Please answer the following questions:

Please answer the following questions:1. is this your first semester at BMCC?nnnnnnnnnnnnnnnyynynyes: 3no: 172. how many math classes have you taken at BMCCbefore this one?0 classes: 101 class: 92 classes: 03 classes: 14+ classes: 03. what was the last grade you got in a math class?A, B, C, D, F ... pass4. what is your major?hum.serv, biz, acc, hum.serv, crj, LA, pharm, crj,mmp, LA...5. I am comfortable taking a math class. (yes/no)yes: 16no: 46. what is the name of your favorite math teacher?phil, ?, patricia, bern, ...The following questions are optional. Answer yes/no:7. I like ice creamyes: 19no: 08. Bush was a good president.yes: 2no: 179. I am in a committed relationship.yes: 10no: 10

1.2 different types of studythere are two major types of studies1. observational studyex) just ask 'who is comfortable taking a math class?'2. experiment (or designed experiment)ex) one class i say every gets $1000 and an A, in another class i say everyone is stupidand will failthen i ask 'who is comfortable taking a math class?'observational study- observe and measure (watch, ask question, etc)ex) "l like ice cream" yes/noex) watch ice cream shop, see how many people go inexperiment - apply a "treatment" to your subject and measure resulttreatment: what you do to the subjectsubject: each person in the studyex) what flavor of ice cream do people like?observational study- sit in a store, watch what flavor people buy OR surveyexperiment - make a suggestion, see what they buyQ - does making a suggestion change the flavor people order?-> need to make a comparison to see the effect of the treatmentexperimental group:make suggestion (this group gets the treatment)control group:do nothingsubjects:customers in the storetreatment:make a suggestionobserved behavior: the result you want to look atwhich flavor they buy (this is also called the response variable)question/ research objective:does suggestion affect flavor they buyobservational studygood - watch subjects in natural environment(realistic)bad - never same conditionsexperimentbad - may be unwanted experimenterinfluencegood - can do exact same thing for eachsubject(control)

@ samplinghow do you get your subjects?@ (simple) random sample - every member of population has equal chance of beingin the sampleex) roll a dieex) have computer randomly generate phone #'snote: we will not discuss the difference between a random sample and a simplerandom sample@ convenience sampling - choose sample that is convenientex) reporter chooses people on a corner@ systematic sampling- choose a starting point select every k th person (could have alist of your population)ex) choose every 100th name in the phone book@ stratified sampling - choose different groups within your population, and select fromeach oneex) presidential panel includes women, minority, Dem, Repprecisely, choose one or more characteristics (wealth, race, gender, etc) and selectpeople with different values of that characteristicnote: which characteristics do you choose? how do you break that up into subgroups?@ cluster sampling - break your population into regions, choose a couple of thoseregions, then sample from each of those selected regionsex) exit polls after an electionreason - cant travel everywhereex) at a high school, when voting for where to go on theschool trip, a newspaper reporter asks 20 freshmen, 20sophmores, 20 juniors, 20 seniorsQ: what kind of sampling is this?stratifiedcharacterisic:values:

hw questions1.1 #42 US Census Bureau survey 50000 US households for characteristics like incomepopulation: US householdssample: 50000 households surveyed1.2 #20 get a simple random sample -> names from a hat

1.4 sources of errorself-selecting sample (or) nonresponseex) announcer on radio station asks callers to voice their opinion,it will appeal to people who care stronglyex) web pollex) customer service surveywho is going to respond? people who had bad customer service, so survey resultswill always be badwhat is a good sample?a sample which is representative of the target populationis this a good survey question: (to find out people's opinion of his leadership)?Q: John McCain's honored military experience makes him a good leader. (yes/no)yesno* loaded questionleads you to answer one wayex) people usually associate military service with leadershipfix it:is John McCain a good leader? ...now its neutralex) abortion (pro-choice V pro-life) ... find out where someone stands on the issueask a question that is: neutral ... loaded towards pro-choice ... loaded towards pro-lifeneutral:loaded towards pro-choice:loaded towards pro-life:Q: (neutral) do you think abortion should be legal?Q: (neutral) would you say you are pro-choice or pro-life?Q: (biased towards pro-choice) should the government be allowed to take away awoman's constitutional right to choose whether she can have a child?Q: (favoring prochoice) Should the woman have the right to choose to end herpregnancy taken away by politicians?Q: (biased towards pro-life) Should a woman be allowed to kill her baby before itsborn?

* self-interest studystudy conducted by someone who has an interest inThe results turning out a particular wayex) be skeptical about a survey about GM cars conducted by GM(every car is #1 ?)* small sampleex) a Newsweek reporter goes to the corner and asks 12 peoplewhat they think about what happened with Tiger Woods?Q: would you go buy products he endorsed, given what he did?* interviewer error- when the interviewer is inappropriate in some wayex) cops ask questions about opinion of police dept@ word order / question orderex) Q: what do you think causes more pollution in the US, traffic, or industry?even more biased: Do you think traffic causes more pollution in the US,or industry?solution: half the time "traffic or industry", half the time "industry or traffic"@ correlation V. causationex) survey results show that people who drive BMW's live longer than driving other cars- if you want to live longer, should you buy a BMW?no, BMW owners are - on average - more wealthy, so they can afford better healthinsurance and take better care of their healththe only way to determine causation is with an experiment, where you have control, andyou can apply a treatment and measure the effect (response variable)ex) take 100 people, give 50 BMWs and 50 Toyotas, and see what their lifespans are

1.5 designing an experimentex) find out the effect of which car you drive on your lifespan:get a sample of 100 people, give 50 BMWs and 50 Toyotas, and see whattheir lifespans areobjective: find out the effect of which car you drive on your lifespantreatment: BMWs, Toyotasfactor: what type of car subjects drive (explanatory variable) [input]response variable: lifespan [output]sample: 100 people (driving cars)the steps:1. identify the issue to be studied2. determine the factors that (you think) affect the response variableex) one factor- which car they drive3. deterine the level of each factor- levels: different values each factor takes onex) levels are BMW, Toyota (this is similar to specifying the treatments)4. determine the number of subjects5. conduct the experiment6. analyze the data and reach a conclusion (see if your claim is true or not)after you have selected your sample, you must assign subjects (to different treaments)and conduct the experimentex) who gets BMWs, who gets Toyotas?- completely randomex) 50 names from a hat get BMWs- randomized block designex) take the rich men, give half BMWs and half Toyotastake the rich women, give half BMWs and half Toyotastake the poor women, give half BMWs and half Toyotas, etcdef: divide your sample into blocks of subjects who are similar according to certaincharacteristics, then from each block randomly assign subjects to each treatment- matched pairs designex) pair up similar people (e.g. two 37-year old rich women), and give one a BMWand one a Toyotaall ways of making sure that both treatment groups are the same, except for the factorwhose effect you want to measurenote, sampling: starting with a population and selecting subjectsassigning subjects: once you have your sample, choosing which treatment eachgets

ex) suppose you give subjects a combination of st johns wort and gingko tosee the effect on how much they sleepsjw: you either give them 0mg, 10mg, 20mggingko: you give them either 0mg, 25mg, 40mgwhat could we do/give to a subject?25mg gingko, 10mg sjw ...this is one treatmentfactors: gingko, sjwtreatments: 0gi,20sjw 40gi,10sjwgingko: 3 different levelssjw: 3 different levelstreatments: 9 possibilitiesnote: medicine is an example of a field where you would use two factors, notjust one- why?because they might interactex) gingko does nothing by itself, sjw does nothing, but together they solve theproblem

suggested experiments1 objective: effect of coffee and redbull on energy levelfactors: coffee, redbullsample: 160 peopletreatments: 16 of themresponse variable: amount of energy(number of jumping jacks they can do)(ask them how energetic they feel)2objective: which shampoo is most effective on dandrufffactor: shampoosample: 40 people (stratified sampling - 20 men, 20 women)treatments: Pantene, Head and Shouldersresponse variable: amount of dandruff OR opinion about how effective shampoo wasis your objective "effectiveness" or "are they happy and will they buy it again"?3objective: how does breakfast affect students gradestreatments: breakfast, no breakfastfactor: mealsample: 40 students in the same classresponse variable: grade in that classassigning subjects: randomized block design: 20 students who are good at math (10skip breakfast, 10 eat it) 20 students who are not good at math (10 skip breakfast, 10eat it)if you want to generalize, you have to have students from different schools and differentsubjects4 objective: how much caf/decaf coffee affects how long you stay upfactors: caf, decaflevels: 1,2,3,4 cups of coffeerandomized block design based on body type (big, small) and caffeine tolerance (priordrinking: 1,2,3,4 cups a day)response variable: time falling asleep5 objective: what gets you more drunkfactors: beer pong, flip cuplevels: 1, 5, 10 gamesrandomized block design: 100-149 pounds, 150+ poundsresponse variable: result of breathalyzerpopulation: BMCC students, 21+sample: stratified sample using weight (12: 100-149, 12: 150+)

objective: which alcohol gets you more drunkfactors: rum, vodkalevels: 0,1,2,3,4 shotssample: 100 mentreatments: 25 diff (1 rum, 2 vodka), etcnote: should control for body weightobjective: which condom prevents pregnancy betterfactor: condomtreatments: nyc condom, trojan condomsample: 50 couples(they use their condoms for a year, see who gets preg)completely random designnote: should control for how often couples have sex

suggested experimentsex)objective: find out the best laptop for college studentssample: 1000 bmcc studentsgive each student a laptop for 60 days. at the end, fill out a survey to rate theperformance of their laptoptreatments: HP, toshiba, panasonic, dellfactor: which laptop they useresponse variable: the rating on the surveycomments: maybe a full semester is a better trial periodwhat exactly are you comparing? ...maybe we will compare the operating systemex) objective: does the time people eat dinner affect their weighthalf the subjects get dinner 4 hours before sleep, half right before sleep. same food,continue routine for 1 monthsubjects: 10 people all the same age, 2 groups of 5factor: time dinner is eatentreatments: bedtime, 4 hours before bedtimeresponse variable: change in weightcomments: same bedtime? - ned to consider sleeping habitsare there other activities affecting weight gain? different metabolism?ex)objective: which shampoo causes an allergic reaction for dogstreatments: scented, unscentedsample: 200 dogsfactor: which shampooresponse variable: allergic reaction (yes/no)assign subjects: randomlyex) have women take a kind of birth control for a year, then check their weight gainobjective: which birth control makes women gain weighttreatments: Yaz, Orthosample: 100 women age 18-35factor: which birth controlresponse variable: weight gainassign subjects: randomlycomment: other things affecting weight gain: diseases, different levels of exercise,stress, diet, etc - particularly for medical studies, make sure each treatment group iscomparableex) 250 students, age 18+, half get tylenol, half get placebo, 2.5 hours later is theirheadache goneobjective: does tylenol get rid of headaches?sample: 250 students, simple random sampletreatments: tylenol, placebofactor: the medicationresponse variable: presence of headache (yes/no)comments: how do you get a sample of people with a headache?maybe the headache will go away anywaythe severity of the headache might affect the result

hw questions ch11.3#29 survey about commuter rail, how do you choose sample? why?stratified sample...and stratify by different neighborhood, since rail will affect peopleif different neighborhoods differentlycluster v stratified samplingex) survey condition of wildlife preservescan you go to everyone?no...to make it easier to get around, go to a cluster in Washington, in New York, inWyoming, etc (once youre in Wyoming, drive around to all or most)i.e. out of 50 states, pick 4 randomly, then visit all sites in each stateex of randomized block design)experiment to find out the effectiveness of Title 5 advising program for bmcc LAstudentsget a sample of 40 bmcc LA studentsquestion:- is the Title 5 advising program effective?experimental group: goes into the programcontrol group: does not go into programtreatment: go into the advising programresponse variable:results of student surveyrate of graduationsampling method: randomrandom block design:- split them into male and female- half the males go into the program, half dont- half the females go into the program, half dont

ch2 - organizing and presenting dataif you cant share your work with other people so they understand it, then what was thepoint2.1 presenting qualitative data (words)- frequency distribution (table)...how often does each data value appearex) is this your first semester at BMCC?yes 3no 17frequency: total number of occurrences of a data valuerelative frequency:acc offop humserv LA humserv acc vat LA bizLA humserv child.ed, acc ...note: relative freqency adds up to 1sometimes, because of rounding, that might nothappen on the chart

Bar Graphwhat is your major?-Pie Chartex) # of people with a particular majorlabel each slice with:category; freq (or relative freq or percentage)

2.2 presenting quantitative datafrequency distribution (table)- looks the same when you have each data value gets its own frequencywhen might it look different?ex) what is your age? i take a sample from all BMCC students19,22,56,20,21,27,32,22,47,25,24,42break data up into data classes (from low to high)(what class you choose is up to you)here, break it up into 10s, 20s, 30s 40s 50s ...this is typical for dealing with ageseach age range or "data group" is called a data classthe class width is the distance between classesex) here its 10 20-10=10 or 30-20=10note: the class width must stay the same (in general)the class limits (or class boundaries) are the highest and lowest values in theclassex) here the class limits on the first class are 10 and 19(note: there are other technical definitions for this, but we will use this simplehow do you make a frequency distribution table with continuous data?...adjust class limits according to your precision (number of decimal places)ex) if you record people's ages with 2 decimal places (20.47years, 21.65, etc)then write your class limits with 2 decimal places: 10-19.99, 20-29.99, 30-30.99 etc

- histogram..bar graph for quantitative datanote: there is no gap between the barswhat if each class containsseveral values?note: for all these presentations,anywhere you put frequency,you can instead put relative frequency(you get same shape, 1 replaced by 1 /12 )what if your data has a big gap?ex) 19,22,56,20,21,27,32,22,47,25,24,42, 105,107,115,117,

19,22,56,20,21,27,32,22,47,25,24,42great for small data sets: it is visual(like a histogram)and it preserves all your dataex) 105, 107, 112, 114

* frequency polygonhistogram, but instead of bars, connect with a line10 20 30 40 50 60* ogive (oh-jive)-uses cumulative frequencythere are lots of creative ways to represent data visually. for one guy, thats his full-time job:http://www.informationisbeautiful.net/want to see a better visual representation we can all use? how about this NYC subway map:http://www.kickmap.com/comparison/

2.4 misleading representations- a graph which is correct, but gives the wrong impression> 3-dimensional pictures exaggerate the appearance of changeex) money pile* misleading graphex) salary graph"look at the raises we give!"short y-axis range - dont start from zeroexaggerates changewe can make:bar graph using Excelpie chart using Excelhistogram using StatDisk

hw questions ch22.2 list original data(from a stem-and-leaf)4 | 0 4 7 --> 40, 44, 47etc

ch 3: numerically summarizing data3.1 measure of central tendencyor, give me one number that represents all the datacomment:2 uses of the word "control"control groupcontrolled factorconsider the number of math classes taken by math 150 students. how canwe sum it up in one number?average: add up all the numbers and divide by the amount of numbers thatthere areex) suppose you score on three tests 71,75,84. what is your test average?also called the meanex) for number of math classes, mean =median: the middle numberex) suppose you score on three tests 71,75,84. what is your median testscore?median is 75interpretation: half the time the score is above 75, half the time the score isbelow 75note: you must put data in ascending order to determine the median0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2...median isex) heights of students (in inches):59,61,62,64,64,64,65,66,66,66,67,68,68,69,70,70,71,71,73what is the median height?...find middle number: there are 19 numbers (19+1)/2=10 ...so its thenumber in the 10th position ...the median is 66or: 59,61,62,64,64,64,65,66,66,66,67,68,68,69,70,70,71,71,73what do you do if there are two middle numbers? add together, divide bytwo (i.e. take the average)..this will happen when there is an even amount of datanote that, using the "+1" method, you would get (20+1)/2 = 10.5...this means the median is between the 10 th and 11 th numbers, so take theiraverage

mode: most common numberex) heights: two modes 64 and 66ex) number of math classes: 1ex) test scores: no mode (all the same frequency)Question: which of these should we use, and why?ex) number of credits taken at BMCC in math150 class:0,0,9,12,21,22,27,32,35,38,44,50,52,56mean =median =mode = 0ex) there can be a problem with the meanthe average salary in this class is around $15,000if Bill Gates (and his $1,000,000,000 salary) walk into the room,the average salary is now around $35,000,000. does this make us allmillionaires? ...nothe median salary is still around $15,000, because at most you go to thenext number on the list"the Bill Gates effect"Bill Gates' salary is an outlier: it is a value far away from most of the datathe average is not robust with respect to an outlierthe median is robust with respect to an outlierrobust: not affected by [also known as resistant]

3.2 Measures of Dispersionhow spread out is the databecause mean & median do not tell the whole storyex) group of 5 men, heightsgroup 1: 5'8,5'10, 5'11, 6', 5'9 ... in inches: 68,69,70,71,72group 2: 4'6,7'4,4'2,6'8,6'6 ... in inches: 50,54,78,80,88find mean:group 1: 68+69+70+71+72 = 350 = 70" (or 5'10)5 5group 2: 54+88+50+80+78 = 350 = 70" (or 5'10)5 5- range(highest) - (lowest)ex) group #1: 72" - 68"=4"group #2: 88" - 50" = 38"note: affected by an outlierex) our salary range is 30000-0 = 30000with Bill Gates, range is 1000000000 - 0 = 1000000000

standard deviationex) group 1 (inches) 68,69,70,71,72" mean = 70standard deviation =you do:ex)group #2: 54, 88, 50, 80, 78 ... mean = 70find the standard deviationex) var = 4 ... st.dev. =ex) st.dev. = 9 ... var =

sample populationmean x µ "mu"st.dev. s σ "sigma"variance s 2 σ 2size n Ndepends on fixedyour samplea "statistic" a "parameter"also: "data value" = xthe way that you calculate the sample mean and thepopulation mean are exactly the same.the difference is the kind of information it gives youex) find the standard deviation of the sample 7,10,16 (and the variance)note for standard dev:for a population, divide by thenumber of datafor a sample, divide by thenumber - 1

3.3 calculating that stuff from a table(measures of central tendency and dispersion)or, what to do if we have only the table of data and not the raw dataex)whats the mean??note: the table is anapproximation, so the resultwill be an approximationFormula for aweightedmean:note: divide by 12, not 5, because12 is the total frequency (e.g. 25appears 7 times)this is similar to a weighted meanex) get three scores, 80, 95, 70but the first score is your hw grade (that counts 20%) the second score is your midtermgrade (that counts 30%) the third score is your final exam grade (that counts 50%)mean = Σ x · rel.freq(x)x or µ

whats the standard deviation? [extra credit material]s =

hw questions3.2 #14 population: 1,19,25,15,12,16,28,13,6 ... find st.dev, variance

3.4, 3.5 measures of position- rankex) New York marathon, 12,635 people run, you finished 586your rank is 586 (out of 12635)- quartilemarked off by: quarter point, half-way point, three-quarter pointhalfway: 6318quarter-mark: 3159- 5-number summarymin--Q1--Q2--Q3--max"Q1": data value that separates first quartile and second quartile5-number summary: Q 2 is in the n+1 / 2 position (then find the data value)Q 1 is in the n+1 / 4 position (then find the data value)Q 3 is in the (n+1)( 3 / 4 ) position (then find the data value)ex) 14,15,16,17,18,19,20,21,22 (n=9)using the formula:Q 1 appears in which position? Q 1 =Q 2 appears in which position? Q 2 =Q 3 appears In which position? Q 3 =why do we need the "+1" ? well, if we didnt have it then for Q 2 we wouldcalculate(9)(1/2) = 4.5but we know thats not right, its too low...the "+1" fixes that problemBoxplot- a visual representation of the 5 number summary

- percentileyou are above ? % of the datapercentile --> valueex) 3,7,9,12,15,15,16,18,19,21,24,26,28,29find the 37th percentile:(n=14)you do: find the 58th percentileex) find the 78th percentilevalue --> percentileex) at what percentile is x=24? [recall: "x" means data value]x=24 is above 10 data values (out of 14)percentile: 10/14 = .71 or 71st percentile (above 71% of the data)notation: the 71st percentile is 24P 71 = 24note that, for both problems,the middle step is to find therank (position)note: the "+1" formula has some glitches for small data sets. this comesfrom the fact that one data value represents a large chunk of your dataset (e.g. if you have 20 numbers, each one represents 5%)...just follow the formulaex) make a 5-number summary for the example [3,7,9,12,15,15,16,18,19,21,24,26,28,29]we have modified our formula: since Q1 is at the 25th percentile, the position is (n+1)(.25)

oxplot- picture of the 5 numbersummarythis distribution shape is called "symmetric"here are some other shapes:

- z-score"the number of standard deviations from the mean"ex) the mean score is 77, you got an 85. is that good? how good?it depends.suppose the standard deviation is 4. how many standard dev's above the mean are you?you are 8 points above...that is 2 standard deviations (since st.dev. is 4)Jerry got a 88. how many standard deviations above the mean is his score?what is each numbercalled?Formulafor a z-score: z = x - µ(population) σfor a sample, same formula:different notationz = x - xsex) find the z-score for 47 if µ=38, σ=5what does that mean, in words? ...ex) find the z-score for 68 if µ=78, σ=4note that a positive z-score means your data value is above the meanand a negative z-score means your data value is below the meanex) which exam score is relatively better, a 75 when the class average was68 and the standard deviation was 4, or a 89 when the class average was 76 andthe standard deviation was 12 ? (use the z-score)ex) find the data value which is 2 standard deviations above the mean if µ=32, σ=6Exam #1ch1,2,3formula for x: x = µ + z·σsame as the formula for z, but you solve for x

hw 3.2 questions3.2 online #1) find the sample variance and standard deviation: 19, 10, 2, 9 ,113.2#1 online) find the sample variance and standard deviation: 22 14 5 9 8

3.2 #12ex) 83,65,91,87,84find variance, standard deviation (sample)

a man is 68" tall, for men the mean is 69.6 with a st.dev. of 2.7". a woman is 62"tall, for women the mean is 64.1" and a stdev of 2.6"who is taller relative to their gender?3.4 #4 online: concentration in 10 soil samples. find the z-score for "10.79"...find mean and st.dev. from data, etc

the average 20-29 yearold man is 69.6" with a standard deviation of 2.7"the average 20-29 yearold woman is 64.1" with a standard deviation of 2.6"who is relatively taller, a 68" man or a 62" woman ?3.4#832-35-week gestation: mean is 2600g, st.dev. is 670g40-week gestation: mean is 3500g, st.dev. is 475g34-week baby is 3000g, 40-week baby weighs 3900gwhich baby weighs relatively less?3.3#12R1 #4f ... 20 38 38 45 48 48 49 51 53 54 59 61 62 64 65 68 71 765-number: min-Q1-Q2-Q3-max

hw questions3.4online #4

ex) z-score18, 16, 5, 9, 8

45 47 48 56 57 59 60 62 63 64 64 65 68 6869

a bit on probability and random variables (6.1)- might also fill in some other info, essentials of what we needa bunch of data (e.g. people's heights)some variable takes on different values...those different values are our data"x" represents our data values"x" is called a random variable(we'll be talking about quantitative data)discrete, or continuous:we have a discrete random variablewe have a continuous random variablebefore: "relative frequency"now: "probability"before: "frequency distribution"now: "probability distribution"probability:whats a probability value?- each value is between 0 and 1(if probability=0, it never happens) e.g. 0 = 0/6 "impossible event"(if probability=1, it always happens) e.g. 1 = 7/7 "certain event"- total probability = 1 (or 100%)notation:ex) the probability that a student's age is over 100= P(x > 100)ex) the probability that student's age is between 0 and 100= P(0 < x < 100)ex) the probability a student's age is 23= P(x = 23)how do you know if you have a discrete probability distribution?

what is the mean in our example?note: total probability is 1, so dontneed to divide by thatthe mean of a random variable is called itsexpected valueex) suppose there is a game where you roll one die. if you roll a 6, youwin $5, otherwise, you lose $1.what is the expected value of this game?ex) lets flip cards! when you pull a king from thedeck, you win $10. for everything else, you lose$1. what is the expected value of this game?deck of cards: 52 cards4 each of A,2,3,4...10,J,Q,K (rank)13 each of spade,heart,diamond,club (suit)ex) P(x="8 of spades") =1/52ex) P(x="heart") = 13/52 or 1/4this also means...if you play through the whole deck, youwin $40 but lose $48 ... so you lose $8 on every tripthrough the deck (52 cards)

ch7: normal distributionwhat does a probability distribution look like for a continuous variable??how do we think about this?....convert a frequency polygon to being "continuous"how can we figure out probabilities (relative frequencies) from this?assume every temperature from 70 to 75 is equally likelywhat is the probability thatthe temperature is less than 72?notation: P(x < 72)that shades in 2/5 (or 40%) of the rectangleso, the probability is 2/5 (or .4 or 40%)note: total area = 1 ... because the total area represents thetotal probabilitythis is called the uniform distributionwhat is the probability that the temp isbetween 71 and 74?P(71

the normal probability distribution(also known as normal distribution, normal density, ..."bell curve"on the web page, click 'activities' ... 'Galton Machine'properties:- area = 1 (because this is the total probability)- each probability value is from 0 to 1- symmetric- highest frequency is in the middle......mode = median = meanwhat is the z-score for a male height of 72"?z=ex) if you have a man who is 70" and a woman who is 67", who is taller?...in inches? ...the man...relative to their gender?normal distribution, using only z-scores: the *standard* normaldistribution

within 1 standard deviation of the meanwhat percent of men are between 68" and 72" ?.....between 0 and 2 (z-values)thats 47%, or .47OR: the probability that a man is between 68" and 72" is .47notation:P(68 < x < 72)or P(0 < z < 2)

ex) find P(z 1.37) ?the TOTAL has to be 1so the area is 1 - .9147= .0853you do: [draw the picture and find the answer, to four decimal places]ex) P(z < -1.77) ex) P(z > .58) ex) P(z < 2.33) ex) P(z > 2.33) ex) P(z < -2.33)note: what determines which side youshade?note that its thesame as theprevious answer,because ofsymmetry

how do you find "between" area??ex) find P(1.26 < z < 2.18)the big area (purple) minus the little area (orange)gives you the area you want (blue)you do:ex) find P(-2.67

finding a z-score (or data value)given the percentile (probability)ex) what z-score is above 90% of your data?given probability,find z-scorein Excel:=NORMSINV(ex) find the z-score at the 85th percentileex) what female height is at the 90th percentile? [recall µ=64 and σ=1.5]note: we need to find the z score for the 90thpercentile, but we found it in the previousproblem, how nice for us

when you use Excel, the software will prompt you forwhich values you need to enter, and in which orderex) type =normdist( ..and it says "x,mean,st.dev,cum"...always use "true" for last value

ex) find P(z < 1.75)have:want:ex) find P(z>1.75)if µ=38 and σ=5, find the z-score of 31ex) find the probability that a man is shorter than 73.2"recall that µ=68 and σ=231 is 1.4 standard deviations belowthe mean of 38ex) find the z-scores which determine the middle 76% of your datanote: when the text asks if something is"unusual"to them, that means a probability less than .05we have covered 7.1, 7.2, 7.3

hw questions

hw questions 6.1#8

hw questions ch7

hw questions 7.2, 7.37.3 #7

"area to the left is .05"or "5th percentile"note: take z to be positive

hw questions ch7

hw questions7.3 online)

hw questions ch77.2 #4

8.1 What about when you take a sample?when you take a sample size 2 or 3 or whatever, you expect a lower standard deviation(the mean is still the same)we call that the standard deviation of x (also called the standard error)notation: σthere is also the mean of xnotation: µCentral Limit Theorem:ex) if σ=8 and you take a sample of size n=16, what standard deviation do you get? (i.e.what is σ_ ?)ex) If σ=1.9 and sample size n=20, find your standard deviation σ_note: dont round too much, that error will get bigger in each stepthis is the mean, for surethis is the standard deviation, for surebut...is the distribution of x *normal* (is it a bell curve) ?we want this because then we can figure out all the probabilities.......lots of experimenting.....population is normal -> distribution for x is normalpopulation is not normal -> take sample of size 30 (to be safe) thendistribution for x is normal

for adult salmon length, µ=42" and σ=6" (normal distribution), find:a) the probability that one salmon is longer than 46"...b) the probability that four salmon average longer than 46"formula for z-score: value - meanst.devyou do:ex) µ=78, σ=6, sample size n=10population has normal distributionfind P(x

8.2 distribution of sample related to proportionsex) 62% of mothers want increased athletic programs in the schoolswe write: .62p represents proportion of the populationrepresents the proportion of the sampleex) from a census, 51% of US residents are womennotation: p = .51ex) from a survey of 800 teenagers, 673 like Justin Timberlake's musicnotation:note: you wouldnt say "what is the average number ofpeople who like his music"what proportion of teenagers do NOT like his music?notation and formulas ... for proportions:what isnote that this is different than with the sample mean x

17% of Americans have high cholesterol. suppose we will take a survey of 80people. what is the probability that less than 12% of your sample has high cholesterol?what is µ^ ?what is σ^ ?is the distribution of p normal?so, what is the probability that less than 12% of your sample has high cholesterol?there are lots of decimal numbers herethere are values the proportionthere are values for probabilityin this problem,proportion is .12probability is .1170be careful as you are doing problems not to confuse them

ex) p = .47 n = 60what is P(p < .43) ?P(p

ex) if p=.28, find P( > .32) with a sample size 180ex) 62% of mothers want increased athletic fundingwhat is the probability that more than 40 out of 60 mothers will vote for funding?

ex) find P(x>32.7) ... µ=29.6 σ=8.2 n=34 ex) find P(p< .6) ... p=.55 n=80ex) a 2003 study found that medicalresidents work an average of 81.7 hoursper week. suppose the number of hoursworked is normally distributed with astandard deviation of 6.9what is the probability that the meannumber of hours worked by a team of 8residents is less than 75 hours a week?

hw questions ch8

hw quetions 8.1, 8.28.2 22a

hw questions ch88.1probability that 10 time intervals have a mean of longer than 111 minutesmean time is 96min, normally distributed, st.dev is 30minthe probability is .0569

hypothesis testing OR testing a claim (ch10)we will do the exact same calculations, but we will use it to answer a differentquestionfrom before:ex) p = .17 (high cholesterol)in the sample, with 80 people: p < .12 ... what is the probability of this happeningso far, we said- we know the population mean- we know the standard deviationOR- we know the population proportion...but then why would we take a sample???we will now flip our perspectivewe will take a "guess" about the population mean (or proportion) andconsider it, given the results of the samplee.g. what if our sample had .02 high cholesterol....does it seemreasonable that the whole population has .17 high cholesterol?ex) recall that: p=.17 ... n=80 ... ^=.17 ... σ^ = .0420 ... normal distributionfind P(p < .02)hey! was p = .17 correct??

hypothesis testing (formally)1. a claim is made about the population2. get data from a sample3. do calculations and assess the plausibility of the claimhypothesis = claimwhat is a claim?(note: here, we will make claims about µ or p... could make lots of different claims...we will, but later)CLAIM:p = valueex) p = .17 "i think that 17% of Americans have high cholesterol"µ = valueex) µ =40,000 "the average income in US is $40,000"µ > valueex) µ>76 "the average basketball player is over 76"consider the claim µ = 40000if the claim is wrong, then µ ≠ 40000set up a hypothesis test....3 waysnote: for technical reasons, H o is always "="what about "≥" ... it gets confusing, so skip itnote that we identify the type of hypothesistest (and the picture) using H 1

Dick Vitale claims that the average college basketball player is over 76"what do you conclude about this claim, from your sample of 40 players with x = 77" (σ =in a hypothesis test,your probability is calleda "level of confidence"if µ=76, then the probability that x is 77" or more is .0008 ... thats unlikelyso probably, µ > 76how probably?.9992 probablyconclusion:we are .9992 confident that µ > 76(99.92% confident)

ex) pollution level in a riveran environmental group takes 30 samples from a river, and they get anaverage pollution level of 4.5 cc/L of a certain pollutant (σ = 1.2)they want to claim that the average pollution level of the river exceedsthe EPA regulation of 4 cc/Lwhat can you conclude about the claim, and at what confidence?heres the idea:

ex) we want to to find out if the average American family has more than 1.8kids. (because if it does, then there is a strain on the school system)we take a sample of 500 families, and we get x = 1.92 σ = .9what can we conclude?conclusion:confidence:we are .9986 (99.86%) confident that the average number of kids in an Americanfamily is greater than 1.8p-value:we conclude that the average number of kids is above 1.8 with p-value = .0014accept/rejectwe reject Ho, accept H 1 with .9986 confidencenote: strongest results are "100% confidence" or "pvalue=0"note: the industry standard for results:you want confidence of .95 (95%) or more ... p-value of .05, or lessQ: in this example, what can you conclude at 95% confidence? (or 5% significance)A: at 95% confidence, reject Ho & accept H 1note that .9986 confidence beats .95 confidence.0014 p-value beats .05 significanceterminology:significanceconfidencep-value = 1 - (confidence)reject/accept[its the shaded area]you reject H o and accept H 1 if:(your confidence) > (requested confidence)(your p-value) < (requested significance)

ex) a drug company is trying to make 200mg pillsthey want to make sure there are exactly 200mg in each pillthey take a sample of 500 pills, and find that x = 199.88 σ = .9is the average pill different from 200mg? what can they conclude?conclusion:the average pill is different from 200mg with .9972 condfidence[pvalue = .0028]reject Ho, accept H1 with 99.72% confidenceQ: what can you conclude at .999 confidence (or .001 significance)?A: at .999 confidence, do not reject Ho, do not accept H 1note: we never "accept Ho"if we think the mean is 200 (µ=200), well it could also be 200.0001the best we can do is support the claim that it is not equal or lessthan or greater than (≠, )

ex) is the average fashion model under 102lbs?we asked 50 models, and their average weight was 97lbs. (σ = 11lbs)what can we conclude?conclusion:we conclude that the average model weighs less than 102lbs with 99.93%confidence[p-value = .0007]OR:reject Ho (and accept H1) with 99.93% confidence

hypothesis test about µ, what if we dont know the population st.dev σ ?we have to use the sample standard deviation s instead...but theres a price to paythe distribution is no longer normal..so what is the distribution? and how do we find probabilities?the distribution of x in this case is a Student t distribution..."t distribution"its similar to the normal distribution- symmetric- middle value is 0- graph kind of looks like a bell curvewrinkle: the distribution changes depending on nwe need "degrees of freedom" = df = n-1note: the t-distribution is similar to the normal distribution in shape,but it is not the samenote: Excel does not like negative t-valuesenter the positive value, the answer is the same dueto symmetry

ex) how many M&Ms in a bag?supposed to be 1000, but I claim its more than thatsample 80 bags, get x = 1036 s = 97.3what can you conclude?conclusion:we are 99.93% confident that the average number ofM&Ms in a bag is more than 1000 [p-value = .0007]note:when we use s instead of σ,s replaces it in our formulasnote: for t-distribution, Excel requires t-value, cant use xExcel does NOT take negative t-scoresnote: the t (or z) value which you calculate from your datais known as the test statistic

σhave σ dont have σ, have snote: Excel and the t-distributionExcel calculates the area in the tail (or tails) given a positive t-valuesometimes, you will be asked "what can you conclude at 98% confidence?" OR"what can you conclude at 2% significance?"ex) we are 99.93% confident that the average number of M&Ms in a bag is morethan 1000 [p-value = .0007]are you 98% confident? ....yesat 2% significance (or at 98% confidence), reject Ho, accept H 1 , the averagenumber of M&Ms is more than 1000ex) we are 89.94% confident that a majority of Americans support the deathpenalty[p-value = .1006 or 10.06%]are we 98% confident? ....no, we cannot conclude at 98% confidence (or 2%significance) that a majority supports the death penalty - do not reject Ho, do notaccept H 1

ex) does the average person own more than 3 hats?you survey 15 people, who respond: 2 5 0 8 3 3 4 3 9 1 2 2 4 12 5what can you conclude, and at what confidence?...what do you need?you can find the mean and standard deviation using Excel:type the data values into Excel, then...=average(=stdev(...then highlight your data valuesconclusion: we are 91.41% confident that µ>3, the average person owns morethan 3 hats. [p-value = .0859]followup:at 10% significance, do you reject H 0 ?at 5% significance, do you reject H 0 ?

ex)H 0 : µ=4H 1 : µ

10.4 hypothesis testing...with proportionsex) H 0 : p = .62H 1 : p < .62p = .54 ... n = 300whats your conclusion about the claim, and at what confidence?

ex) does the public believe the death penalty should be used?we want to find out if a majority of Americans support the death penaltywe survey 120 people, and 67 say yes.what can we conclude, and at what confidence?follow-up question: what can you conclude at 5% significance?conclusion:we are 89.97% confident that a majority ofAmericans support the death penaltyp > .5 with.8997 confidence[p-value = .1003, or 10.03%]reject H o , accept H 1 with .8997 confidence

ex) do more than 60% of people support legalizing gay marriage?out of 180 people surveyed, 119 said yes.what can you conclude, and at what confidence?conclusion: we are 95.25% confident that p > .6(more than 60% support legalizing gay marriage)reject Ho, accept H 1 with .9525 confidencep-value = .0475

hw questions - ch 10

hw questions10.3 #2

hw questions - ch10

hw questions 10.2#15Ho: = 20H1: < 20n=18population is normal= 3x = 18.3we are .9919 confident that < 20reject Ho accept H1 with .9919 confidencep-value = .0081also, with .05 significance, we can conclude < 20 (reject H 0 , accept H 1 )(.05 significance = .95 confidence)

10.6 summary of ch10we have to identify, given a problem, what approach to useex) the Health and Safety Board recommends at most 4 hours of tv each day.you think people watch more. you survey 130 people, who watch an average of4.5 hours a day, with a standard deviation of 1.7what can you conclude, and at what confidence?ex) you need to know if proposition 285 will pass with 2 / 3 of the vote.you survey 200 people and 141 say yeswhat can you conclude, and at what confidence?ex) gasoline regulations state that gas sold at the pump should contain exactly.18 liters of ethanol in each liter of gas. you survey 65 gas stations and find anaverage of .161 liters with a standard deviation of .06what can you conclude and at what confidence?ex) what are the average number of large-screen tv's sold at a Best Buy storeyearly?management insists they sell more than 100 tvs. 45 stores are surveyed, andthey sell an average of 107.3 (in the industry, σ = 25.2)what can you conclude at 95% confidence?ex) the US RDA for calcium is 1000mg. the Dairy Food Association isconcerned that teens are not getting enough. they conduct a survey of 500teens, who consume an average of 989mg of calcium. from the sample,standard deviation s = 110what can you conclude about the DFA claim, and at what confidence?ex) is the average fashion model under 102lbs?we asked 50 models, and their average weight was 97lbs. (σ = 11 lbs)what can we conclude?

ex) the Health and Safety Board recommends at most 4 hours of tv eachday. you think people watch more. you survey 130 people, who watch anaverage of 4.5 hours a day, with a standard deviation of 1.7what can you conclude, and at what confidence?we are .9995 confident that µ > 4reject H o , accept H 1 with .9995 confidence

ex) you need to know if proposition 285 will pass with 2/3 of the vote.you survey 200 people and 141 say yeswhat can you conclude, and at what confidence?we are 87.08% confident that proposition 285 will pass with 2/3 of the votereject Ho, accept H1 with .8708 confidencepvalue = .1292

ex) gasoline regulations state that gas sold at the pump should contain exactly .18liters of ethanol in each liter of gas. you survey 65 gas stations and find an average of.161 liters with a standard deviation of .06what can you conclude and at what confidence?ex) what are the average number of large-screen tv's sold at a Best Buy store yearly?management insists they sell more than 100 tvs. 45 stores are surveyed, and they sellan average of 107.3 (in the industry, σ = 25.2)what can you conclude at 95% confidence?on the exam you will be given this:Excel commands:NormdistNormsdistNorminvNormsinvTdistTinvAverageStdevsqrt

hw questions ch1010.3 #2 online

hw questions - ch1010.6 online-5)we are 99.22% confident that µ < 6.4 (p-value = .0078)in particular, at .05 significance we accept the claim that µ < 6.4 because pvalue < .05

10.3-online #2)Ho: µ = 1.67H 1 : µ = 1.67x = 1.669s = .0051n = 12 ... df=11sex) what is the average age of alllawyers?is it 39, or is it higher?you survey 110 lawyers, who have anaverage age of 40.2, with astandard deviation of 4.1what can you conclude, and at whatconfidence?= tdist(.6667,11,2)= .5187pvalue = .5187confidence = .4813we are .4813 confident that = 1.67we are .9986 confident that > 39pvalue = .0014reject Ho, accept H1 with .9986 confidence, with.0014 significance

ch9now, ask a "two-tail" type question:find P(80 < x < 100) µ=90, σ=20, n=16 population has normal distribution)

why might we wind up with two equal-size tails?- perhaps you are concerned with being, not less, or not more,but rather "a certain amount away"ex) what is the probability that is no more than 10 away from the pop.meane.g. could be 10 more, or could be 10 less (but no more than that)with 95.45% confidencehow do we interpret this??suppose we do not know µ but we know xwe can give a range of values for µ at a certain level of confidenceex) suppose you test drive 80 Hondas, and they have an average gas mileage of23mpg. (let σ=4.5)- what is the range of values centered around x which with 95% confidence containsµnote!!!for confidence intervals,x is the middle valuenote!!!confidence invervals alwayshave two tailshave: zwant: data valuewe are 95% confident that the population mean µ is between 22.0139 and 23.9861OR: the 95% confidence interval for µ is 22.0139 < µ < 23.9861 OR (22.0139, 23.9861)

ex) what is the 98% confidence interval for the average age of college students?you do a survey, and x=21.9 σ=6.1 n=120we are 98% confident that the population mean is between20.6024 and 23.1976OR: the 98% confidence interval for the population mean is20.6024 < µ < 23.197621.9 ± 1.2976x is called an estimate or "point estimate" for µex) what is the 95% CI for µ (with the same information)x=21.9 σ=6.1 n=120ex) if you want a 96% confidence interval, what is your z-score?when you know your level of confidence and you find a z-score,that z-score is called the critical value

lets analyze the margin of error:CI:what effect does E (the margin of error) have on the confidence interval?- bigger E -> bigger intervalyou'd like to be as precise as possible saying something about µ...precise means smaller intervalin other words, you'd like E to be smallwhat can we do to make E small?can we make z smaller?→yes, but...higher confidence -> bigger Esmaller E -> lower confidence

suppose you want 95% confidence. then the onlything you control is nalso suppose you want E to be a certain amount:first, recall our confidence interval...x = 21.9n = 120σ = 6.1σ x =the 95% CI was:E was:now, what if you still want 95% confidence, but you needyour margin of error to be .4 ... what does n have to be?we can figure this out in general:so in this problem:....parameterx ...point estimate

ex) find the 96% CI for µ when x = 38.2 σ = 7.1 n = 60follow-up: what if you want 98% confidence and a margin of error of .8what does n need to be? (still have x = 38.2, σ = 7.1 )

what if you dont know σ ?...use t-distributionex) find the 95% CI for µ if we have from our sample, x =124 s=8.6 n=81we are 95% confident that µ is between 122.0983 and 125.9017note!!!in Excel, use "=tinv"type probability, dfget t-scoreBUT you must enter theprobability for two tailscombined!step 0: check which distribution and formula you need to useex) find the average age of all BMCC students. you ask 16 people and get:19 24 45 30 19 18 22 31 39 27 25 20 28 32 18 77find the 95% confidence interval ... then do it again without the outlier

lets do a confidence interval....for proportionsex) find a 95% CI for p when you take a survey and get:we are 95% confident that p (the population proportion) isbetween .57 and .67 (between 57% and 67%)OR p is 62% with a margin of error of 5%note: the values for p [.62, .67, .57] are values for the proportion, which has to do with ourdata. it does NOT have to do with area/probability. keep track of which is which.note: margin of error. in this example, p = 62% (with a margin of error of ±5%)ex) find the CI for p at 96% confidence if p = .38 and n=600

for a CI:if you know the estimate (x or p), and you know the margin of error E,then find the upper and lower valuesex) p = .74, E = .04, then what are the upper and lower values?but what about: given the upper and lower values, find the estimate (x or p) and themargin of errorex) your CI is (.52, .58). find the estimate p and margin of error E

elationship between confidence level and margin of error:ex)confidence ↑ ... E ↑(it was true for µ,it is true for p)what if you want high confidence AND low margin of error?...have to increase nex) what if you want E = .03 with 95% confidence (take = .5)what does n need to be ?first, we solve the general case:we take to be .5 because that isthe "worst-case scenario" (it givesthe biggest possible value for n )so: to find n, if you are given p, useit. if not, use p=.5so, in our problem,note: this problem reflects the polling industry standardsevery professional poll follows the guideline of: 95% confidence and margin of error 3%what do we realize from this?almost every pollster needs 1068 people to askso: go find a newspaper, look for a poll where it has the fine print: confidence, E, #people

Confidence Interval Summaryfor mean µ: know σ dont know σ for proportion p:x ± z·σ x ± t·s p ± z·σnote: remember to think about if it is a mean problem or a proportion problem...sometimes you have to calculate the proportion, it is not given explicitly[e.g. 67 out of 120 people]given E, find nfor mean:for proportion:

ex) are the people in support of stricter gun control laws?out of 400 people surveyed, 296 say yes.what is your 96% confidence interval for p?as a percent: what is the point estimate for p? what is your margin oferror?we are 96% confident that between 69.5% and 78.5%of the population want stricter gun control lawspoint estimate p = 74%margin or error E = 4.5%ex) if you want E=.05 with 99% confidence (take p = .5)how big does your sample need to be?

Q: we went from using p to using p in the formulawhy?...because we have no value for p, only for pdoes it matter?...not much. here's an example why.ex) compare margin of error E at 95% confidence, n = 800, and different values for p:so, does it matter?...not much (as long as the values are kind of close)

hw questions ch99.3#1 construct a CI for the proportionx=125, n=250, 90% confidence

hw questions ch9

hw questions ch9online 9.1#2confidence interval

hw questions ch9

hw questions ch9online 9.2#4construct a confidence interval for p if:we are 99% confident that the population proportion(p) is between .5271 and .6729

ch11 inferences on two samples (from two populations)compare means, matched pairs (dependent samples)...recall: two different groups, testing the outcome for eachpair subjects so that each pair is as identical as possible, then put one in eachtreatmentevery subject (in each treatment) goes through the same procedurehow do you organize your data after the study?- take the average (life span) from group 1 (bmw) and compare it with group 2 (toyota)- go pair by pair, and take difference for each pair, and let the difference be your dataclaim: BMW owners live longerex) data pairs:bmw: 81 78 77 85toyota: 79 82 74 78we calculate each differencethe difference for the population (average) is... d (thats like µ)the (average) difference from the sample is... d (thats like x)so, what are the hypotheses?Ho:H 1 :in words, thats like: bmw > toyota(bmw) - (toyota) > 0so, d > 0conclusion: we are 77.82% confident that we reject Ho, accept H1, that d >0bmw owners live longer [p-value = .2218]solve it by STATDISK ... instead of by handnote: if the claim is "morethan 5 years longer", then H 1

ex) are men paid a higher salary (for the same job) than women?Ho: d = 0H1: d > 0conclusion: we are 97.74% confident that we rejectH o , accept H 1 , d>0, men get paid a higher salary thanwomen [p-value = .0226]also: Confidence Intervalex, continued) find the 95% CI for the difference between male andfemale salariesrecall: x ± z·σ_now:d ± t·s_the 95% confidence interval for the difference betweenmale and female salaries is (69.73, 5130.27)note that you must find the t-score for a 95% CIyou DO NOT use the t-score you found before - that was for adifferent confidence level!

hypothesis test for two samples...proportionsex) claim that proportion of men who are smokers is higher than proportionof women who are smokersp 1: proportion of men who smoke ... p 2: proportion of women who smokeHo: p 1 = p 2H1: p 1 > p 2out of 2103 men, 547 smokeout of 1671 women, 368 smokethere will be indices (little 1's and 2's) everywhere,because we need to indicate which population we arerefering tonote: compare as a whole population, notperson - theres no such thing as aproportion for one person=SQRT(0.2601*(1-0.2601)/2103+0.2202*(1-0.2202)/1671)we conclude with .9979 confidence that p 1 > p 2OR we reject Ho and accept H1 with 99.79% confidenceOR the proportion of men who smoke is greater than the proportion of womenwho smoke [pvalue = .0021]

to do a confidence interval for the difference between proportions:we do CI for (p 1 - p 2 )recall: the CI for p is p ± z·σnow: the CI for p 1 - p 2 is: p 1 - p 2 ± z·σex) p 1 = .2601 p 2 = .2202 find the 95% CI for p 1 - p 2our 95% CI for p 1 - p 2 is (.0127, .0671)OR we are 95% confident that p 1 - p 2 is between .0127 and.0671

you do:ex) out of 200 men surveyed, 55% believe abortion should be legalout of 250 women surveyed, 62% believe abortion should be legala) how confident are you that a greater proportion of women believeabortion should be legal?b) find a 95% CI for the difference in proportionswe are .9332 confident that p 1 > p 2 (or p 1 - p 2 > 0 )a higher proportion of women believe abortion should be legal[pvalue is .0668]we reject H o with 93.32% confidence

hw questions - ch1111.1 #14x: .582 .481 .841 .267 .685 .450 (blue)y: .408 .407 .542 .402 .456 .533 (red)is the blue time different from the red time?Ho: b = r b - r = 0H1: b ≠ r b - r ≠ 0-> matched pairs designd: .174 .074 .299 -.135 .229 -.083mean for d = d = .093st.dev for d = s = .1737st.dev for d = s d = .1737 = .0709t = .093 - 0 = 1.3117.0709use StatDisk to get probabilities...conclusion: we are 75.34% confident that the blue time is different from the red timereject Ho, accept H1 with 75.34% confidencep-value = .2466at 1% significance (which is 99% confidence) we cannot reject Hoc) make a 95% confidence intervalCI: d + ts d.093 + (2.5706)(.0709).093 + .1823OR (-.0893, .2753)use Statdisk to get t-score=SQRT(0.1076*(1-0.1076)/251+0.0987*(1-0.0987)/314)

hw questions ch11

linear correlation and regression [ch4.1,2]describing the relationship between two variables ...highlightswhat does correlation mean?...it means that you have two variables and one is related to the otherspecifically in statistics, correlation means you have two quantitative variables,and saying they are (linearly) correlated means that as one variable goes up, the othervariable tends to go up (or down)[recall, linear means straight line]what two quantitative variables might be correlated?ex) years of employment & salary....the more years of employment, the higher your salarywhat is a positive correlation? if one goes up, the other tends to go upwhat is a negative correlation? if one goes up, the other tends to go downnote: no calculations by hand (because theyre very long) - we will use technologywhat is a scatter plot?when you have a scatter plot, what is the line of best fit?go to my website, click on 'activities', click on 'regression' and play the game

scatter plot: the graph of (x,y) data pointsr: linear correlation coefficientit measures how strong the (linear) relationship is between X and Yr will be positive if: as X goes up, Y goes up (positive correlation)r will be negative if: as X goes up, Y goes down (negative correlation)if the relationship is exactly linear - all points are on a straight linethen r = 1 (or -1)if there is no linear relationshipthen r = 0-1 ≤ r ≤ 1go to my website, click on 'activties', click on 'correlation' and play the gamenote: r measures the strength of the correlationslope measures the slope of the liner-squared: coefficient of determinationit measures the proportion (percentage) of the variation from the meanpredicted by the line of best fitif r 2 = 1, then our line predicts our data perfectly ... in other words thedata is in a straight lineif r 2 = 0, the data is scattered all over the place ... in other words, nolinear relaionship at all...we will say more about r 2 later

Doing Regression and Correlation using StatDiskand interpreting the resultscorrelationrr 2y ("y-hat", the line of best fit)slopeinterceptpredictionresidualto use StatDisk:1 type in your data2 click 'analysis'3 click 'correlation and regression'4 make sure you have the proper columns selected for the x-variable and y-variable5 click 'evaluate'ex) bears: x=age y=weightwhat is r ? r 2 ?line of best fit:predict what a 31-month-old bear would weigh:x: explanatory variable (input)y: response variable (output)the average bear weighs 207kgi found a 31-month old bear!it weighs 143 kg143 - 207 = -64 ... total variation = (actual) - (mean)149.16 - 207 = -57.84 ... explained variation = (prediction) - (mean)143 - 149.16= -6.16 ... unexplained variation or residual = (actual) - (prediction)you do: for bears "headwidth" (x) and "neck" (y), find r, r 2 , line of best fit.if headwidth=7.5, predict the neck sizefor bear #7, find the total variation, explained variation, unexplained variationalso, what does the slope tell you?how much y increases as x increasesex) here, y = 1.64 + 3.05xso that tells you that the rate of increase is 3.05... in other words every time x increases 1, y increases 3.05

a bit about r 2 , and the 3 variationsex) 4.1 #32 ... scatter plot and (approximate) line of best fitto find r 2 by hand,square each of the explained variations, and add them upsquare each of the total variations, and add them updivide explained/total - thats r 2(and thats why we dont do it by hand!)

Probability [ch5]...from a more mathematical point of viewquestion:suppose a family has two kids. if one kid is a boy, what is the probability that theother is a girl?guess:...we will come back and answer this.....i promise the answer will surprise youprobability measures the likelihood of an outcome (or event)ex) if you flip a coin, what isP(head) =if you roll a die, what is P(1 or 2) ?what is the probability that if you guess someone's birthday, you are correct?P(correct) =x = the number of ways an event can happenn = the total number of possibilities0 ≤ prob ≤ 1prob = 0 ... impossible eventex) flip a coin, get 'blue'prob = 1 ... certain eventex) roll a die, P(roll is less than 7)in a given scenario, all probabilities must add to 1

ex) pick a card from a 52-card deckP(king) =P(heart)P(spade)P(spade OR heart)addition rule for disjoint eventsdisjoint: no overlapP(king OR spade)P( E or F ) = P(E) + P(F)P( not a heart )general addition ruleP( E or F ) = P(E) + P(F) - P( E and F )P(heart) + P(not heart) = 1P(not heart) = 1 - P(heart)complement ("not")notation:

finding probabilities from real-world dataex) P(a person in the US was never married)ex) P(a person in the US was widowed)P(person in the US not widowed)you do:ex) from a 52-card deck, pull 1 cardP(jack or "10")=P("2" or "3" or "heart")=ex) P(someone is born on weekend)=

ex) flip a coin *twice*P(two heads)...guess:compound eventif you can say "first event THEN second event"that means that you have a compound event[different from: draw one card and see if it isa king OR spade ... that is one event]how do you find probabilities in a compound event?first lets look at finding "n"ex) n = 4 comes from (2)(2) ....two possible outcomes on the first flip,two possible outcomes on the second flipso it makes sense that to find the probability, we are going to multiplyP(HH)ex) flip a coin 4 times...find P(HHTH)*** flip a coin 4 times...P(3 heads) =

ex) roll a dieP(roll a 6, then a 3, then a 2...in that order)ex) flip a coin 3 times. what is the probability you getno heads?at least one head?"at least one" is the opposite of "none"you do:ex) P(from two people, no one born on a tuesday)ex) P(from two people, at least one born on a tuesday)ex) flip a coin, roll a die ... P(head, 5)ex) suppose P(sunny day) = .6 ... over three days, P(sun, no sun, sun)***ex) roll two dice ... P(sum is 2)***ex) P(roll a die four times, get four different numbers)recall: suppose a family has two kids. if one kid is a boy, what is theprobability that the other is a girl?moral: probability can be very tricky, even counter-intuitive