Applied Statistics Using SPSS, STATISTICA, MATLAB and R

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B.2 Continuous Distributions 449 Properties: 2 1. χ df ( x) = γ df / 2, 2 ( x) ; in particular, df = 2 yields the exponential distribution with λ = ½. 2. 3. 4. 5. 6. Example B. 14 n 2 = ∑ X independent ~ i 1 i X n0, 1 ⇒ X = i X , ~ χ n n i= 1 2 X = ∑ ( X i − x) , X i independent ~ n0, 1 ⇒ X ~ χ n− X 1 = 2 σ n ∑ ( X i= 1 i 2 − µ ) , X i independent ~ nµ , σ ⇒ X ~ χ X 1 = 2 σ n ∑ ( X i= 1 i 2 − x) , X i independent ~ nµ , σ ⇒ 2 X ~ χ n X ~ 2 df 2 , Y ~ χ df ⇒ 2 X + Y ~ χ df + df χ 1 2 1 (convolution of two χ 2 2 results in a χ 2 ). Q: The electric current passing through a 10 Ω resistance shows r<strong>and</strong>om fluctuations around its nominal value that can be well modelled by n0,σ with σ = 0.1 Ampere. What is the probability that the heat power generated in the resistance deviates more than 0.1 Watt from its nominal value? A: The heat power is p = 10 i 2 , where i is the current passing through the 10 Ω resistance. Therefore: 2 P ( p > 0. 1) = P( 10 i > 0. 1) = P( 100 i > 1) . But: i ~ n0 , 0. 1 ⇒ 1 2 2 2 i = 100i ~ χ 2 1 . σ 2 Hence: P( p > 0. 1) = P( χ > 1) = 0.317. B.2.8 Student’s t Distribution 1 2 Description: the mean deviations over the sample st<strong>and</strong>ard deviation. It was derived by the 20 th The Student’s t distribution is the distribution followed by the ratio of English brewery chemist W.S. Gosset (pen-name “Student”) at the beginning of the century. Sample space: ℜ . Density function: t df 2 Γ(( df + 1) / 2) ⎛ ⎞ ( ) ⎜ x x = 1+ ⎟ dfπ Γ( df / 2) ⎜ ⎟ ⎝ df ⎠ −( df + 1) / 2 2 2 1 2 n −1 , with df degrees of freedom. B. 28
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Applied Statistics Using SPSS, STAT
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E d itors Prof. Dr. Joaquim P. Marq
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Contents Preface to the Second Edit
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Contents ix 5.2.3 The Chi-Square Te
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Contents xi Appendix A - Short Surv
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Contents xiii E.26 Soil Pollution .
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Preface to the First Edition This b
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Symbols and Abbreviations Sample Se
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|A| determinant of matrix A tr(A) t
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Σ covariance matrix x arithmetic m
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1 Introduction 1.1 Deterministic Da
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18 h 16 14 12 10 8 6 4 2 0 1.1 Dete
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1.2 Population, Sample and Statisti
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Table 1.3 1.2 Population, Sample an
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Table 1.4 1.3 Random Variables 9 Da
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1.4 Probabilities and Distributions
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1.5 Beyond a Reasonable Doubt... 13
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1.5 Beyond a Reasonable Doubt... 15
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1.6 Statistical Significance and Ot
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1.8 Software Tools 19 book we will
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1.8 Software Tools 21 In the follow
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1.8 Software Tools 23 illustrates t
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1.8 Software Tools 25 On-line help
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1.8 Software Tools 27 Figure 1.12.
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2 Presenting and Summarising the Da
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2.1 Preliminaries 31 The data can t
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» meteo=[ 181 143 36 39 37 % Pasti
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2.1 Preliminaries 35 are interested
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2.1 Preliminaries 37 Besides the in
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2.2 Presenting the Data 39 Sorting
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2.2 Presenting the Data 41 In Table
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2.2 Presenting the Data 43 With SPS
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2.2 Presenting the Data 45 Figure 2
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2.2 Presenting the Data 47 Figure 2
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2.2 Presenting the Data 49 Let X de
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2.2 Presenting the Data 51 Commands
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2.2 Presenting the Data 53 A: The c
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2.2 Presenting the Data 55 The s, c
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2.2 Presenting the Data 57 histogra
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2.3 Summarising the Data 59 type da
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2.3 Summarising the Data 61 delimit
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2.3 Summarising the Data 63 The sam
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Note that: 2.3 Summarising the Data
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where sXY, the sample covariance of
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2.3 Summarising the Data 69 STATIST
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2.3 Summarising the Data 71 A: The
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2.3.6 Measures of Association for N
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2.3 Summarising the Data 75 with th
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Exercises 77 A: We use the N, S and
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Exercises 79 2.13 Determine the box
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3 Estimating Data Parameters Making
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3.1 Point Estimation and Interval E
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3.2 Estimating a Mean 85 In Chapter
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3.2 Estimating a Mean 87 There are
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3.2 Estimating a Mean 89 A: Using M
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3.2 Estimating a Mean 91 Figure 3.5
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3.3 Estimating a Proportion 93 esti
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3.4 Estimating a Variance 95 is to
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3.5 Estimating a Variance Ratio 97
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3.6 Bootstrap Estimation 99 i. F df
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3.6 Bootstrap Estimation 101 about
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3.6 Bootstrap Estimation 103 The bi
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3.6 Bootstrap Estimation 105 In the
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Exercises 107 In order to obtain bo
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Exercises 109 3.14 Consider the CTG
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112 4 Parametric Tests of Hypothese
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5 Non-Parametric Tests of Hypothese
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5.1 Inference on One Population 173
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s = npq = 224× 0. 75× 0. 25 = 6.4
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5.1.3 The Chi-Square Goodness of Fi
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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 F
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5.1.5 The Lilliefors Test for Norma
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5.2 Contingency Tables 189 fewer mi
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2 1 5.2 Contingency Tables 191 degr
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5.2 Contingency Tables 193 An alter
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5.2 Contingency Tables 195 male and
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5.2 Contingency Tables 197 first ca
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5.2 Contingency Tables 199 very low
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5.3.1 Tests for Two Independent Sam
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5.3 Inference on Two Populations 20
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Example 5.19 5.3 Inference on Two P
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3 ( N − N ) 5.4 Inference on More
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5.4 Inference on More Than Two Popu
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5.4 Inference on More Than Two Popu
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Exercises 219 5.7 Several previous
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Exercises 221 5.23 Run the non-para
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6 Statistical Classification Statis
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x 2 o o o o o o o o oo o o o o o o
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6.2 Linear Discriminants 227 Figure
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6.2 Linear Discriminants 229 Figure
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6.2 Linear Discriminants 231 Let us
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Table 6.5. Summary of minimum dista
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6.3 Bayesian Classification 235 Not
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6.3 Bayesian Classification 237 Fig
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6.3 Bayesian Classification 239 Let
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6.3 Bayesian Classification 241 Not
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0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15
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6.3 Bayesian Classification 245 For
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Reality A N Decision A N a b c d 6.
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6.4 The ROC Curve 249 Figure 6.17.
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6.4 The ROC Curve 251 In order to o
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6.5 Feature Selection 253 A: The RO
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6.5 Feature Selection 255 comfortab
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6.6 Classifier Evaluation 257 Resub
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259 A: Table 6.12 shows the leave-o
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6.7 Tree Classifiers 261 Figure 6.2
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6.7 Tree Classifiers 263 or not a g
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6.7 Tree Classifiers 265 The classi
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6.7 Tree Classifiers 267 The classi
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Exercises 269 6.2 Repeat the previo
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7 Data Regression An important obje
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7.1.2 Estimating the Regression Fun
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7.1 Simple Linear Regression 275 Fi
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2 2 2 7.1 Simple Linear Regression
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7.1 Simple Linear Regression 279 Co
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Example 7.3 7.1 Simple Linear Regre
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7.1 Simple Linear Regression 283 Th
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7.1 Simple Linear Regression 285 Fi
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7.1 Simple Linear Regression 287 Le
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7.2 Multiple Regression 7.2.1 Gener
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7.2 Multiple Regression 291 For the
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YYˆ = ∑ ∑ ( y ( y − y)( yˆ
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7.2 Multiple Regression 295 Figure
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7.2 Multiple Regression 297 Table 7
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7.2.5.1 Tests for Regression Coeffi
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The Yi can also be linearly modelle
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3000 2000 1000 0 -1000 -2000 Raw Re
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7.3 Building and Evaluating the Reg
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There are other ways to detect outl
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a 4 3 2 1 0 -1 -2 -3 Expected Norma
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Example 7.19 7.4 Regression Through
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6 y 4 2 0 -2 -4 -6 -8 -10 x 7.5 Rid
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7.5 Ridge Regression 319 The b vect
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Example 7.20 7.5 Ridge Regression 3
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7.5 Logit and Probit Models 323 Let
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7.5 Logit and Probit Models 325 sec
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Exercises 327 Commands 7.7. SPSS an
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8 Data Structure Analysis In the pr
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8.1 Principal Components 331 where
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8.1 Principal Components 333 » % E
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8.1 Principal Components 335 A: The
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8.2 Dimensional Reduction 8.2 Dimen
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12000 10000 8000 6000 4000 2000 eig
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0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 U2 8.
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8.3 Principal Components of Correla
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8.3 Principal Components of Correla
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8.4 Factor Analysis 8.4 Factor Anal
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8.4 Factor Analysis 349 The main be
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Exercises 351 c) The scatter plot o
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9 Survival Analysis In medical stud
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9.2 Non-Parametric Analysis of Surv
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1. The Log-Rank Test. 9.3 Comparing
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9.4 Models for Survival Data 367 Th
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9.4 Models for Survival Data 369 ST
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9.4 Models for Survival Data 371 Ex
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1.0 0.9 0.8 0.7 0.6 0.5 0.4 Cumulat
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10 Directional Data The analysis an
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10.1 Representing Directional Data
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10.1 Representing Directional Data
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10.2 Descriptive Statistics 381 com
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10.3 The von Mises Distributions 38
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10.3 The von Mises Distributions 38
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10.4 Assessing the Distribution of
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Example 10.10 10.4 Assessing the Di
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10.5 Tests on von Mises Distributio
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10.6 Non-Parametric Tests 10.6 Non-
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10.6 Non-Parametric Tests 399 Examp
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Exercises 401 10.8 Compare the two
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404 Appendix A - Short Survey on Pr
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Page 906: B.1 Discrete Distributions 437 Prob
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Page 914: B.2.2 Normal Distribution B.2 Conti
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Page 922: Properties: 1. w1,1/λ(x) ≡ ελ(
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Page 932: 450 Appendix B - Distributions Dist
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Page 940: 454 Appendix B - Distributions 0.6
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Page 952: 460 Appendix D - Tables p n k 0.05
Page 964: 466 Appendix D - Tables D.3 Student
Page 968: 468 Appendix D - Tables D.5 Critica
Page 972: 470 Appendix E - Datasets The varia
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474 Appendix E - Datasets E.9 FHR T
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476 Appendix E - Datasets E.14 Fore
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478 Appendix E - Datasets DATE_REOP
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480 Appendix E - Datasets CG: Conic
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482 Appendix E - Datasets E.26 Soil
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484 Appendix E - Datasets E.29 VCG
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Appendix F - Tools F.1 MATLAB Funct
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Appendix F - Tools 489 r
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References Chapters 1 and 2 Anderso
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References 493 Gardner MJ, Altman D
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References 495 Raudys S, Pikelis V
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References 497 Mardia KV, Jupp PE (
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500 Index 5.9 (two paired samples t
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502 Index H hazard function, 353 ha
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504 Index S sample, 5 mean, 416 siz
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