Statistical Inference

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 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 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 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 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 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 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