 44 

pasuxebi: 

1. 0.714 . 2. 40. 3. 0.75. 

237

Tavi XIX 

dispersiuli analizi (ANOVA) 

mowmdeba hipoTeza sami an meti populaciis saSualos tolobis 

Sesaxeb. aRebulia R SerCeva ( x1,1 ,..., x 

1, n 

), . . . , ( x 

1 

R,1 ,..., x 

R, 

n R 

) moculobebiT 

n1, n2,..., n 

R 

; N n1 n2 nR 

; i -uri SerCevis SerCeviTi saSualo da Sesworebuli 

SerCeviTi dispersia aRniSnulia Sesabamisad x i da s '2 

i 

simboloebiT; 

x x 

2 

i, j i, 

j 

erT. -- aris erToblivi saSualo; SBB : ni 

( X i X 

erT. ) , 

i 

N 

2 2 

'2 

2 2 

SBB / 

( R 1) ; SSW : ( ni1) 

S 

i 

i 

, SSW / 

( N R) 

( N R ( ni 

1) 

). 

i 


: a 1 

a 2 

a R 

. 


: erTi saSualo mainc gansxvavdeba danarCenebisag 

an. 


kriteriumis statistika: 

2 

SB 

F F( R 1, N R) 

. 

2 

S 

238 

W 

2 

sB 

kriteriumis mniSvneloba T.V. f , sadac 

2 

s 2 B 

(Sesabamisad, s 2 W 

) aris 

jgufTa Soris variacia (Sesabamisad, jgufebSi variacia), 

sW 

2 

'2 

( ) 

2 ni 

xi 

xerT. 

i 

2 ( ni 

1) 

s 

i 

i 

sB 

: 

, sW 

: 

. 

R1 

( n 1) 


kritikuli mniSvneloba CV .. FR 1, NR, 

. 


-is uaryofis are) = [ FR 

1, NR, 

, ) . 

P - mniSvneloba = P{ F( R1, N R) T. V.} 

. 

SezRudvebi: 1. populaciebi unda iyos normaluri an daaxloebiT 

normaluri; 2. SerCevebi unda iyos erTmaneTisagan damoukidebeli; 3. populaciaTa 

dispersiebi unda iyos toli. 





0 



i 

i


magaliTi 1. mkvlevars surs gamoscados maRali arteriuli wnevis 

dawevis sami gansxvavebuli meTodi. SemTxveviT SeirCa pacientTa sami 

jgufi. I jgufs aZlevdnen garkveul preparats, II jgufs utardeboda specialuri 

varjiSebi, xolo III jgufi icavda specialur dietas. oTxi kviris 

Semdeg TiToeul pacients gauzomes arteriuli wneva (monacemebi moyvanilia 

qvemoT). 0.05 mniSvnelovnebis doniT SeamowmeT hipoTeza imis 

Sesaxeb, rom populaciaTa saSualoebs Soris ar arsebobs gansxvaveba. 

preparati varjiSi dieta 

---------------------------------------------------------------------- 

10 6 5 

12 8 9 

9 3 12 

15 0 8 

13 2 4 

amoxsna. 


H 

0 

: a 1 

a 2 

a 3 

, H 1 

: erTi saSualo mainc gansxvavdeba danarCenebisagan. 

nabiji 2. vipovoT kritikuli mniSvneloba. radganac Tavisuflebis 

xarisxebia: R1 3 1 da N R153 12 , amitom fiSeris ganawilebis 

cxrilidan vpoulobT, rom CV . . FR 

1, NR, F2,12,0.05 

3.89 . 

nabiji 3. gamovTvaloT kriteriumis mniSvneloba. 

a). TiToeuli SerCevisaTvis vipovoT SerCeviTi saSualo da Sesworebuli 

SerCeviTi dispesia. gvaqvs: 

x1 11.8 , s '2 

1 

5.7 ; x 2 3.8 , s '2 

2 

10.2 ; x 3 7.6 , s '2 

3 

10.3 . 

b). gamovTvaloT erToblivi saSualo: 

 

x i, j i, 

j 10 12 4 

xerT. 7.73 . 

N 15 

g). gamovTvaloT jgufTa Soris variacia: 

2 

( ) 

2 2 2 

2 ni 

xi 

xerT. 

i 

5 (11.8 7.73) 5 (3.8 7.73) 5 (7.6 7.73) 

sB 

: 80.07 

. 

R1 31 

d). gamovTvaloT jgufebSi variacia: 

'2 

( 1) 

2 ni 

s 

i 

i (51) 5.7 (51) 10.2 (51) 10.3 

sW 

: 8.73 . 

( n 1) 3 (5 1) 

i 

i 

e). gamovTvaloT kriteriumis mniSvneloba: 

2 

sB 

80.07 

f 

2 9.17 . 

sW 

8.73 

nabiji 4. gadawyvetilebis miReba: vinaidan 9.17 > 3.89, amitom 

nulovani hipoTeza unda ukuvagdoT, anu 0.05 mniSvnelovnebis doniT 

239

Cven gvaqvs sakmarisi safuZveli davaskvnaT, rom sul cota erTi saSualo 

mainc gansxvavdeba danarCenebisagan. 

magaliTi 2. marketingis specialists surs gaarkvios aris Tu ara 

gansxvaveba momxmareblebis mier sami sxvadasxva supermarketis salarosTan 

rigSi dgomis droebis saSualoebs Soris (monacemebi moyvanilia qvemoT). 

0.05 mniSvnelovnebis doniT aris Tu ara arsebiTi gansxvaveba rigSi 

dgomis droebis saSualoebs Soris? 

maRazia A maRazia B maRazia C 

-------------------------------------------------------------------- 

3 5 1 

2 8 3 

5 9 4 

6 6 2 

3 2 7 

1 5 3 

amoxsna. 


H 

0 

: a 1 

a 2 

a 3 

, H 1 

: erTi saSualo mainc gansxvavdeba danarCenebisagan. 

nabiji 2. vipovoT kritikuli mniSvneloba. aq Tavisuflebis xarisxebia: 

R1 3 1 da N R183 15 , amitom fiSeris ganawilebis cxrilidan 

vpoulobT, rom CV . . FR 

1, NR, F2,15,0.05 

3.68 . 

nabiji 3. gamovTvaloT kriteriumis mniSvneloba. 

a). TiToeuli SerCevisaTvis vipovoT SerCeviTi saSualo da Sesworebuli 

SerCeviTi dispesia. gvaqvs: 

x1 3.33 , s '2 

1 

3.47 ; x 2 5.83 , s '2 

2 

6.17 ; x 3 3.33 , s '2 

3 

4.27 . 

b). gamovTvaloT erToblivi saSualo: 

 

x i, j i, 

j 3 23 xerT. 4.17 . 

N 18 

g). gamovTvaloT jgufTa Soris variacia: 

2 

( ) 

2 2 2 

2 ni 

xi 

xerT. 

i 

6 (3.33 4.17) 6 (5.83 4.17) 6 (3.33 4.17) 

sB 

: 12.5 

. 

R1 31 

d). gamovTvaloT jgufebSi variacia: 

'2 

( 1) 

2 ni 

s 

i 

i (61) 3.47 (61) 6.17 (61) 4.27 

sW 

: 4.64 . 

( n 1) 3 (61) 

i 

i 

e). gamovTvaloT kriteriumis mniSvneloba: 

2 

sB 

12.5 

f 2.69 . 

2 

sW 

4.64 

nabiji 4. gadawyvetilebis miReba: vinaidan 2.69 < 3.68, amitom nulovani 

hipoTezis uaryofis safuZveli ara gvaqvs, anu 0.05 mniSvnelovnebis 

doniT Cven gvaqvs sakmarisi safuZveli davaskvnaT, rom saSualoebi arsebiTad 

ar gansxvavdebian. 

240

amocanebi 

igulisxmeT, rom: populaciebi normaluri an daaxloebiT normaluria; 

SerCevebi erTmaneTisagan damoukidebelia da populaciaTa dispersiebi 

tolia. 

1. menejers ainteresebs aris Tu ara respublikuri saavadmyofos eqimebis, 

eqTanebisa da teqnikuri personalis saSualo asaki gansxvavebuli.. 

saavadmyofos SemTxveviT SerCeuli TanamSromlebis qvemoTmoyvanili monacemebis 

mixedviT, 0.05 mniSvnelovnebis doniT, SeuZlia Tu ara menejers 

daaskvnas, rom am sami jgufis saSualo asaki gansxvavebulia? 

eqimebi eqTnebi teqnikuri personali 

-------------------------------------------------------------------------------------- 

60 23 33 

36 25 28 

29 26 35 

56 35 29 

32 42 23 

54 22 41 

58 

2. kompiuteris sami sxvadasxva tipis SemTxveviT SerCeul diskebSi 

aRmoCenili defeqtebis raodenobis qvemoTmoyvanili monacemebis mixedviT, 

0.05 mniSvnelovnebis doniT, SegviZlia Tu ara daaskvnaT, rom defeqtebis 

raodenobis saSualo diskebis am sam jgufSi gansxvavebulia? 

diski A diski B diski C 

--------------------------------------------------------- 

0 2 1 

1 0 0 

0 3 1 

2 5 1 

3 3 0 

2 4 2 

0 6 0 

1 0 0 

1 2 1 

0 5 2 

3. kolejis sportul programebSi monawile studentebis mier mopovebuli 

qulebis qvemoTmoyvanili monacemebis mixedviT, 0.1 mniSvnelovnebis 

doniT, SegviZlia Tu ara daaskvnaT, rom am sam jgufSi qulebis 

saSualo gansxvavebulia? 

241

fexburTi kalaTburTi xelburTi 

---------------------------------------------------------------------------- 

3.2 3.8 2.6 

2.6 3.1 1.9 

2.4 2.6 1.7 

2.4 3.9 2.5 

1.8 3.3 1.9 

4. SemTxveviT SerCeul pacientTa sam jgufs utardeboda mkurnaloba 

stresis donis dawevis sami gansxvavebuli meTodiT. specialuri mowyobilobiT 

gazomil iqna (procentebSi gamosaxuli) TiToeuli pacientis 

daweuli stresis done (monacemebi moyvanilia qvemoT). 0.05 mniSvnelovnebis 

doniT, SegviZlia Tu ara daaskvnaT, rom procentebis saSualo am 

jgufebSi gansxvavebulia? 

I meTodi II meTodi III meTodi 

---------------------------------------------------------------------------- 

3 12 15 

10 12 14 

5 17 18 

1 13 14 

13 18 20 

3 9 22 

4 14 16 

5. SemTxveviT SerCeuli aTletebi dayves sam jgufad da dauniSnes 

sami saxis dieta erTi Tvis ganmavlobaSi. erTi Tvis gasvlis Semdeg TiToeuli 

aTletis mier dietebis mixedviT daklebuli kilogramebi moyvanilia 

qvemoT. 0.05 mniSvnelovnebis doniT, SegviZlia Tu ara daaskvnaT, 

rom dietebi gansxvavebulia? 

dieta A dieta B dieta C 

------------------------------------------------------------ 

3 10 8 

6 12 3 

7 11 2 

4 14 5 

8 

6 

6. samomxmareblo JurnalSi WurWlis sarecxi manqanebi iyofa sam kategoriad: 

umaRlesi, Zalian kari da kargi. am sami kategoriis manqanebis 

qvemoTmoyvanili fasebis mixedviT, 0.1 mniSvnelovnebis doniT, aris 

Tu ara manqanebis saSualo fasebi gansxvavebuli? 

242

umaRlesi Zalian kargi kargi 

-------------------------------------------------------------------------- 

565 330 350 

400 840 379 

369 510 280 

550 470 320 

460 380 400 

375 400 450 

290 

319 

7. manqanaTmSenebeli qarxnis muSebi SemTxveviT mimagres oTx asawyob 

xazs (“konveirs”). TiToeuli muSis mier gakeTebuli defeqturi nawilebis 

raodenobis qvemoTmoyvanili raodenobebis mixedviT, 0.05 mniSvnelovnebis 

doniT, SegviZlia Tu ara daaskvnaT, rom defeqturi nawilebis 

raodenobis saSualo am xazebze erTidaigivea? 

I xazi II xazi III xazi IV xazi 

---------------------------------------------------------------------------------- 

3 8 10 9 

2 6 9 15 

0 2 8 3 

6 0 11 0 

4 1 12 2 

3 9 15 0 

5 7 17 1 

8. qvemoTmoyvanili magnitofonis sami tipis kasetis muSaobis xangr- 

Zlivobis monacemebis mixedviT, 0.1 mniSvnelovnebis doniT, SegviZlia 

Tu ara davaskvnaT, rom muSaobis xangrZlivobis saSualoebi gansxvavebulia? 

I tipi II tipi III tipi 

---------------------------------------------------------- 

196 98 94 

183 91 106 

112 101 85 

107 99 102 

189 84 101 

9. mkvlevars ainteresebs aris Tu ara gansxvaveba oTxi sxvadasxva 

tipis balaxis sakreWi manqanis wonebs Soris. 0.1 mniSvnelovnebis doniT 

SegviZlia Tu ara davaskvnaT, rom manqanebis saSualo wonebi gansxvavebulia? 

243

gazze benzinze eleqtronuli meqanikuri 

-------------------------------------------------------------------------------------------------- 

95 73 55 37 

101 69 52 24 

108 72 51 25 

97 71 37 29 

101 67 57 22 

62 54 17 

68 34 17 

71 45 22 

41 20 

53 18 

21 

10. qvemoT moyvanilia dReRamis sam sxvadasxva monakveTSi 

gadaudebeli samedicino daxmarebis oTaxSi SemTxveviT SerCeuli 

pacientis mier gatarebuli drois monacemebi (gazomili wuTebSi). 0.05 

mniSvnelovnebis doniT, SegviZlia Tu ara daaskvnaT, rom pacientis 

daxmarebaze daxarjul droTa saSualoebi am sam jgufSi mniSvnelovnad 

gansxvavdebian erTmaneTisagan? 

dila (7--15) naSuadRe (15--23) Rame (23—7) 

------------------------------------------------------------------------------------------ 

12 9 6 

18 8 15 

18 16 8 

21 20 9 

19 15 5 


11. kompaniis mflobels ainteresebs aris Tu ara gayidvaTa raodenoba 

erTnairad ganawilebuli regionebis mixedviT. man SemTxveviT SearCia 

Tveebi da daTvala 5 sxvadasxva regionSi gayidvaTa raodenoba. 0.05 

mniSvnelovnebis doniT, SeuZlia Tu ara kompaniis mflobels daaskvnas, 

rom gayidvaTa raodenoba TiToeul regionSi erTidaigivea? 

regioni soxumi zugdidi gori Telavi quTaisi 

gayidvebi 324 236 182 221 365 

12. jandacvis saministros ainteresebs vin daTanxmdeba nebayofilobiT 

Tavis Tavze gamoscados arteriuli wnevis dasawevi axali preparati. 

regionebis mixedviT miRebuli monacemebis qvemoTmoyvanili cxrilis 

mixedviT, 0.01 mniSvnelovnebis doniT, SegviZlia Tu ara daaskvnaT, 

rom TiToeul regionSi eqsperimentSi monawileobis msurvelTa raodenoba 

erTidaigivea? 

244

egioni I II III IV 

Tanaxmaa 87 62 56 93 

13. iusticiis saministros ainteresebs srulwlovani mosaxleobis 

Sexeduleba nafic msajulTa institutis SemoRebis mimarT. Catarebuli 

gamokiTxvis qvemoTmoyvanili Sedegebis mixedviT, 0.1 mniSvnelovnebis 

doniT, SegviZlia Tu ara daaskvnaT, rom Sexeduleba dakavSirebulia adamianis 

sqesTan? 

sqesi eTanxmeba ar eTanxmeba gaurkvevelia 



14. swrafi kvebis obieqtis menejerma gadawyvita gaarkvios Tu romel 

souss aniWeben momxmareblebi upiratesobas hamburgeris Wamisas. Catarebuli 

gamokiTxvis qvemoTmoyvanili Sedegebis mixedviT, 0.1 mniSvnelovnebis 

doniT, aris Tu ara momxmareblis sasurveli sousis saxeoba 

damokidebuli mis sqesze? 

sousi 

sqesi maionezi “ketCupi” mdogvi 

----------------------------------------------------------------------------------------------- 



15. gamokiTxul iqna 40 da 50 wlis investorebi fulis ganTavsebis 

gzebis Sesaxeb. miRebuli manacemebis mixedviT, 0.05 mniSvnelovnebis 

doniT, aris Tu ara kavSiri investoris asaksa da fulis ganTavsebis 

gzebs Soris? 

didi mcire saerTaSoriso fuladi 

saaqcio saaqcio saaqcio bazris 

fondi fondi fondi fondi obligacia 

----------------------------------------------------------------------------------------------------------------------- 

40 wlis 20 10 10 15 45 

50 wlis 42 24 24 6 24 

16. zoologiuri maRaziis mepatrones ainteresebs dakavSirebulia 

Tu ara arCeuli cxovelis saxeoba adamianis sqesTan. qvemoTmoyvanili 

monacemebis mixedviT, 0.1 mniSvnelovnebis doniT, SeamowmeT Sesabamisi 

hipoTeza. 

cxovelis saxeoba 

---------------------------------------------------- 

ZaRli kata Citi 



17. gamokiTxul iqna 3 maRaziis 200 – 200 momxmarebeli, romlebic 

sargebloben sakredito baraTebiT. maT dausves kiTxva “qondaT Tu ara 

245

sakredito baraTiT sargeblobisas problemebi?” 0.01 mniSvnelovnebis 

doniT SeiZleba Tu ara davaskvnaT, rom proporciebi erTidaigivea? 

maRazia I MmaRazia II maRazia III 

diax 87 56 43 

ara 113 144 157 

jami 200 200 200 

18. SemTxveviT SearCies kvira da daTvales qalaqis sam sxvadasxva 

avtosadgomze yovedRiurad gaCerebul avtomobilTa raodenoba. 

SeamowmeT hipoTeza imis Sesaxeb, rom 0.01 mniSvnelovnebis doniT ar 

aris gansxvaveba am avtosadgomebze gaCerebul avtomobilTa raodenobebs 

Soris. 

avtosadgomi I avtosadgomi II avtosadgomi III 

203 319 89 

162 321 126 

190 271 115 

219 194 92 

188 342 106 

209 423 100 

212 199 94 

19. mamakacis sami tipis fexsacmelis qvemoTmoyvanili fasebis mixedviT, 

0.05 mniSvnelovnebis doniT, SegviZlia Tu ara davaskvnaT, rom 

fasebis saSualoebs Soris mniSvnelovani gansxvavebaa? 

oficialuri yoveldRiuri sportuli 

110 80 64 

95 100 66 

95 135 70 

265 90 92 

59 80 

70 

50 

20. firmis direqtors ainteresebs aris Tu ara gansxvaveba im droebis 

saSualoebs Soris, rac misi TanamSromlebis sam jgufs sWirdeba samsaxurSi 

misasvlelad. 0.05 mniSvnelovnebis doniT SeuZlia Tu ara firmis 

direqtors daaskvnas, rom saSualoebs Soris arsebiTi gansxvavebaa? 

246

menejeri gamyidveli mesawyobe 

35 9 15 

18 3 6 

27 12 27 

24 6 22 

14 

8 

21 

21. qvemoT moyvanilia nawlavis Cxirebis raodenoba sami sxvadasxva 

tbis garkveul farTobze 5 dRiani periodis ganmavlobaSi. 0.05 mniSvnelovnebis 

doniT aris Tu ara gansxvaveba Cxirebis raodenobis saSualoebSi? 

kus tba lisis tba faravnis tba 

45 97 33 

53 82 35 

41 99 31 

38 84 28 

55 79 26 

22. studentebi SemTxveviT gadaanawiles sam jgufSi da TiToeul 

jgufs SesTavazes swavlebis gansxvavebuli meTodi. kursis damTavrebis 

Semdeg Catarebuli gamocdis Sedegebi moyvanilia qvemoT. 0.05 mniSvnelovnebis 

doniT aris Tu ara mniSvnelovani gansxvaveba gamocdis Sedegebis 

saSualoebSi jgufebis mixedviT? 

I jgufi II jgufi III jgufi 

87 82 97 

92 78 90 

61 41 83 

83 65 92 

47 63 91 

247

pasuxebi: 

1. H 

0 

: a 1 

a 2 

a 3 

, H 1 

: erTi mainc gansx.; CV . . F2,16,0.05 

3.63 ; T. V. f 5.96 ; ki -- 

H 

1. 2. H 

0 

: a 1 

a 2 

a 3 

, H 1 


3.35 ; T. V. f 7.456 ; 

ki -- H 

1. 3. H 

0 

: a1 a2 a3 

, H 

1: erTi mainc gansx.; CV . . F2,12,0.1 

2.81 ; 

T. V. f 8.448 ; ki -- H 

1. 4. H 

0 

: a1 a2 a3 

, H 

1: erTi mainc gansx.; 

CV . . F2,18,0.05 

3.55 ; T. V. f 19.05 ; ki -- H 

1. 5. H 

0 

: a 1 

a 2 

a 3 

, H 

1: erTi mainc 

gansx.; CV . . F2,11,0.05 

3.98 ; T. V. f 7.75 ; ki -- H 

1. 6. H 

0 

: a 1 

a 2 

a 3 

, H 

1: erTi 

mainc gansx.; CV . . F2,17,0.1 

2.64 ; T. V. f 1.28 ; ara -- H 

0 

. 7. H 

0 

: a 1 

a 2 

a 3 

a 4 

, 

H 


3.01 ; T. V. f 6.974 ; ara -- H 

1. 8. H 

0 

: 

a1 a2 a3 

, H 


2.81 ; T. V. f 9.16 ; ki -- H 

1. 9. 

H 

0 

: a 1 

a 2 

a 3 

a 4 

, H 1 


2.28 ; T. V. f 234.5 ; ki 

-- H 

1. 10. H 

0 

: a1 a2 a3 

, H 


3.89 ; 

T. V. f 5.877 ; ki -- H 

1. 11. H 

0 

: erTidaigivea, H 

1: 

gansxvavebulia; 

2 

2 

CV . . 4,0.05 

9.488 ; TV . . 

87.14 ; ara -- H 1 

. 12. H 

0 

: erTidaigivea, H 1 

: gansxvavebulia; 

CV . . 3,0.01 

11.345 ; TV . . 

13.38 ; ara -- H 

2 

2 

1. 13. H 

0 

: damoukid., 

2 

2 

H 

1: damokid.; CV . . 2,0.1 

4.605 ; TV . . 

6.163 ; ki -- H 

1. 14. H 

0 

: damoukid., 

2 

2 

H 

1: damokid.; CV . . 2,0.1 

4.605 ; TV . . 

3.05 ; ara -- H 

0 

. 15. H 

0 

: damoukid., 

2 

2 

H 

1: damokid.; CV . . 4,0.05 

9.488 ; TV . . 

28; ki -- H 1 

. 16. H 

0 


1: 

2 

2 

damokid.; CV . . 2,0.1 

4.605 ; TV . . 

7.674 ; H 

1. 17. H 

0 

: p 1 

p 2 

p 3 

, H 

1: erTi 

2 

2 

mainc gansx.; CV . . 2,0.01 

9.21; 

TV . . 

23.89 ; ara -- H 

1. 18. H 

0 

: a 1 

a 2 

a 3 

, 

H 


6.01 ; T. V. f 27.02 ; H 

1. 19. H 

0 

: 

a1 a2 a3 

, H 1 


3.81 ; T. V. f 0.533 ; ara -- H 0 

. 

20. H 0 

: a 1 

a 2 

a 3 

, H 1 


3.89 ; T. V. f 6.141 ; ki 

-- H 

1. 21. H 

0 

: a1 a2 a3 

, H 


3.89 ; 

T. V. f 65.263 ; ki -- H 

1. 22. H 

0 

: a1 a2 a3 

, H 

1: erTi mainc gansx.; 

CV . . F 

3.89 ; T. V. f 3.673 ; ara -- H 0 

. 

2,12,0.05 

248

Tavi XX 

sakontrolo werebisa da Sualeduri da saboloo gamocdebis 

bileTebis nimuSebi 2007—2009 wlebSi 

varianti 8.12.1.1. 

1. dawereT jamis albaTobis formula. (1q.) 

2. ganmarteT maCvenebliani ganawileba. (1q.) 

3. ganmarteT SemTxveviTi sididis mediana. (1q.) 

4. binomialuri ganawilebis lodini da dispersia. (1q.) 

5. dawereT maTematikuri lodinis gamosaTvleli formula. (1q.) 

6. agdeben erT wesier kamaTels da meore iseT kamaTels, romelzec 6 qulis 

mosvlis albaTobaa 1/4. ipoveT albaToba imisa, rom wesier kamaTelze mova 6 

qula da arawesierze ar mova 6 qula. (2q.) 

7. yuTSi moTavsebulia 3 wesieri da imave zomisa da wonis mqone 4 arawesieri 

moneta, romelzec gerbis mosvlis albaTobaa 3/8. SemTxveviT amoiRes erTi 

moneta da aagdes. ipoveT gerbis mosvlis albaToba. (3q.) 

8. mocemulia ori damoukidebeli da SemTxveviTi sidideebis ganawilebis 

kanonebi. a). ipoveT -s standartuli gadaxra; (1q.) b). aageT -s ganawilebis 

funqcia; (1q.) g). ipoveT P{ [ 2,6)} ; (1q.) d). aageT SemTxveviTi sididis 

ganawilebis kanoni. (1q.) 

-2 1 -4 2 5 

P 0.82 0.18 P 0.4 0.1 0.5 

9. mocemulia ori da SemTxveviTi sididis erToblivi ganawilebis kanoni. 

a). gamoTvaleT korelaciis koeficienti; (4q.) b). avagoT min( , ) SemTxveviTi 

sididis ganawilebis kanoni; (1q.) g). damoukidebelia Tu ara es SemTxveviTi 

sidideebi? (1q.) 

-3 -1 2 

-2 0.25 0.15 0.2 

1 0.13 0.05 0.22 

10. dawereT normaluri ganawilebis simkvrive, romlis lodinia 3, xolo 

standartuli gadaxra 4. (1q.) 

1/5, Tu x 

[ 1,4]; 

11. Tanabari ganawilebis simkvrivea f() 

x 

ras udris: a). maTematikuri 

lodini; (1q.) b). standartuli gadaxra. (1q.) 

0, Tu x 

[ 1,4]. 

12. N(0,1) 

-is 0.77-kvantilia 0.74. ipoveT N( 3,16) 

-is: a). 0.77-kvantili; (1q.) b). 0.23- 

kvantili. (1q.) 

Sifri: 

==================================================================== 

ekonomikisa da biznesis fakulteti. kolokviumi albaTobis Teoriasa da 

maTematikur statistikaSi. 

studentis gvari, saxeli _____________________________ 

Sifri: 

kursi _____ 

249

varianti # 2007.1.1 

1. Ada B damoukidebelia, PA ( ) 0.7 ; PB ( ) 0.8. ipoveT P( A\ B ) . 

a). 0.1 b). -0.1 g). 0.56 d). 0.14 e). 0.24 

2. yuTSi 3 TeTri da 5 Savi burTia. ipoveT albaToba imisa, rom dabrunebis 

gareSe SemTxveviT SerCeuli sami burTidan 2 TeTria da 1 Savi? 

a). 1/56 b). 15/56 g). 15/28 d). 5/28 e). 1/28 

3. saTamaSo kamaTels agoreben samjer. rogoria albaToba imisa, rom samivejer 

mova erTi da igive qula? 

a). 1/36 b). 5/9 g). 25/216 d). 1/8 e). 43/216 v). 91/216. 

4. saTamaSo kamaTels agdeben orjer. gamoTvaleT P( AB C) 

da P( AB C) 

, 

Tu: A--qulaTa jami luwia, B --qulaTa jami 7-ze naklebia, C --meore agdebisas 

movida 2. 

a). 11/12, 1/9 b). 31/36, 1/6 g). 25/26, 1/18 d). 2/3, 0 e). 4/9, 1/12 v). 7/18, 0 

5. saTamaSo kamaTels agdeben orjer. arian Tu ara damoukideblebi: Ada B ? 

daC ? B da C ? arian Tu ara erToblivad damoukidebeli A, B daC ? Tu: A -- 

pirveli agdebisas movida 1 qula, B -- meore agdebisas movida 6 qula, C -- 

jamSi movida 7 qula; 

a). ki, ara, ki, ara b). ki, ki, ara, ara g). ara, ara, ki, ki; d). ki, ki, ki, ara 

6. saTamaSo kamaTels agdeben orjer. gamoTvaleT PB 

( A ) da miuTiTeT aris Tu 

ara Ada B damoukidebeli, Tu:: A--movida erTi da igive qula, B --qulaTa jami 

luwia. 

a). 1/3, ara b). 1/6, ki g). 1/2, ara d). 2/5, ara e). 1/6, ki v). 1, ara 

7. I, II, III brigada awarmoebs detalebis 30%, 20%. 50%-s, romelTa Soris Sesabamisad 

1%, 3% da 2% cudia. SemTxveviT aRebuli detali aRmoCnda cudi. rogoria 

alaToba imisa, rom is daamzada I brigadam? 

a). 0.258 b). 0.613 g). 0..316 d). 0.526 e). 0.185 v). 0.158 

#1. 1. albaTobis klasik. ganmarteba. #2. 1. muavr-laplasis lok. form. 

2. sruli alb. da baiesis formulebi. 2. ganaw. funqcia: ganm., Tvisebebi. 

3. maTematikuri lodini: ganm., Tvisebebi. 3. Tanab. ganaw.: simkv. M da D. 

4. P ( A) 

0,3; P( 

B) 

4/9; A|| 

B 

4. PA ( ) 3/7 ; PB ( ) 0,4 ; AB || 

ipoveT P( A B) 

, P ( A\ 

B) 

. ipoveT P( A B) 

, P ( A\ 

B) 

. 

5. vip. monetis 3-jer agdebisas gerbTa 5. S.s. mniS. -2, 1, 3. alb. 0,2; 0,4; x . 

mosvlis g. kan.; gan. f.; M ; D da . vip. x ; g. kan.; gan. f.; M ; D da . 

6. mniS. -2, 1, 3. alb. 0,2; 0,5; 0,3. 6. S.s. mniS. -1, 0, 2. alb. 0,3; 0,3; 0,4. 

S.s. mniS. -1, 0, 2. alb. 0,3;0,3;0,4. S.s. mniS. -2, 1, 3. alb. 0,2; 0,5; 0,3. 

|| . ipoveT 3 

|| . ipoveT min{ , 

} 

S.s. gan. kan. 

4 

3 

7. f ( x) 

ax ,1 x 4; =0 sxvagan. 7. f ( x) 

ax ,2 x 5; =0 sxvagan. : 

ipoveT: a, P{| 

| 2}, F ( x) 

, M da D ipoveT: a, P{| 

| 3}, F ( x) 

, M da D. 

 

8. SerCeviTi dispersia. 8. asimetriis koeficienti. 

9. -s mniS. -2;0;3. -s mniS. -2;-1;1. 9. -s mniS. -1;1;2. -s mniS. -1; 0; 2. 

p 

11 

0,2; 

p12 

0,1; p13 

01; p21 

0; 

p22 

p 

11 

0,1; 

p12 

0; p13 

0,2; p21 

0; 

p22 

 

/6; p 0; p 0,1; p 0 . ? /7; p 0,1; p 0; p 0, 2 . ? 

1 

23 31 32 

 

1 

23 31 32 

 

 

250

# 2008.4.1. 1. sxvaobis albaTobis formula. 2. bernulis formula. 3. xdomileb- 

Ta damoukidebloba. 4. manqanaSi 5 adgilia. ramden sxvadasxvanairad SeiZleba 

ganTavsdes manqanaSi 5 adamiani, romelTagan erTs ara aqvs ufleba daikavs 

adgili saWesTan? 

5. monetas agdeben xuTjer. rogoria alabaToba imisa, rom gerbi mova kent rixvjer, 

xolo safasuri luw ricxvjer? 


da P( AB 

C) 

albaTobebi A, B daC xdomilebaTa Semdegi sameulisaTvis: A--mosul qulaTa 

jami luwia, B --mosul qulaTa jami 7-ze naklebia, C --meore agdebisas movida 2 

qula; 

7. I brigada awarmoebs detalebis 30%-s, romelTa Soris 3% wundebulia. II 

briga-da awarmoebs imave detalebis 35%-s, romelTa Soris 4% wundeblia. III 

brigada awarmoebs detalebis 15%-s, romelTa Soris 1% wundebulia. IV brigada 

amzadebs detalebis 20%-s, romelTa Soris 2% wundebulia. erT sawyobSi 

mogrovili am detalebidan SemTxveviT aiRes erTi detali da is aRmoCnda 

wundebuli. rogoria albaToba imisa, rom es detali daamzada I brigadam? 

Sefaseba: #1—6 or-ori qula, #7 – 3 qula. sul 15 qula 

#28.10.1. 1. marjvena jibeSi 3 ocTeTriani da 4 aTTeTriani monetaa, marcxena jibSi 

ki 6 ocTeTriani da 3 aTTeTriani moneta. Mmarjvena jibidan marcxenaSi 

SemTxveviT 2 moneta gadades. ipoveT alb-aToba imisa, rom amis Semdeg marcxena 

jibidan SemTxveviT amoRebuli moneta aTTeTriania. (3 q.) 

2. yuTSi 7 wiTeli da 5 lurji burTia. ipoveT albaToba imisa, rom SemTxveviT 

amoRebuli sami burTi erTi ferisaa. (3 qula) 

3. Ada B damoukidebeli xdomilobebia, P ( A) 

0.5, P( 

B) 

0. 3. GgamoTvaleT P( A B) 

. 

(2 qula) 

4. agoreben 2 wesier kamaTels, ras udris albaToba imisa, rom musuli 

cifrebis jami tolia 8-is, Tu cnobilia rom es jami luwi ricxvia. (2 

qula) 

5. aagdes sami simetriuli moneta, damoukidebelia Tu ara xdomilobebi 

A={pirvel monetaze movi-da gerbi}, B={movida erTi safasuri mainc}. (2 qula) 

6. cifrebidan 1,2,3,4,5 jer irCeven erTs, xolo Semdeg darCenili oTxidan 

meores. ipoveT albaToba imisa, rom orive amoirCeuli ricxvi kentia. (3 qula) 

sagamocdo bileTi #1611. 2.1. 

1. qvemoT moyvanili X -is ganawilebis kanonis mixedviT ipoveT: ganawilebis 

funqcia (0.5q.) da aageT grafiki (0.5q.); P(4 X 

8) (0.25q.); ganawilebis kanonebi: 

max(5, X) 

-isa (0.25q.) da 4X 9-is (0.25q.); EX (0.25q.); E(5 3 X) 

(0.25q.); DX (0.5q.); 

D( 5 4 X) 

(0.25q.); sul 3 qula 

X 3 4 5 10 

P P 0.25 0.1 ? 0.3 

2. damoukidebel X da Y SemTxveviT sidideTa ganawilebis kanonebis mixedviT 

ipoveT: EX , EY (0.25q.); DX , DY (0.25q.); E( 2 XY ) (0.25q.); D( 3X 4 Y) 

(0.25q.); 

max( XY , ) -is ganawilebis kanoni (0.5q.). 

sul 1.5 qula 

X -3 -1 2 Y -2 3 

P 0.3 0.35 ? P 0.4 0.6 

3. ipoveT ganawilebis kanonebi: X -is,Y -is (0.5q.) da 2X3Y 

-is (0.5q.); EX , 

EY (0.25q.), E(3X 5 Y) 

(0.5q.); DX DY (0.5q.); E( XY )(0.5q.); cov( XY , ) (0.25q.); 

251

( XY , ) (0.5q.); cov(5 X, 4Y 7) (0.25q.);Dcov(3 X 2 Y, Y) 

(0.25q.);D D(4X 2 Y) 

(0.25q.); 

( 3X 4, X) 

(0.25q.); damoukidebelia Tu ara X daY ?(0.5q.). sul 5 qula 

Y \ X -2 -1 1 

-3 0.17 ? 0.15 

1 0.23 0.1 0.05 

4. dawereT eqsponencialuri ganawilebis simkvrive, Tu EX 0.5. 0.5 qula 

4 

5. mocemulia X -is ganawilebis funqcia FX ( x) 0, Tu x 2; ax , Tu 2 

x < 4; =1, 

Tu x > 4 . ipoveT ganawilebis simkvrive da a (1.5q.); EX (0.5q.); DX (0.5q.); 

PX ( [2,3)) (0.5q.). sul 3 qula 

2 

( x5) 

 

6. f ( x) e 

18 

/ 18 

.ras udris EX (0.25q.), DX (0.25q.), PX ( 4) (0.25q.), 

PX ( 0) (0.25q.). sul 1 qula 

k k 10k 

7. P( X k) C10(0.2) (0.8) ), k 0,1,...10 . ras udris EX (0.25q.), DX (0.25q.), 

PX ( 4) (0.25q.), PX ( [3,5)) (0.25q.). sul 1 qula. 

8. mocemulia SerCeva: -1, 2, 7, 9, 3, -7, 5, 9, 8, -9. ipoveT populaciis saSualos 

Caunacvlebeli Sefaseba. (0.5q.) gamoTvaleT eqscesis koeficienti. (0.5q.) 

sul 1 qula. 

9. normaluri populaciidan aRebulia SerCeva: 18, 12, 22, 15, 30, 16, 39, 38, 35. 

2 

64. avagoT 0.99 saimedoobis ndobis intervali ucnobi saSualosaTvis (normaluri 

ganawilebis 0.995-kvantilia 2.58). 

2 qula 

10. normaluri populaciidan Na ( ,25) aRebulia SerCeva: 10, 12, -15, 6, 18, -9, 14, -20, 

16. 0.03 mniSvnelovnebis doniT SevamowmoT H0 : a 5 hipoTeza H1 : a 7 -is 

winaaRmdeg ( x0.97 1.89 ). 2 qula 

yoveli Teoriuli kiTxva fasdeba 2 quliT: 

11. ras ewodeba gadanacvleba, wyoba, jufTeba. dawereT maTi gamosaTvleli formulebi. 

12. vTqvaT 1, 2, 

, 

n 

aris mocemuli elementarul xdomilebaTa sivrce 

da aris am sivrceze gansazRvruli albaToba. ra ZiriTad Tvisebebs akmayofilebs 

albaToba? 

13. dawereT ori xdomilebis damoukideblobis gansazRvreba. Tu ori A da B 

xdomilebisaTvis A B, ( 

A) 

1/ 

4 da ( B ) 1/ 

2 , daasabuTeT damoukidebelia 

Tu ara A da B xdomileba. 

14. ras ewodeba ualbaTesi ricxvi. dawereT ualbaTesi ricxvis gamosaTvleli 

formula. 

15. SemTxveviTi sididis ganawilebis funqcia da misi Tvisebebi. 

16. diskretuli SemTxveviTi sididis magaliTebi: bernulis, binomuri da 

puasonis ganawilebebi. maTi maTematikuri molodini da dispersia. 

17. arakorelirebuli SemTxveviTi sidideebi. kavSiri SemTxveviT sidideTa 

arakorelirebulobasa da damoukideblobas Soris. 

18. SerCeviTi saSualos maTematikuri molodini da dispersia. SerCeviTi 

dispersiis maTematikuri molodini. 

19. SefasebaTa agebis meTodi _ momentTa meTodi. 

20. intervaluri Sefaseba _ ndobis intervali bernulis sqemaSi warmatebis 

ucnobi albaTobisaTvis. 

252

sakontrolo wera: varianti 24.2008.12.1. 

1. albaTobis klasikuri ganmarteba. 2. sruli albaTobisa da baiesis formulebi. 

3. maTematikuri lodini: ganmarteba, Tvisebebi. 4. eqscesis koeficienti. 5. 

SemTxveviT sidideTa damoukidebloba. 

6. PA ( ) 0.3 ; PB ( ) 4/9 ; A da B damoukidebelia. ipoveT P( A B) 

, P ( A\ 

B) 

. 

0, Tu x 2; 

0.4, Tu 2 x 0; 

7. mocemulia SemT. sididis ganawilebis funqcia: F 

() x 

0.75, Tu 0 x 

3; 

 

1, Tu x 3. 

ipoveT SemTxveviTi sididis: a). ganawilebis kanoni; b). dispersia. 

8. da damoukidebeli SemTxveviTi sidideebia ganawilebis kanonebiT: 

-2 1 3 -1 0 2 

P 0.2 0.5 0.3 P 0.3 0.3 0.4 

ipoveT: 

2 3 

 

-s: a). ganaw. kanoni; b). maTematikuri lodini; g). 

2 

P{ 3 0} =? 

3 

ax 

, Tu x[2,5]; 

9. mocemulia SemT. sididis ganawilebis simkvrive: f 

() x 

0, Tu x[2,5]. 

ipoveT: a). a ; b). P{| | 2} ; g). ganawilebis funqcia F () x ; d). E ; e). D . 

10. SemTxveviTi sididis mniSvnelobebia: -2; 0 da 6, xolo -si: -2; -1 da 1. ma- 

Ti erToblivi ganawilebis kanonia: p 11 

0.2 ; p 12 

0.1 ; p 13 

0.1; p 21 

0 ; p 22 

1/ 6; 

p23 0 ; p 31 

0.1 ; p 32 

? p 33 

0 . a). gamoTvaleT korelaciis koeficienti; b). aageT 

max( / 2, ) 

-s ganawilebis kanoni; g). gamoTvaleT E(max{ / 2, }) ; d). damoukidebelia 

Tu ara es SemTxveviTi sidideebi? 

2009 wlis sagamocdo bileTis amocanebis nawili 

D. SemTxveviTi sididis maTematikuri molodini aris 11, xolo dispersia 9- 

is tolia. CebiSevis utolobis gamoyenebiT ipoveT mudmivis mniSvneloba, romlisTvisac 

sruldeba utoloba: P{| 11| } 0.09 . 

E. nika eZebs samuSaos. igi imyofeboda gasaubrebaze bankSi da sadazRvevo 

kompaniaSi. misi SefasebiT bankSi mas miiReben 0.5 albaTobiT, xolo sadazRvevo 

kompaniaSi ki 0.6 albaTobiT. garda amisa, mas miaCnia, rom 0.3-is toli 

albaTobiT mas orive es organizacia miiwvevs. vipovoT albaToba imisa, rom nika 

samuSaos iSovis. 

F. studentma 25 sakiTxidan icis 20. maswavlebeli mas aZlevs 3 SekiTxvas. ra 

aris albaToba imisa, rom studenti upasuxebs: a) mxolod or SekiTxvas, b) erT 

SekiTxvas mainc. 

G. fizkulturis maswavlebelma 9 moswavle, romelTagan 7 gogonaa, Caayena 

mwkrivSi. ra aris albaToba imisa, rom laSa da giorgi aRmoCndebian erTmaneTis 

gverdiT? 

H. yuTSi, romelSic 2 burTulaa, CauSves 2 TeTri burTula, ris Semdegac 

alalbedze amoiRes 1 burTula. ra aris albaToba imisa, rom amoRebuli 

burTula TeTria, Tu tolalbaTuria yvela hipoTeza sawyisi 2 burTulis 

ferebis Sesaxeb (igulisxmeba, rom feri aris TeTri an Savi). 

253

danarTi (statistikuri cxrilebi) 

k 

 

puasonis ganawilebis cxrilebi ( P() 

k e ) 

k! 

= 1.0 = 1.5 = 2.0 = 2.5 = 3.0 = 3.5 = 4.0 = 4.5 = 5.0 

p(0) 0.3679 0.2231 0.1353 0.0821 0.0498 0.0302 0.0183 0.0111 0.0067 

p(1) 0.3679 0.3347 0.2707 0.2052 0.1494 0.1057 0.0733 0.0500 0.0337 

p(2) 0.1839 0.2510 0.2707 0.2565 0.2240 0.1850 0.1465 0.1125 0.0842 

p(3) 0.0613 0.1255 0.1804 0.2138 0.2240 0.2158 0.1954 0.1687 0.1404 

p(4) 0.0153 0.0471 0.0902 0.1336 0.1680 0.1888 0.1954 0.1898 0.1755 

p(5) 0.0031 0.0141 0.0361 0.0668 0.1008 0.1322 0.1563 0.1708 0.1755 

p(6) 0.0005 0.0035 0.0120 0.0278 0.0504 0.0771 0.1042 0.1281 0.1462 

p(7) 0.0001 0.0008 0.0034 0.0099 0.0216 0.0385 0.0595 0.0824 0.1044 

p(8) 0.0001 0.0009 0.0031 0.0081 0.0169 0.0298 0.0463 0.0653 

p(9) 0.0002 0.0009 0.0027 0.0066 0.0132 0.0232 0.0363 

p(10) 0.0002 0.0008 0.0023 0.0053 0.0104 0.0181 

p(11) 0.0002 0.0007 0.0019 0.0043 0.0082 

p(12) 0.0001 0.0002 0.0006 0.0016 0.0034 

p(13) 0.0001 0.0002 0.0006 0.0013 

p(14) 0.0001 0.0002 0.0005 

p(15) 0.0001 0.0002 

standartuli normaluri ganawilebis zeda -kritikuli 

wertilebi ( z ) 

0.1 0.05 0.025 0.125 0.01 0.005 0.0025 0.001 

z 1.28 1.64 1.96 2.24 2.33 2.57 2.81 3.08 

254

1 

N(0.1) 

-is simkvrivis ( () 

z e 

2 

z 

2 /2 

) mniSvnelobebi 

Z 0 1 2 3 4 5 6 7 8 9 

0.0 .398942 .398922 .398862 .398763 .398623 .398444 .398225 .397966 .397668 .397330 

0.1 .396953 .396536 .396080 .395585 .395052 .394479 .393868 .393219 .392531 .391806 

0.2 .391043 .390242 .389404 .388529 .387617 .386668 .385683 .384663 .383606 .382515 

0.3 .381388 .380226 .379031 .377801 .376537 .375240 .373911 .372548 .371154 .369728 

0.4 .368270 .366782 .365263 .363714 .362135 .360527 .358890 .357225 .355533 .353812 

0.5 .352065 .350292 .348493 .346668 .344818 .342944 .341046 .339124 .337180 .335213 

0.6 .333225 .331215 .329184 .327133 .325062 .322972 .320864 .318737 .316593 .314432 

0.7 .312254 .310060 .307851 .305627 .303389 .301137 .298872 .296595 .294305 .292004 

0.8 .289692 .287369 .285036 .282694 .280344 .277985 .275618 .273244 .270864 .268477 

0.9 .266085 .263688 .261286 .258881 .256471 .254059 .251644 .249228 .246809 .244390 

Z 0 1 2 3 4 5 6 7 8 9 

1.0 .241971 .239551 .237132 .234714 .232297 .229882 .227470 .225060 .222653 .220251 

1.1 .217852 .215458 .213069 .210686 .208308 .205936 .203571 .201214 .198863 .196520 

1.2 .194186 .191860 .189543 .187235 .184937 .182649 .180371 .178104 .175847 .173602 

1.3 .171369 .169147 .166937 .164740 .162555 .160383 .158225 .156080 .153948 .151831 

1.4 .149727 .147639 .145564 .143505 .141460 .139431 .137417 .135418 .133435 .131468 

1.5 .129518 .127583 .125665 .123763 .121878 .120009 .118157 .116323 .114505 .112704 

1.6 .110921 .109155 .107406 .105675 .103961 .102265 .100586 .098925 .097282 .095657 

1.7 .094049 .092459 .090887 .089333 .087796 .086277 .084776 .083293 .081828 .080380 

1.8 .078950 .077538 .076143 .074766 .073407 .072065 .070740 .069433 .068144 .066871 

1.9 .065616 .064378 .063157 .061952 .060765 .059595 .058441 .057304 .056183 .055079 

Z 0 1 2 3 4 5 6 7 8 9 

2.0 .053991 .052919 .051864 .050824 .049800 .048792 .047800 .046823 .045861 .044915 

2.1 .043984 .043067 .042166 .041280 .040408 .039550 .038707 .037878 .037063 .036262 

2.2 .035475 .034701 .033941 .033194 .032460 .031740 .031032 .030337 .029655 .028985 

2.3 .028327 .027682 .027048 .026426 .025817 .025218 .024631 .024056 .023491 .022937 

2.4 .022395 .021862 .021341 .020829 .020328 .019837 .019356 .018885 .018423 .017971 

2.5 .017528 .017095 .016670 .016254 .015848 .015449 .015060 .014678 .014305 .013940 

2.6 .013583 .013234 .012892 .012558 .012232 .011912 .011600 .011295 .010997 .010706 

2.7 .010421 .010143 3z98712 3z96058 3z93466 3z90936 3z88465 3z86052 3z83697 3z81398 

2.8 3z79155 3z76965 3z74829 3z72744 3z70711 3z68728 3z66793 3z64907 3z63067 3z61274 

2.9 3z59525 3z57821 3z56160 3z54541 3z52963 3z51426 3z49929 3z48470 3z47050 3z45666 

255

1 x 

N(0.1) 

-is ganawilebis funqciis ( ( x) 

e 

2 

dt ) mniSvnelobebi 

2 

 

2 

t 

x (x) x (x) x (x) x (x) x (x) x (x) 

0.00 0.500 0.33 0.629 0.66 0.745 0.99 0.838 1.32 0.906 1.65 0.950 

0.01 0.503 0.34 0.633 0.67 0.748 1.00 0.841 1.33 0.908 1.66 0.951 

0.02 0.507 0.35 0.636 0.68 0.751 1.01 0.843 1.34 0.909 1.67 0.952 

0.03 0.511 0.36 0.640 0.69 0.754 1.02 0.846 1.35 0.911 1.68 0.953 

0.04 0.515 0.37 0.644 0.70 0.758 1.03 0.848 1.36 0.913 1.69 0.954 

0.05 0.519 0.38 0.648 0.71 0.761 1.04 0.850 1.37 0.914 1.70 0.955 

0.06 0.523 0.39 0.651 0.72 0.764 1.05 0.853 1.38 0.916 1.71 0.956 

0.07 0.527 0.40 0.655 0.73 0.767 1.06 0.855 1.39 0.917 1.72 0.957 

0.08 0.531 0.41 0.659 0.74 0.770 1.07 0.857 1.40 0.919 1.73 0.958 

0.09 0.535 0.42 0.662 0.75 0.773 1.08 0.859 1.41 0.920 1.74 0.959 

0.10 0.539 0.43 0.666 0.76 0.776 1.09 0.862 1.42 0.922 1.75 0.959 

0.11 0.543 0.44 0.670 0.77 0.779 1.10 0.864 1.43 0.923 1.76 0.960 

0.12 0.547 0.45 0.673 0.78 0.782 1.11 0.866 1.44 0.925 1.77 0.961 

0.13 0.551 0.46 0.677 0.79 0.785 1.12 0.868 1.45 0.926 1.78 0.962 

0.14 0.555 0.47 0.680 0.80 0.788 1.13 0.870 1.46 0.927 1.79 0.963 

0.15 0.559 0.48 0.684 0.81 0.791 1.14 0.872 1.47 0.929 1.80 0.964 

0.16 0.563 0.49 0.687 0.82 0.793 1.15 0.874 1.48 0.930 1.81 0.964 

0.17 0.567 0.50 0.691 0.83 0.796 1.16 0.876 1.49 0.931 1.82 0.965 

0.18 0.571 0.51 0.694 0.84 0.799 1.17 0.879 1.50 0.933 1.83 0.966 

0.19 0.575 0.52 0.698 0.85 0.802 1.18 0.881 1.51 0.934 1.84 0.967 

0.20 0.579 0.53 0.701 0.86 0.805 1.19 0.882 1.52 0.935 1.85 0.967 

0.21 0.583 0.54 0.705 0.87 0.807 1.20 0.884 1.53 0.936 1.86 0.968 

0.22 0.587 0.55 0.708 0.88 0.810 1.21 0.886 1.54 0.938 1.87 0.969 

0.23 0.590 0.56 0.712 0.89 0.813 1.22 0.888 1.55 0.939 1.88 0.969 

0.24 0.594 0.57 0.715 0.90 0.815 1.23 0.890 1.56 0.940 1.89 0.970 

0.25 0.598 0.58 0.719 0.91 0.818 1.24 0.892 1.57 0.941 1.90 0.971 

0.26 0.602 0.59 0.722 0.92 0.821 1.25 0.894 1.58 0.942 1.91 0.971 

0.27 0.606 0.60 0.725 0.93 0.823 1.26 0.896 1.59 0.944 1.92 0.972 

0.28 0.610 0.61 0.729 0.94 0.826 1.27 0.897 1.60 0.945 1.93 0.973 

0.29 0.614 0.62 0.732 0.95 0.828 1.28 0.899 1.61 0.946 1.94 0.973 

0.30 0.617 0.63 0.735 0.96 0.831 1.29 0.901 1.62 0.947 1.95 0.974 

0.31 0.621 0.64 0.738 0.97 0.833 1.30 0.903 1.63 0.948 1.96 0.975 

0.32 0.625 0.65 0.742 0.98 0.836 1.31 0.904 1.64 0.949 1.97 0.975 

256

t 

1 

z e 

2 

dt 

z 

0() 

 

2 

 

0 

funqciis cxrilebi 

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 

0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359 

0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753 

0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141 

0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 

0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879 

0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224 

0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549 

0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852 

0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133 

0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389 

1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621 

1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830 

1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015 

1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177 

1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319 

1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441 

1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545 

1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633 

1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706 

1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767 

2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 

2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857 

2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890 

2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916 

2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936 

2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952 

2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964 

2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974 

2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981 

2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986 

3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990 

2 

257

t (stiudentis) ganawilebis zeda -kritikuli wertilebi ( t ,n 

) 

 

n 0.1 0.05 0.025 0.01 0.005 0.0025 0.001 

1 3.078 6.314 12.706 31.821 63.656 127.321 318.289 

2 1.886 2.920 4.303 6.965 9.925 14.089 22.328 

3 1.638 2.353 3.182 4.541 5.841 7.453 10.214 

4 1.533 2.132 2.776 3.747 4.604 5.598 7.173 

5 1.476 2.015 2.571 3.365 4.032 4.773 5.894 

6 1.440 1.943 2.447 3.143 3.707 4.317 5.208 

7 1.415 1.895 2.365 2.998 3.499 4.029 4.785 

8 1.397 1.860 2.306 2.896 3.355 3.833 4.501 

9 1.383 1.833 2.262 2.821 3.250 3.690 4.297 

10 1.372 1.812 2.228 2.764 3.169 3.581 4.144 

11 1.363 1.796 2.201 2.718 3.106 3.497 4.025 

12 1.356 1.782 2.179 2.681 3.055 3.428 3.930 

13 1.350 1.771 2.160 2.650 3.012 3.372 3.852 

14 1.345 1.761 2.145 2.624 2.977 3.326 3.787 

15 1.341 1.753 2.131 2.602 2.947 3.286 3.733 

16 1.337 1.746 2.120 2.583 2.921 3.252 3.686 

17 1.333 1.740 2.110 2.567 2.898 3.222 3.646 

18 1.330 1.734 2.101 2.552 2.878 3.197 3.610 

19 1.328 1.729 2.093 2.539 2.861 3.174 3.579 

20 1.325 1.725 2.086 2.528 2.845 3.153 3.552 

21 1.323 1.721 2.080 2.518 2.831 3.135 3.527 

22 1.321 1.717 2.074 2.508 2.819 3.119 3.505 

23 1.319 1.714 2.069 2.500 2.807 3.104 3.485 

24 1.318 1.711 2.064 2.492 2.797 3.091 3.467 

25 1.316 1.708 2.060 2.485 2.787 3.078 3.450 

26 1.315 1.706 2.056 2.479 2.779 3.067 3.435 

27 1.314 1.703 2.052 2.473 2.771 3.057 3.421 

28 1.313 1.701 2.048 2.467 2.763 3.047 3.408 

29 1.311 1.699 2.045 2.462 2.756 3.038 3.396 

30 1.310 1.697 2.042 2.457 2.750 3.030 3.385 

258

t ganawilebis zeda /2-kritikuli wertilebi t n, 

/2 

(orkudiani) 

n 

A 0.80 

0.20 

0.90 

0.10 

0.95 

0.05 

0.98 

0.02 

0.99 

0.01 

0.995 

0.005 

0.998 

0.002 

0.999 

0.001 

1 3.078 6.314 12.706 31.820 63.657 127.321 318.309 636.619 

2 1.886 2.920 4.303 6.965 9.925 14.089 22.327 31.599 

3 1.638 2.353 3.182 4.541 5.841 7.453 10.215 12.924 

4 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610 

5 1.476 2.015 2.571 3.365 4.032 4.773 5.893 6.869 

6 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.959 

7 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408 

8 1.397 1.860 2.306 2.897 3.355 3.833 4.501 5.041 

9 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781 

10 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587 

11 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.437 

12 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.318 

13 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.221 

14 1.345 1.761 2.145 2.625 2.977 3.326 3.787 4.140 

15 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.073 

16 1.337 1.746 2.120 2.584 2.921 3.252 3.686 4.015 

17 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3.965 

18 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.922 

19 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.883 

20 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.850 

21 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3.819 

22 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3.792 

23 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.768 

24 1.318 1.711 2.064 2.492 2.797 3.090 3.467 3.745 

25 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3.725 

26 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.707 

27 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.690 

28 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3.674 

29 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.659 

30 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3.646 

1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291 

259

2 

(xi kvadrat) ganawilebis zeda -kritikuli wertilebi ( 

) 

2 

,n 

 

n 0.995 0.975 0.20 0.10 0.05 0.025 0.02 0.01 0.005 0.002 0.001 

1 0.000039 0.00098 1.642 2.706 3.841 5.024 5.412 6.635 7.879 9.550 10.828 

2 0.0100 0.0506 3.219 4.605 5.991 7.378 7.824 9.210 10.597 12.429 13.816 

3 0.0717 0.216 4.642 6.251 7.815 9.348 9.837 11.345 12.838 14.796 16.266 

4 0.207 0.484 5.989 7.779 9.488 11.143 11.668 13.277 14.860 16.924 18.467 

5 0.412 0.831 7.289 9.236 11.070 12.833 13.388 15.086 16.750 18.907 20.515 

6 0.676 1.237 8.558 10.645 12.592 14.449 15.033 16.812 18.548 20.791 22.458 

7 0.989 1.690 9.803 12.017 14.067 16.013 16.622 18.475 20.278 22.601 24.322 

8 1.344 2.180 11.030 13.362 15.507 17.535 18.168 20.090 21.955 24.352 26.124 

9 1.735 2.700 12.242 14.684 16.919 19.023 19.679 21.666 23.589 26.056 27.877 

10 2.156 3.247 13.442 15.987 18.307 20.483 21.161 23.209 25.188 27.722 29.588 

11 2.603 3.816 14.631 17.275 19.675 21.920 22.618 24.725 26.757 29.354 31.264 

12 3.074 4.404 15.812 18.549 21.026 23.337 24.054 26.217 28.300 30.957 32.909 

13 3.565 5.009 16.985 19.812 22.362 24.736 25.472 27.688 29.819 32.535 34.528 

14 4.075 5.629 18.151 21.064 23.685 26.119 26.873 29.141 31.319 34.091 36.123 

15 4.601 6.262 19.311 22.307 24.996 27.488 28.259 30.578 32.801 35.628 37.697 

16 5.142 6.908 20.465 23.542 26.296 28.845 29.633 32.000 34.267 37.146 39.252 

17 5.697 7.564 21.615 24.769 27.587 30.191 30.995 33.409 35.718 38.648 40.790 

18 6.265 8.231 22.760 25.989 28.869 31.526 32.346 34.805 37.156 40.136 42.312 

19 6.844 8.907 23.900 27.204 30.144 32.852 33.687 36.191 38.582 41.610 43.820 

20 7.434 9.591 25.038 28.412 31.410 34.170 35.020 37.566 39.997 43.072 45.315 

21 8.034 10.283 26.171 29.615 32.671 35.479 36.343 38.932 41.401 44.522 46.797 

22 8.643 10.982 27.301 30.813 33.924 36.781 37.659 40.289 42.796 45.962 48.268 

23 9.260 11.689 28.429 32.007 35.172 38.076 38.968 41.638 44.181 47.391 49.728 

24 9.886 12.401 29.553 33.196 36.415 39.364 40.270 42.980 45.559 48.812 51.179 

25 10.520 13.120 30.675 34.382 37.652 40.646 41.566 44.314 46.928 50.223 52.620 

26 11.160 13.844 31.795 35.563 38.885 41.923 42.856 45.642 48.290 51.627 54.052 

27 11.808 14.573 32.912 36.741 40.113 43.195 44.140 46.963 49.645 53.023 55.476 

28 12.461 15.308 34.027 37.916 41.337 44.461 45.419 48.278 50.993 54.411 56.892 

29 13.121 16.047 35.139 39.087 42.557 45.722 46.693 49.588 52.336 55.792 58.301 

30 13.787 16.791 36.250 40.256 43.773 46.979 47.962 50.892 53.672 57.167 59.703 

31 14.458 17.539 37.359 41.422 44.985 48.232 49.226 52.191 55.003 58.536 61.098 

32 15.134 18.291 38.466 42.585 46.194 49.480 50.487 53.486 56.328 59.899 62.487 

33 15.815 19.047 39.572 43.745 47.400 50.725 51.743 54.776 57.648 61.256 63.870 

34 16.501 19.806 40.676 44.903 48.602 51.966 52.995 56.061 58.964 62.608 65.247 

35 17.192 20.569 41.778 46.059 49.802 53.203 54.244 57.342 60.275 63.955 66.619 

260

F( n, m ) (fiSeris) ganawilebis zeda -kritikuli wertilebi ( Fnm 

, , 

) 

n 0.1 

m 1 2 3 4 5 7 10 15 20 

1 39.864 49.500 53.593 55.833 57.240 58.906 60.195 61.220 61.740 

2 8.5264 8.9999 9.1618 9.2434 9.2926 9.3491 9.3915 9.4248 9.4413 

3 5.5384 5.4624 5.3907 5.3426 5.3092 5.2661 5.2304 5.2003 5.1845 

4 4.5448 4.3245 4.1909 4.1073 4.0505 3.9790 3.9198 3.8704 3.8443 

5 4.0605 3.7798 3.6194 3.5202 3.4530 3.3679 3.2974 3.2379 3.2067 

7 3.5895 3.2575 3.0740 2.9605 2.8833 2.7850 2.7025 2.6322 2.5947 

10 3.2850 2.9244 2.7277 2.6054 2.5216 2.4139 2.3226 2.2434 2.2007 

15 3.0731 2.6951 2.4898 2.3615 2.2729 2.1582 2.0593 1.9722 1.9243 

20 2.9746 2.5893 2.3801 2.2490 2.1582 2.0397 1.9368 1.8450 1.7939 

30 2.8808 2.4887 2.2761 2.1423 2.0493 1.9269 1.8195 1.7222 1.6674 

60 2.7911 2.3932 2.1774 2.0409 1.9457 1.8194 1.7070 1.6034 1.5435 

n 0.05 

m 1 2 3 4 5 7 10 15 20 

1 161.45 199.50 215.71 224.58 230.16 236.77 241.88 245.95 248.01 

2 18.513 19.000 19.164 19.247 19.296 19.353 19.396 19.429 19.446 

3 10.128 9.5522 9.2766 9.1172 9.0135 8.8867 8.7855 8.7028 8.6602 

4 7.7086 6.9443 6.5915 6.3882 6.2560 6.0942 5.9644 5.8579 5.8026 

5 6.6078 5.7862 5.4095 5.1922 5.0504 4.8759 4.7351 4.6187 4.5582 

7 5.5914 4.7375 4.3469 4.1202 3.9715 3.7871 3.6366 3.5108 3.4445 

10 4.9645 4.1028 3.7082 3.4780 3.3259 3.1354 2.9782 2.8450 2.7741 

15 4.5431 3.6823 3.2874 3.0556 2.9013 2.7066 2.5437 2.4035 2.3275 

20 4.3512 3.4928 3.0983 2.8660 2.7109 2.5140 2.3479 2.2032 2.1241 

30 4.1709 3.3159 2.9223 2.6896 2.5336 2.3343 2.1646 2.0149 1.9317 

60 4.0012 3.1505 2.7581 2.5252 2.3683 2.1666 1.9927 1.8365 1.7480 

261

F( n, m ) (fiSeris) ganawilebis zeda -kritikuli wertilebi ( Fnm 

, , 

) 

n 0.01 

m 1 2 3 4 5 7 10 15 20 

1 4052.2 4999.5 5403.4 5624.6 5763.6 5928.4 6055.8 6157.3 6208.7 

2 98.503 99.000 99.166 99.249 99.299 99.356 99.399 99.433 99.449 

3 34.116 30.817 29.457 28.710 28.237 27.672 27.229 26.872 26.690 

4 21.198 18.000 16.694 15.977 15.522 14.976 14.546 14.198 14.020 

5 16.258 13.274 12.060 11.392 10.967 10.455 10.051 9.7222 9.5526 

7 12.246 9.5467 8.4513 7.8466 7.4605 6.9929 6.6201 6.3143 6.1554 

10 10.044 7.5594 6.5523 5.9944 5.6363 5.2001 4.8492 4.5582 4.4055 

15 8.6831 6.3588 5.4169 4.8932 4.5557 4.1416 3.8049 3.5223 3.3719 

20 8.0960 5.8489 4.9382 4.4306 4.1027 3.6987 3.3682 3.0880 2.9377 

30 7.5624 5.3903 4.5098 4.0179 3.6990 3.3046 2.9791 2.7002 2.5486 

60 7.0771 4.9774 4.1259 3.6491 3.3388 2.9530 2.6318 2.3522 2.1978 

n 0.005 

m 1 2 3 4 5 7 10 15 20 

1 16211 19999 21615 22500 23056 23715 24224 24630 24836 

2 198.50 199.00 199.17 199.25 199.30 199.36 199.40 199.43 199.45 

3 55.552 49.799 47.467 46.195 45.392 44.434 43.686 43.085 42.777 

4 31.333 26.284 24.259 23.155 22.456 21.622 20.967 20.438 20.167 

5 22.785 18.314 16.530 15.556 14.940 14.200 13.618 13.146 12.903 

7 16.235 12.404 10.882 10.050 9.5221 8.8853 8.3803 7.9677 7.7539 

10 12.826 9.4270 8.0807 7.3428 6.8723 6.3025 5.8467 5.4706 5.2740 

15 10.798 7.7007 6.4760 5.8029 5.3721 4.8473 4.4235 4.0697 3.8826 

20 9.9439 6.9865 5.8176 5.1744 4.7616 4.2569 3.8470 3.5020 3.3178 

30 9.1796 6.3547 5.2387 4.6233 4.2275 3.7416 3.3439 3.0058 2.8231 

60 8.4946 5.7950 4.7290 4.1399 3.7599 3.2911 2.9042 2.5705 2.3872 

262

damateba 

sagamocdo amocanebi xdomilebebisaTvis 

xdomilebebi 

1. aageT elementarul xdomilebaTa sivrce saTamaSo kamaTlis orjer gagorebisas. 

gamoyaviT xdomilebebi: A -- mosul qulaTa jami metia 5-ze, 

magram naklebia 9-ze; B -- mosul qulaTa jami naklebia 6-ze an metia 8- 

ze; C -- mosul qulebSi erTi mainc oTxiania; D -- mosul qulaTa jami 

kentia; E -- mosul qulaTa sxvaoba tolia 4-is. miuTiTeT Tavsebad da 

uTavsebad xdomilebaTa yvela wyvili. 

2. ramdeni elementisagan Sedgeba elementarul xdomilebaTa sivrce: saTamaSo 

kamaTlis samjer agdebisas? 

3. a). Tu A da B uTavsebadi xdomilebebia, da A da C agreTve uTavsebia, 

maSin gamodis Tu ara aqedan, rom B da C uTavsebadia? b). Tu A da 

B Tavsebadi xdomilebebia, da A da C agreTve Tavsebadia, maSin gamodis 

Tu ara aqedan, rom B da C Tavsebadia? 

4. aucilebeli da SeuZlebeli xdomilebebis: a). gadakveTa b). gaerTianeba. 

5. daamtkiceT, rom Tu AB A C , maSin B C araa marTebuli. 

6. daamtkiceT, rom Tu AB A C da A , maSin B C araa marTebuli. 

7. iwvevs Tu ara xdomileba A( B C) 

xdomilebas A( B C) 

? 

8. samarTliania Tu ara CarTva [( BC) B] C [( C C) B] ( B B) 

? 

9. daamtkiceT Tanafardobebi: a). A( BA) 

; b). A( BA) 

A B ; g). 

A( BA) 

A B ; d). A( BA) 

A B ; e). A 

( A) 

( 

A 

A) 

; v). 

A( A) ( A) 

A. 

10. rogor SeiZleba martivad Caiweros Semdegi xdomilebebi: 

a). A( B A) 

; b). A( B A) 

; g). A[ B( A B)] 

; d). B[ A( B A)] 

; 

e). ( BBB) ( A A A) 

; v). ( BBB) ( A A A) 

; z). B[ A( B A)] 

; 

T). ( BA) ( A A) 

; i). ( AB) 

A; k). ( BA) 

B ; l). ( BA) 

A; 

m). ( BA) 

B . 

11. daamtkiceT Semdegi Tanafardobebi: 

a). B( A A) A( B B) 

; b). ( A) ( A) A( A B) 

; 

g). B( AB) ( BB) ( A) 

; d). ( A A) ( A) ( A) 

A; 

e). [ B( AB)] A ( BB) ( A A) 

; v). ( AB) 

A A B ; 

z). ( AB) ( AB) ( A) 

B; T). [ A( BA)] [( AB) B] 

A B. 

kombinatorika 

12. manqanaSi 5 adgilia. ramden sxvadasxvanairad SeiZleba ganTavsdes manqanaSi 

5 adamiani, Tu erTs ara aqvs ufleba daikavos adgili saWesTan? 

13. manqanaSi 5 adgilia. ramden sxvadasxvanairad SeiZleba ganTavsdes manqanaSi 

4 adamiani, Tu saWesTan adgilis dakaveba SeuZlia or maTgans? 

263

14. studentma unda Caabaros 4 gamocda 7 dRis ganmavlobaSi. ramden sxvadasxvanairad 

SeiZleba gamocdebis cxrilis Sedgena, Tu erT dRes ar SeiZleba 

erTze meti gamocdis Cabareba? 

15. studentma 6 dReSi unda Caabaros 3 gamocda: maTematika, fizika da 

eleqtronika. ramden sxvadasxvanairad SeiZleba gamocdebis cxrilis 

Sedgena, Tu erT dRes ar SeiZleba erTze meti gamocdis Cabareba da jer 

unda Caabaros maTematika, Semdeg fizika da bolos, eleqtronika? 

16. 7 adamianidan, romelTa Soris 3 doqtoria, unda Sedges 4 kaciani 

komisia, im pirobiT, rom komisiaSi Sediodes erTi mainc doqtori. ramden 

sxvadasxvanairad SeiZleba Sedges aseTi komisia? 

17. 10 adamiani unda ganTavsdes 3 oTaxSi, ise, rom TiToeul oTaxSi iyos 

aranakleb 3 adamiani. ramden sxvadasxvanairad SeiZleba amis gakeTeba? 

18. 8 adamiani unda ganTavsdes 3 oTaxSi, ise, rom TiToeul oTaxSi iyos 

aranakleb 2 adamiani. ramden sxvadasxvanairad SeiZleba amis gakeTeba? 

19. monetas agdeben 6-jer. ramdeni variantia safasuris an 2-jer an 3-jer 

mosvlis? 

20. monetas agdeben 7-jer. ramdeni variantia an safasuris 2-jer an gerbis 

3-jer mosvlis? 

21. monetas agdeben 8-jer. ramdeni variantia erTdroulad safasuris 5- 

jer da gerbis 3-jer mosvlis? 

22. monetas agdeben 9-jer. sul ramdeni variantia gerbebisa da safasurebis 

mosvlis? 

23. monetas agdeben 8-jer. ras ufro meti Sansebi aqvs: safasuris 4-jer 

mosvlas, Tu gerbis 5-jer mosvlas da ramdenjer? 

albaTobis klasikuri ganmarteba 

24. yuTSi 10 wiTeli, 5 TeTri da 4 lurji burTia. SemTxveviT iReben erT 

burTs. vipovoT albaToba imisa, rom: a). amoRebulia lurji burTi? b). 

araa amoRebuli wiTeli burTi? g). araa amoRebuli TeTri burTi? d). amo- 

Rebulia wiTeli an TeTri burTi? 

25. rva adamianisagan, romelTa Soris 3 doqtoria da 5 kandidati, unda 

SevadginoT sam kaciani komisia. komisiis wevrebi SeirCeva SemTxveviT. 

rogoria albaToba imisa, rom komisiaSi moxvdeba: a). sami doqtori? b). 

ori doqtori da erTi kandidati? g). erTi doqtori da ori kandidati? 

d). sami kandidati? 

26. 9 wignidan, romelTa Soris 4 xelovnebazea da 5 deteqtivia, 

SemTxveviT irCeven sams. rogoria albaToba imisa, rom: a). samive wigni 

iqneba xelovnebaze? b). samive iqneba deteqtivi? g). erTi an ori wigni 

iqneba xelovnebaze? d). zustad ori iqneba deteqtivi? 

27. monetas agdeben samjer. rogoria albaToba imisa, rom gerbi mova: a). 

kent ricxvjer? b). luw ricxvjer? g). arc erTxel? d). orjer mainc? 

28. monetas agdeben 4-jer. rogoria albaToba imisa, rom gerbi mova: a). 

kent ricxvjer? b). luw ricxvjer? g). aranakleb orjer? d). erT-jer mainc? 

264

29. monetas agdeben oTxjer. rogoria alabaToba imisa, rom safasuri 

mova: a). aranakleb samjer? b). aranakleb orjer? g). aranakleb erT-jer? 

d). arc erTxel? 

30. monetas agdeben oTxjer. rogoria alabaToba imisa, rom gerbi mova: a). 

erTjer an arc erTjer? b). an orjer an samjer? g). an erTjer an orjer? 

d)/ an erTjer an samjer? 

31. monetas agdeben xuTjer. rogoria alabaToba imisa, rom: a). gerbi mova 

kent ricxvjer, xolo safasuri luw ricxvjer? b). gerbi mova kent ricxvjer? 

g). gerbi mova luw ricxvjer? d). gerbi mova luw ricxvjer, xolo 

safasuri kent ricxvjer? 

32. monetas agdeben xuTjer. rogoria alabaToba imisa, rom gerbi mova: a). 

aranakleb samjer? b). aranakleb oTxjer? g). an erTjer an orjer? d). an 

erTjer an oTxjer? e). an orjer an samjer? 

33. saTamaSo kamaTels agoreben orjer. ra ufro albaTuria – jamSi mova: 

a). 6 qula Tu 8 qula? b). 5 qula Tu 9 qula? g). 5 qula Tu 6 qula? d). 5 

qula Tu 8 qula? e). 6 qula Tu 7 qula? v). 7 qula Tu 5 qula? 

34. rogoria albaToba imisa, rom ori saTamaSo kamaTlis gagorebisas mova 

iseTi qulebi, romlebic orjer gansxvavdebian erTmaneTisagan? 

35. rogoria albaToba imisa, rom saTamaSo kamaTlis orjer gagorebisas 

erTjer mainc mova: a). erTiani? b). oriani? g). samiani? d). oTxiani? e). xu- 

Tiani? v). eqvsiani? 

36. romelia meti da ramdeniT: albaToba imisa, rom saTamaSo kamaTlis 

erTjer gagorebisas mova 6 qula Tu orjer gagorebisas jamSi mova 6 

qula? 


erTjer gagorebisas mova luwi qula Tu orjer gagorebisas jamSi mova 

luwi qula? 

38. saTamaSo kamaTels agoreben orjer. rogoria albaToba imisa, rom mosul 

qulaTa jami aRmoCndeba: a). kenti ricxvi? b). 4-ze meti da 9-ze naklebi? 

g). 3-is jeradi? d). 4-is jeradi? e). 5-is jeradi? v). 6-is jeradi? 


erTjer gagorebisas mova 3-is jeradi qula Tu orjer gagorebisas jamSi 

mova 3-is jeradi qula? 


erTjer gagorebisas mova 4-is jeradi qula Tu orjer gagorebisas jamSi 

mova 4-is jeradi qula? 

41. saTamaSo kamaTels agoreben orjer. rogoria albaToba imisa, rom mosul 

qulaTa namravli aRmoCndeba: a). 12-is toli? b). 3-is jeradi? g). 4-is 

jeradi? d). 5-is jeradi? e). 10-is jeradi? 

42. saTamaSo kamaTels agoreben samjer. rogoria albaToba imisa, rom mova 

sami waxnagi: a). erTi da igive qulebiT? b). sxvadasxva qulebiT? g). 9-is 

toli qulaTa jamiT? d). 10-is toli qulaTa jamiT? e). 5-is jeradi qula- 

Ta jamiT? 

43. saTamaSo kamaTels agoreben samjer. rogoria albaToba imisa, rom mova 

erTi mainc erTiani? 

265

44. saTamaSo kamaTels agoreben samjer. rogoria albaToba imisa, rom mova: 

a). erTjer mainc oriani? b). erTjer mainc samiani? g). erTjer mainc 

oTxiani? d). erTjer mainc xuTiani? e). erTjer mainc eqvsiani? 

45. saTamaSo kamaTels agoreben samjer. rogoria albaToba imisa, rom 

arc erTjer ar mova erTiani? igive kiTxva 2, 3, 4, 5 da 6-saTvis. 

46. romeli albaTobaa meti da ramdeniT: a). 5 qulis mogroveba saTamaSo 

kamaTlis orjer gagorebisas Tu samjer gagorebisas? b). 6 qulis mogroveba 

saTamaSo kamaTlis orjer gagorebisas Tu samjer gagorebisas? g). 11 

qulis mogroveba saTamaSo kamaTlis orjer gagorebisas Tu samjer gagorebisas? 

00 

30 

47. ori adamiani SeTanxmda, rom erTmaneTs Sexvdes 19 -dan 19 -mde. rogoria 

Sexvedris albaToba, Tu a). erTi adgilze mosvlis Semdeg icdis 30 

wuTs, xolo meore saerTod ar icdis? b). erTi icdis 10 wuTs, xolo meore 

5 wuTs? 

albaTobis gamoTvlis ZiriTadi formulebi 

48. rogori C xdomilebebisaTvis miiRebs jamis albaTobis formula Semdeg 

saxes: P( ABC) P( A) P( B) P( C) P( A B) 

? gamosaxeT venis diagramiT. 

49. rogori C xdomilebebisaTvis miiRebs jamis albaTobis formula Semdeg 

saxes: P( AB) P( A) P( B) P( A B) 

? gamosaxeT venis diagramiT. 

50. saTamaSo kamaTels agdeben orjer. gamoTvaleT P( A B) 

da P( A 

B) 

albaTobebi A da B xdomilebaTa Semdegi wyvilebisaTvis: 

a). A -- mosul qulaTa jami kentia, B -- movida erTi da igive qula; 

b). A -- mosul qulaTa jami luwia, B -- movida erTi da igive qu-la; 

g). A -- mosul qulaTa jami 7-ze metia, B -- mosul qulaTa jami 7-ze 

naklebia; 

d). A -- mosul qulaTa jami 10-ze naklebia, B -- mosul qulaTa jami 5-ze 

metia; 

e). A -- mosul qulaTa jami luwia, B -- mosul qulaTa jami 8-ze naklebia; 

v). A -- mosul qulaTa jami kentia, B -- mosul qulaTa jami 3-ze naklebia. 


da 

P( AB C) 

albaTobebi A, B da C xdomilebaTa Semdegi sameulebisaTvis: 

a). A -- mosul qulaTa jami luwia, B -- mosul qulaTa jami 7-ze naklebia, 

C -- meore agdebisas movida 2 qula; 

b). A -- mosul qulaTa jami kentia, B -- mosul qulaTa sxvaobaa 1, C -- 

mosul qulaTa jami 8-ze naklebia; 

g). A -- movida sxvadasxva qula, B -- mosul qulaTa jami 7-ze naklebia, 

C -- mosul qulaTa jamia 6; 

d). A -- movida erTi mainc erTiani, B -- movida erTi mainc eqvsiani, C -- 

movida erTi da igive qula; 

266

e). A -- movida erTi mainc oriani, B -- pirveli agdebisas movida 2 qula, 

C -- mosul qulaTa jami 6-ze naklebia; 

v). A -- pirveli agdebisas movida 1 qula, B -- movida erTi da igive qula, 

C -- mosul qulaTa jamia 6. 

52. saTamaSo kamaTels agdeben orjer. gamoTvaleT pirobiTi albaToba 

PB( A ) A da B xdomilebaTa Semdegi wyvilebisaTvis, da miuTiTeT arian 

Tu ara isini damoukideblebi: 

a). A -- movida erTi da igive qula, B -- mosul qulaTa jami luwia; 

b). A -- pirveli agdebisas movida 3 qula, B -- meore agdebisas movida 3 

qula; 

g). A -- mosul qulaTa sxvaobaa 1, B -- mosul qulaTa jamia 5; 

d). A -- mosul qulaTa sxvaobaa 2, B -- mosul qulaTa jamia 8; 

e). A -- mosul qulaTa jamia 7, B -- meore agdebisas movida 1 qula; 

v). A -- mosul qulaTa jami naklebia 6-ze, B -- mosul qulaTa jami naklebia 

10ze. 

53. monetas agdeben 4-jer. daamtkiceT, rom A, B da C xdomilebaTa Semdegi 

sameulebi erToblivad damoukidebelia da gamoTvaleT albaTobebi 

gadakveTebis: A B, A C, B C, AB C : 

a). A -- pirveli agdebisas movida gerbi, B -- mesame agdebisas movida gerbi, 

C -- meoTxe agdebisas movida gerbi; 

b). A -- pirveli agdebisas movida gerbi, B -- meore agdebisas movida gerbi, 

C -- mesame agdebisas movida gerbi; 

g). A -- pirveli ori agdebisas movida gerbi, B -- mesame agdebisas movida 

gerbi, C -- meoTxe agdebisas movida gerbi; 

d). A -- pirveli agdebisas movida gerbi, B -- meore agdebisas movida gerbi, 

C -- ukanaskneli ori agdebisas movida gerbi. 

54. monetas agdeben 4-jer. daamtkiceT, rom A, B da C xdomilebaTa Semdegi 

sameulebi ar arian erToblivad damoukidebelebi: 

a). A -- pirveli ori agdebisas movida gerbi, B -- mesame agdebisas movida 

gerbi, C -- ukanaskneli ori agdebisas movida gerbi; 

b). A -- pirveli agdebisas movida gerbi, B -- meore agdebisas movida gerbi, 

C -- pirveli ori agdebisas movida gerbi; 

g). A -- pirveli agdebisas movida gerbi, B -- pirveli ori agdebisas movida 

gerbi, C -- oTxive agdebisas movida gerbi; 

d). A -- pirveli agdebisas movida gerbi, B -- pirveli ori agdebisas movida 

gerbi, C -- ukanaskneli ori agdebisas movida gerbi. 

55. amocana #54-Si moyvanil TiToeuli sameulisaTvis arian Tu ara damoukideblebi 

A da B ? A da C ? B da C ? 

56. saTamaSo kamaTels agdeben orjer. daamtkiceT, rom A, B da C xdomilebaTa 

Semdegi sameulebisaTvis adgili aqvs wyvil-wyvilad damoukideblobas, 

magram ara aqvs adgili erToblivad damoukideb-lobas: 

a). A -- pirveli agdebisas movida 1 qula, B -- meore agdebisas movida 6 

qula, C -- jamSi movida 7 qula; 

267

). A -- pirveli agdebisas movida 1 qula, B -- meore agdebisas movida 1 

qula, C -- movida erTi da igive qula; 

g). A -- pirveli agdebisas movida 2 qula, B -- meore agdebisas movida 5 


d). A -- pirveli agdebisas movida 1 qula, B -- meore agdebisas movida 6 

qula, C -- movida erTi da igive qula; 

e). A -- pirveli agdebisas movida 6 qula, B -- meore agdebisas movida 6 


v). A -- pirveli agdebisas movida 6 qula, B -- meore agdebisas movida 1 

qula, C -- movida erTi da igive qula. 

57. saTamaSo kamaTels agdeben samjer. rogoria albaToba imisa, rom mosul 

qulaTa jami iqneba; a). 4; b). 5; g). 6; d). 7; e). 8; v). 9; z). 10? 

58. eqvsi monadire erTdroulad esvris gadamfren ixvs. sami 

maTganisaTvis mizanSi moxvedris albaTobaa 0.4, xolo danarCeni 

samisaTvis – 0.6. rogoria albaToba imisa, rom mizanSi moaxvedrebs erTi 

monadire mainc? 

59. Tu A da B damoukidebeli xdomilebebia, aCveneT, rom maSin damoukidebelia 

A da B xdomilebebi. 

baiesis formula 

60. I brigada awarmoebs detalebis 30%-s, romelTa Soris 1% wundebulia. 

II brigada awarmoebs imave detalebis 20%-s, romelTa Soris 3% wundebulia. 

III brigada awarmoebs detalebis 50%-s, romelTa Soris 2% wundebulia. 

erT sawyobSi mogrovili am detalebidan SemTxveviT aiRes erTi detali 

da is aRmoCnda wundebuli. rogoria albaToba imisa, rom es detali 

daamzada: a). I brigadam; b). II briga-dam; g). III brigadam? 

61. amocana #60-is pirobebSi SemTxveviT amoiRes ori detali da orive 

aRmoCnda wundebuli. rogoria albaToba imisa, rom es detalebi 

daamzada: a). I brigadam; b). II brigadam; g). III brigadam, Tu cnobilia, rom 

sawyobSi iyo 1000 detali? 

62. amocana #60-is pirobebSi SemTxveviT amoiRes sami detali da samive 

aRmoCnda wundebuli. rogoria albaToba imisa, rom es detalebi 

daamzada: a). I brigadam; b). II brigadam; g). III brigadam, Tu cnobilia, rom 

sawyobSi iyo 1000 detali? 

63. avadmyofis gamokvlevis Sedegad gakeTda daskvna, rom mas aqvs erT-er- 

Ti ori daavadebidan – an A 

1 

(hipoTeza A 

1), an A 

2 

(hipoTeza A 

2 

). 

PA ( 

1) 0.4 da PA ( 

2) 0.4 . eqimma gadawyvita diagnozis dazusteba da daniSna 

analizis Catareba, romlis Sedegs warmoadgens dadebiTi an uaryofiTi 

reaqcia. Tu pacients aqvs A 1 

avadmyofoba, analizma unda mogvces dadebi- 

Ti reaqcia albaTobiT 0.9 da uaryofiTi reaqcia – albaTobiT 0.1. Tu ki 

pacients aqvs A 2 

avadmyofoba, maSin dadebiTi da uaryofiTi reaqciebi 

tolalbaTuria. analizi Catarda da man mogvca uaryofiTi reaqcia. rog- 

268

oria albaToba imisa, rom es Sedegi ganpirobebulia: a). A 1 

avadmyofobiT; 

b). A 2 

avadmyofobiT? 

64. amocana #63-is pirobebSi analizi Catarda orjer da orive 

SemTxvevaSi mogvca uaryofiTi reaqcia. rogoria albaToba imisa, rom es 

Sedegebi ganpirobebulia: a). A 1 

avadmyofobiT; b). A 2 

avadmyofobiT? 

65. amocana #63-is pirobebSi analizi Catarda orjer da pirvel 

SemTxvevaSi mogvca uaryofiTi reaqcia, xolo meore SemTxvevaSi – 

dadebiTi. rogoria albaToba imisa, rom es Sedegebi ganpirobebulia: a). 

A 

1 

avadmyofobiT; b). A 2 

avadmyofobiT? 

ganmeorebiTi cdebi 

66. rogoria albaToba imisa, rom saTamaSo kamaTlis oTxjer gagorebisas 

erTiani mova zustad orjer? 


erTiani mova araumetes orjer? 


erTiani mova aranakleb erTjer? 


erTiani mova aranakleb samjer? 


samjer mova waxnagi luwi quliT? 


orjer mova waxnagi kenti quliT? 


aranakleb orjer mova waxnagi kenti quliT? 


aranakleb samjer mova waxnagi luwi quliT? 


orjer mova waxnagi 3-is ara jeradi quliT? 


araumetes orjer mova waxnagi 3-is ara jeradi quliT? 

76. albaToba imisa, rom mocemul sawarmoSi eleqtroenergiis dRiuri danaxarji 

ar gadaaWarbebs dadgenil normas, tolia 0.8-is. rogoria 

albaToba imisa, rom kviris ganmavlobaSi eleqtroenergiis danaxarji ar 

gava normidan oTxi dRis ganmavlobaSi? 

77. 76-e amocanaSi vipovoT albaToba imisa, rom kviris ganmavlobaSi eleqtroenergiis 

danaxarji ar gava normidan aranakleb xuTi dRe? 

78. 76-e amocanaSi vipovoT albaToba imisa, rom kviris ganmavlobaSi eleqtroenergiis 

gadaxarjva aRiniSneboda araumetes sami dRis ganmavlobaSi? 

79. albaToba imisa, rom naTura ar gadaiwveba 1000 saaTis muSaobis Sedegad, 

tolia 0.95-is. risi tolia 500 naTuraSi im naTurebis ualbaTesi ricxvi, 

romlebic ar gadaiwva 1000 saaTis muSaobis Sedegad? 

80. aCveneT, rom Tu pq 1/ 2 da n -- luwia, maSin warmatebis ualbaTesi 

ricxvi tolia maTematikuri lodinis. 

269

81. msrolelis mier samiznis dazianebis albaTobaa 0.8. rogoria albaToba 

imisa, rom 80 gasrolis Sedegad msroleli daazianebs samiznes: a). zustad 

60-jer? b). zustad 64-jer? g). 60-dan 70-mde? d). aranakleb 60-jer? 

82. Teslis mocemuli partiidan daTeses 10000 Tesli da naxes, rom aRmocenda 

8498 Tesli. amis Sedegad gakeTda daskvna, rom Teslis mocemuli 

partiis aRmocenebadoba Seadgens 85%-s ( p 0.85 ). vipovoT albaToba imisa, 

rom mocemuli partiidan daTesili 100 Teslidan aRmocendeba: a). zustad 

85 Tesli? b). 60-dan 80-mde? g). 70-dan 90-mde? d). aranakleb 80 Tesli? e). 

aranakleb 85 Tesli? v). aranakleb 90 Tesli? z). aranakleb 92 Tesli? 

83. gamoTvaleT albaToba imisa, rom saTamaSo kamaTlis gagorebisas erTi 

qulis mosvlis fardobiTi sixSire gadaixreba 1/6-sagan ufro metad vidre 

0.01 , Tu saTamaSo kamaTels vagorebT Sesabamisad, 600, 3000, 6000, 

12000 da 24000-jer? 

84. davuSvaT, rom garkveuli epidemiis dros daavadebis albaTobaa 

p 0.005 . rogoria albaToba imisa, rom 1000 adamianidan daavaddeba: a). 

araumetes 2 adamiani? b). araumetes 5 adamiani? g). araumetes 10 adamiani? 

d). aranakleb 3 adamiani? 

85. profilaqtikuri RonisZiebebis gatarebam mosalodneli epidemiis 

dros daavadebis albaToba Seamcira p1 0.005 -dan p1 0.003 -mde. amis 

Sedegad ramdenjer gaizarda albaToba imisa, rom 1000 adamianidan daavaddeba: 

a). araumetes 5 adamiani? b). araumetes 2? g). aravin ar daavaddeba? 

86. wundebuli produqciis gamoSvebis alabaTobaa p 0.04 . ramdenjer unda 

SevamciroT wundebuli produqciis procenti imisaTvis, rom 20-jer 

gaizardos albaToba imisa, rom 1000 erTeul produqciaSi ar aRmoCndes 

arcerTi wundebuli? 

270

pasuxebi: 

2. 216. 3. a). ara; b). ara. 4. ; . 7. ki. 8. ara. 10. a). A B; b). A 

B; g). 

A B; d). A B; e). A B; v). A B; z). A B; T). A; i). ( A 

B) 

; k). B ; l). 

( B A) 

; m). B . 12. 96. 13. 48. 14. 840. 15. 20. 16. 34. 17. 4200 (sqema 3-3-4: C 

3 C 

3 ). 

10 7 

18. 980 (ganTavsebis sqema: 2-2-4 an 2-3-3: C 2 2 2 3 

8 

C6 C8 C6 

). 19. 35. 20. 56. 21. 56. 

22. 512. 24. a). 4/19; b). 9/19; g). 14/19: d). 15/19. 25. a). 1/56; b). 15/56; g). 15/28; d). 

5/28. 26. a). 1/21; b). 5/42: g). 5/6 d). 10/21. 27. a). 1/2; b). 3/8; g). 1/8; d). 1/2. 28. a). 

1/2; b). 7/16; g). 11/16; d). 15/16. 29. a). 5/16; b). 11/16; g). 15/16; d). 1/16. 30. a). 5/16; 

b). 5/8; g). 5/8; d). 1/2. 31. a). 15/32; b). 1/2; g). 15/32; d). 15/32. 32. a). 1/2; b). 3/16; g). 

15/32; d). 5/16; e). 5/8. 34. 1/6. 35. yvela SemTxvevaSi 11/36. 36. metia pirvel 

SemTxvevaSi 1/36-iT. 37. orive SemTxvevaSi tolia 1/2-is. 38. a). 1/2; b). 5/9; g). 

1/3; d). 1/4; e). 7/36; v). 1/6. 39. orive SemTxvevaSi tolia 1/3-is. 40. metia 

orjer gagorebisas 1/12-iT. 41. a). 1/9; b). 5/9; g). 5/12; d). 11/36; e). 1/6. 42. a). 

1/36; b). 5/9; g). 25/216; d). 1/8; e). 43/216. 43. 91/216. 44. yvela albaToba tolia 

91/216-is. 45. 125/216. 46. a). orjer gagorebisas 1/12-iT; b). orjer 

gagorebisas 5/54-iT; 3). samjer gagorebisas 5/72-iT. 47. a). 1/2; b). 31/72. 50. a). 

2/3, 0; b). 1/2, 1/6; g). 5/6, 0; d). 1, 5/9; e). 5/6, 1/4; v). 3/4, 1/3. 51. a). 25/26, 1/18; b). 

31/36, 1/6; g). 11/12, 1/9; d). 2/3, 0; e). 4/9, 1/12; v). 7/18, 0. 52. a). 1/3, ara; b). 1/6, ki; 

g). 1/2, ara; d). 2/5, ara; e). 1/6, ki; v). 1, ara. 53. a). 1/4, 1/4, 1/4, 1/8; b). 1/4, 1/4, 

1/4, 1/8; g). 1/8, 1/8, 1/4, 1/16; d). 1/4, 1/8, 1/8, 1/16. 55. a). ki, ki, ara; b). ki, ara, 

ara; g). ara, ara, ara; d). ara, ki, ki. 57. a). 1/72; b). 1/36; g). 5/108; d). 5/72; e). 

7/72; v). 25/216; z). 1/8. 58. 0.986. 60. a). 0.158; b). 0.316; g). 0.526. 61. a). 0.073; b). 

0.439; g). 0.488. 62. a). 0.031; b). 0.557; g). 0.412. 63. a). 1/6; b). 1/12. 64. a). 0.06; b). 

0.94. 65. a). 0.35; b). 0.65. 66. 66. 0.116. 67. 0.984. 68. 0.518. 69. 0.016. 70. 1/4. 71. 3/8. 

72. 11/16. 73. 5/16. 74. 8/27. 75. 11/27. 76. 0.115. 77. 0.853. 78. 0.968. 79. 475. 81. a). 0.06; 

b). 0.11; g). 0.82; d). 0.87. 82. 0.11; 0.08; 0.92; 0.92; 0.5; 0.08; 0.025. 83. 0.73; 0.46; 0.30; 

0.14; 0.04. 84. a). 0.125; b). 0.656; g). 0.986; d). 0.735. 85. a). 1.4-jer; b). 3.4-jer; g). 

7.5-jer. 86. 4-jer. 

271

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272

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