Statistical Inference: A Short Course

CONTENTS
Preface xv
1 The Nature of Statistics 1
1.1 Statistics Defined 1
1.2 The Population and the Sample 2
1.3 Selecting a Sample from a Population 3
1.4 Measurement Scales 4
1.5 Let us Add 6
Exercises 7
2 Analyzing Quantitative Data 9
2.1 Imposing Order 9
2.2 Tabular and Graphical Techniques: Ungrouped Data 9
2.3 Tabular and Graphical Techniques: Grouped Data 11
Exercises 16
Appendix 2.A Histograms with Classes of Different Lengths 18
3 Descriptive Characteristics of Quantitative Data 22
3.1 The Search for Summary Characteristics 22
3.2 The Arithmetic Mean 23
3.3 The Median 26
vii
3.4 The Mode 27
3.5 The Range 27
3.6 The Standard Deviation 28
3.7 Relative Variation 33
3.8 Skewness 34
3.9 Quantiles 36
3.10 Kurtosis 38
3.11 Detection of Outliers 39
3.12 So What Do We Do with All This Stuff? 41
Exercises 47
Appendix 3.A Descriptive Characteristics of Grouped Data 51
3.A.1 The Arithmetic Mean 52
3.A.2 The Median 53
3.A.3 The Mode 55
3.A.4 The Standard Deviation 57
3.A.5 Quantiles (Quartiles, Deciles, and Percentiles) 58
4 Essentials of Probability 61
4.1 Set Notation 61
4.2 Events within the Sample Space 63
4.3 Basic Probability Calculations 64
4.4 Joint, Marginal, and Conditional Probability 68
4.5 Sources of Probabilities 73
Exercises 75
5 Discrete Probability Distributions and Their Properties 81
5.1 The Discrete Probability Distribution 81
5.2 The Mean, Variance, and Standard Deviation of a Discrete
Random Variable 85
5.3 The Binomial Probability Distribution 89
5.3.1 Counting Issues 89
5.3.2 The Bernoulli Probability Distribution 91
5.3.3 The Binomial Probability Distribution 91
Exercises 96
6 The Normal Distribution 101
6.1 The Continuous Probability Distribution 101
6.2 The Normal Distribution 102
6.3 Probability as an Area Under the Normal Curve 104
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6.4 Percentiles of the Standard Normal Distribution and Percentiles
of the Random Variable X 114
Exercises 116
Appendix 6.A The Normal Approximation to Binomial Probabilities 120
7 Simple Random Sampling and the Sampling Distribution
of the Mean 122
7.1 Simple Random Sampling 122
7.2 The Sampling Distribution of the Mean 123
7.3 Comments on the Sampling Distribution of the Mean 127
7.4 A Central Limit Theorem 130
Exercises 132
Appendix 7.A Using a Table of Random Numbers 133
Appendix 7.B Assessing Normality via the Normal Probability Plot 136
Appendix 7.C Randomness, Risk, and Uncertainty 139
7.C.1 Introduction to Randomness 139
7.C.2 Types of Randomness 142
7.C.2.1 Type I Randomness 142
7.C.2.2 Type II Randomness 143
7.C.2.3 Type III Randomness 143
7.C.3 Pseudo-Random Numbers 144
7.C.4 Chaotic Behavior 145
7.C.5 Risk and Uncertainty 146
8 Confidence Interval Estimation of m 152
8.1 The Error Bound on X as an Estimator of m 152
8.2 A Confidence Interval for the Population Mean m (s Known) 154
8.3 A Sample Size Requirements Formula 159
8.4 A Confidence Interval for the Population Mean m (s Unknown) 160
Exercises 165
Appendix 8.A A Confidence Interval for the Population Median MED 167
9 The Sampling Distribution of a Proportion and its Confidence
Interval Estimation 170
9.1 The Sampling Distribution of a Proportion 170
9.2 The Error Bound on ^p as an Estimator for p 173
9.3 A Confidence Interval for the Population Proportion
(of Successes) p 174
9.4 A Sample Size Requirements Formula 176
CONTENTS ix
Exercises 177
Appendix 9.A Ratio Estimation 179
10 Testing Statistical Hypotheses 184
10.1 What is a Statistical Hypothesis? 184
10.2 Errors in Testing 185
10.3 The Contextual Framework of Hypothesis Testing 186
10.3.1 Types of Errors in a Legal Context 188
10.3.2 Types of Errors in a Medical Context 188
10.3.3 Types of Errors in a Processing or
Control Context 189
10.3.4 Types of Errors in a Sports Context 189
10.4 Selecting a Test Statistic 190
10.5 The Classical Approach to Hypothesis Testing 190
10.6 Types of Hypothesis Tests 191
10.7 Hypothesis Tests for m (s Known) 194
10.8 Hypothesis Tests for m (s Unknown and n Small) 195
10.9 Reporting the Results of Statistical Hypothesis Tests 198
10.10 Hypothesis Tests for the Population Proportion
(of Successes) p 201
Exercises 204
Appendix 10.A Assessing the Randomness of a Sample 208
Appendix 10.B Wilcoxon Signed Rank Test (of a Median) 210
Appendix 10.C Lilliefors Goodness-of-Fit Test for Normality 213
11 Comparing Two Population Means and Two Population
Proportions 217
11.1 Confidence Intervals for the Difference of Means when
Sampling from Two Independent Normal Populations 217
11.1.1 Sampling from Two Independent Normal Populations
with Equal and Known Variances 217
11.1.2 Sampling from Two Independent Normal Populations
with Unequal but Known Variances 218
11.1.3 Sampling from Two Independent Normal Populations
with Equal but Unknown Variances 218
11.1.4 Sampling from Two Independent Normal Populations
with Unequal and Unknown Variances 219
11.2 Confidence Intervals for the Difference of Means
when Sampling from Two Dependent Populations:
Paired Comparisons 224
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11.3 Confidence Intervals for the Difference of Proportions when
Sampling from Two Independent Binomial Populations 227
11.4 Statistical Hypothesis Tests for the Difference of Means when
Sampling from Two Independent Normal Populations 228
11.4.1 Population Variances Equal and Known 229
11.4.2 Population Variances Unequal but Known 229
11.4.3 Population Variances Equal and Unknown 229
11.4.4 Population Variances Unequal and Unknown
(an Approximate Test) 230
11.5 Hypothesis Tests for the Difference of Means when Sampling
from Two Dependent Populations: Paired Comparisons 234
11.6 Hypothesis Tests for the Difference of Proportions when
Sampling from Two Independent Binomial Populations 236
Exercises 239
Appendix 11.A Runs Test for Two Independent Samples 243
Appendix 11.B Mann–Whitney (Rank Sum) Test for Two
Independent Populations 245
Appendix 11.C Wilcoxon Signed Rank Test when Sampling from
Two Dependent Populations: Paired Comparisons 249
12 Bivariate Regression and Correlation 253
12.1 Introducing an Additional Dimension to our Statistical Analysis 253
12.2 Linear Relationships 254
12.2.1 Exact Linear Relationships 254
12.3 Estimating the Slope and Intercept of the Population
Regression Line 257
12.4 Decomposition of the Sample Variation in Y 262
12.5 Mean, Variance, and Sampling Distribution of the Least
Squares Estimators ^b0 and ^b1 264
12.6 Confidence Intervals for b0 and b1 266
12.7 Testing Hypotheses about b0 and b1 267
12.8 Predicting the Average Value of Y given X 269
12.9 The Prediction of a Particular Value of Y given X 270
12.10 Correlation Analysis 272
12.10.1 Case A: X and Y Random Variables 272
12.10.1.1 Estimating the Population Correlation
Coefficient r 274
12.10.1.2 Inferences about the Population
Correlation Coefficient r 275
12.10.2 Case B: X Values Fixed, Y a Random Variable 277
CONTENTS xi
Exercises 278
Appendix 12.A Assessing Normality (Appendix 7.B Continued) 280
Appendix 12.B On Making Causal Inferences 281
12.B.1 Introduction 281
12.B.2 Rudiments of Experimental Design 282
12.B.3 Truth Sets, Propositions, and Logical
Implications 283
12.B.4 Necessary and Sufficient Conditions 285
12.B.5 Causality Proper 286
12.B.6 Logical Implications and Causality 287
12.B.7 Correlation and Causality 288
12.B.8 Causality from Counterfactuals 289
12.B.9 Testing Causality 292
12.B.10 Suggestions for Further Reading 294
13 An Assortment of Additional Statistical Tests 295
13.1 Distributional Hypotheses 295
13.2 The Multinomial Chi-Square Statistic 295
13.3 The Chi-Square Distribution 298
13.4 Testing Goodness of Fit 299
13.5 Testing Independence 304
13.6 Testing k Proportions 309
13.7 A Measure of Strength of Association in a
Contingency Table 311
13.8 A Confidence Interval for s2 under Random Sampling from
a Normal Population 312
13.9 The F Distribution 314
13.10 Applications of the F Statistic to Regression Analysis 316
13.10.1 Testing the Significance of the Regression
Relationship Between X and Y 316
13.10.2 A Joint Test of the Regression Intercept and Slope 317
Exercises 318
Appendix A 323
Table A.1 Standard Normal Areas [Z is N(0,1)] 323
Table A.2 Quantiles of the t Distribution (T is tv) 325
Table A.3 Quantiles of the Chi-Square Distribution (X is w2v
) 327
Table A.4 Quantiles of the F Distribution (F is Fv1;v2 ) 329
Table A.5 Binomial Probabilities P(X;n,p) 334
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Table A.6 Cumulative Binomial Probabilities 338
Table A.7 Quantiles of Lilliefors’ Test for Normality 342
Solutions to Exercises 343
References 369
Index 373