Advanced Statistics from an Elementary Point of View

Advanced Statistics from an Elementary Point of View......Page 4
Copyright page......Page 5
Contents......Page 8
Preface......Page 16
1.2 Types of Statistics......Page 20
1.3 Levels of Discourse: Sample vs. Population......Page 21
1.4 Levels of Discourse: Target vs. Sampled Population......Page 23
1.5 Measurement Scales......Page 24
1.7 Exercises......Page 26
2.1 Summarizing Sets of Data Measured on a Ratio or Interval Scale......Page 28
2.2 Tabular Methods......Page 30
2.3 Quantitative Summary Characteristics......Page 35
2.4 Correlation between Variables X and Y......Page 57
2.5 Rank Correlation between Variables X and Y......Page 61
2.6 Exercises......Page 65
3.1 Mathematical Foundations: Sets, Set Relations, and Functions......Page 72
3.2 The Random Experiment, Events, Sample Space, and the Random Variable......Page 78
3.3 Axiomatic Development of Probability Theory......Page 81
3.4 The Occurrence and Probability of an Event......Page 83
3.5 General Addition Rule for Probabilities......Page 84
3.6 Joint, Marginal, and Conditional Probability......Page 85
3.7 Classification of Events......Page 91
3.8 Sources of Probabilities......Page 96
3.9 Bayes’ Rule......Page 98
3.10 Exercises......Page 101
4.1 Random Variables......Page 112
4.2 Discrete Probability Distributions......Page 113
4.3 Continuous Probability Distributions......Page 120
4.4 Mean and Variance of a Random Variable......Page 125
4.5 Chebyshev’s Theorem for Random Variables......Page 130
4.6 Moments of a Random Variable......Page 132
4.7 Quantiles of a Probability Distribution......Page 136
4.8 Moment-Generating Function......Page 138
4.9 Probability-Generating Function......Page 146
4.10 Exercises......Page 151
5.2 Discrete Bivariate Probability Distributions......Page 166
5.3 Continuous Bivariate Probability Distributions......Page 173
5.4 Expectations and Moments of Bivariate Probability Distributions......Page 181
5.6 Joint Moment–Generating Function......Page 188
5.7 Exercises......Page 193
6.1 Introduction......Page 206
6.2 Counting Rules......Page 207
6.3 Discrete Uniform Distribution......Page 213
6.4 The Bernoulli Distribution......Page 214
6.5 The Binomial Distribution......Page 216
6.6 The Multinomial Distribution......Page 222
6.7 The Geometric Distribution......Page 225
6.8 The Negative Binomial Distribution......Page 227
6.9 The Poisson Distribution......Page 231
6.10 The Hypergeometric Distribution......Page 237
6.11 The Generalized Hypergeometric Distribution......Page 244
6.12 Exercises......Page 245
7.1 Introduction......Page 254
7.2 The Uniform Distribution......Page 255
7.3 The Normal Distribution......Page 257
7.4 The Normal Approximation to Binomial Probabilities......Page 272
7.5 The Normal Approximation to Poisson Probabilities......Page 276
7.6 The Exponential Distribution......Page 277
7.7 Gamma and Beta Functions......Page 283
7.8 The Gamma Distribution......Page 285
7.9 The Beta Distribution......Page 289
7.10 Other Useful Continuous Distributions......Page 295
7.11 Exercises......Page 304
8.1 The Purpose of Random Sampling......Page 312
8.2 Sampling Scenarios......Page 313
8.3 The Arithmetic of Random Sampling......Page 320
8.4 The Sampling Distribution of a Statistic......Page 325
8.5 The Sampling Distribution of the Mean......Page 327
8.6 A Weak Law of Large Numbers......Page 335
8.7 Convergence Concepts......Page 338
8.8 A Central Limit Theorem......Page 341
8.9 The Sampling Distribution of a Proportion......Page 345
8.10 The Sampling Distribution of the Variance......Page 352
8.11 A Note on Sample Moments......Page 357
8.12 Exercises......Page 361
9.1 Derived Continuous Parametric Distributions......Page 368
9.2 The Chi-Square Distribution......Page 369
9.3 The Sampling Distribution of the Variance When Sampling from a Normal Population......Page 373
9.4 Student’s t Distribution......Page 376
9.5 Snedecor’s F Distribution......Page 381
9.6 Exercises......Page 387
10.1 Statistics as Point Estimators......Page 392
10.2 Desirable Properties of Estimators as Statistical Properties......Page 394
10.3 Small Sample Properties of Point Estimators......Page 395
10.4 Large Sample Properties of Point Estimators......Page 427
10.5 Techniques for Finding Good Point Estimators......Page 438
10.6 Exercises......Page 450
11.1 Interval Estimators......Page 458
11.2 Central Confidence Intervals......Page 460
11.3 The Pivotal Quantity Method......Page 461
11.4 A Confidence Interval for µ Under Random Sampling from a Normal Population with Known Variance......Page 462
11.5 A Confidence Interval for µ Under Random Sampling from a Normal Population with Unknown Variance......Page 465
11.6 A Confidence Interval for s2 Under Random Sampling from a Normal Population with Unknown Mean......Page 466
11.7 A Confidence Interval for p Under Random Sampling from a Binomial Population......Page 470
11.8 Joint Estimation of a Family of Population Parameters......Page 474
11.9 Confidence Intervals for the Difference of Means When Sampling from Two Independent Normal Populations......Page 477
11.10 Confidence Intervals for the Difference of Means When Sampling from Two Dependent Populations: Paired Comparisons......Page 483
11.11 Confidence Intervals for the Difference of Proportions When Sampling from Two Independent Binomial Populations......Page 489
11.12 Confidence Interval for the Ratio of Two Variances When Sampling from Two Independent Normal Populations......Page 490
11.13 Exercises......Page 492
12.1 Statistical Inference Revisited......Page 502
12.2 Fundamental Concepts for Testing Statistical Hypotheses......Page 503
12.3 What Is the Research Question?......Page 505
12.4 Decision Outcomes......Page 506
12.5 Devising a Test for a Statistical Hypothesis......Page 507
12.6 The Classical Approach to Statistical Hypothesis Testing......Page 510
12.7 Types of Tests or Critical Regions......Page 512
12.8 The Essentials of Conducting a Hypothesis Test......Page 514
12.9 Hypothesis Test for µ Under Random Sampling from a Normal Population with Known Variance......Page 515
12.10 Reporting Hypothesis Test Results......Page 520
12.11 Determining the Probability of a Type II Error β......Page 523
12.12 Hypothesis Tests for µ Under Random Sampling from a Normal Population with Unknown Variance......Page 529
12.13 Hypothesis Tests for p Under Random Sampling from a Binomial Population......Page 531
12.14 Hypothesis Tests for σ2 Under Random Sampling from a Normal Population......Page 535
12.15 The Operating Characteristic and Power Functions of a Test......Page 538
12.16 Determining the Best Test for a Statistical Hypothesis......Page 547
12.17 Generalized Likelihood Ratio Tests......Page 556
12.18 Hypothesis Tests for the Difference of Means When Sampling from Two Independent Normal Populations......Page 565
12.19 Hypothesis Tests for the Difference of Means When Sampling from Two Dependent Populations: Paired Comparisons......Page 572
12.20 Hypothesis Tests for the Difference of Proportions When Sampling from Two Independent Binomial Populations......Page 574
12.21 Hypothesis Tests for the Difference of Variances When Sampling from Two Independent Normal Populations......Page 576
12.22 Hypothesis Tests for Spearman’s Rank Correlation Coefficient .S......Page 578
12.23 Exercises......Page 580
13.1 Parametric vs. Nonparametric Methods......Page 588
13.2 Tests for the Randomness of a Single Sample......Page 591
13.3 Single-Sample Sign Test Under Random Sampling......Page 599
13.4 Wilcoxon Signed Rank Test of a Median......Page 602
13.5 Runs Test for Two Independent Samples......Page 606
13.6 Mann-Whitney (Rank-Sum) Test for Two Independent Samples......Page 609
13.7 The Sign Test When Sampling from Two Dependent Populations: Paired Comparisons......Page 616
13.8 Wilcoxon Signed Rank Test When Sampling from Two Dependent Populations: Paired Comparisons......Page 618
13.9 Exercises......Page 622
14.2 The Multinomial Chi-Square Statistic: Complete Specification of H0......Page 628
14.3 The Multinomial Chi-Square Statistic: Incomplete Specification of H0......Page 635
14.4 The Kolmogorov-Smirnov Test for Goodness of Fit......Page 640
14.5 The Lilliefors Goodness-of-Fit Test for Normality......Page 649
14.6 The Shapiro-Wilk Goodness-of-Fit Test for Normality......Page 650
14.7 The Kolmogorov-Smirnov Test for Goodness of Fit: Two Independent Samples......Page 651
14.8 Assessing Normality via Sample Moments......Page 653
14.9 Exercises......Page 657
15.2 Testing Independence......Page 662
15.3 Testing k Proportions......Page 668
15.4 Testing for Homogeneity......Page 670
15.5 Measuring Strength of Association in Contingency Tables......Page 674
15.6 Testing Goodness of Fit with Nominal-Scale Data: Paired Samples......Page 680
15.7 Exercises......Page 683
16.1 The Regression Model......Page 688
16.2 The Strong Classical Linear Regression Model......Page 689
16.3 Estimating the Slope and Intercept of the Population Regression Line......Page 692
16.4 Mean, Variance, and Sampling Distribution of the Least Squares Estimators β0 and β1......Page 695
16.5 Precision of the Least Squares Estimators β0, β1: Confidence Intervals......Page 698
16.6 Testing Hypotheses Concerning β0, β1......Page 699
16.7 The Precision of the Entire Least Squares Regression Equation: A Confidence Band......Page 703
16.8 The Prediction of a Particular Value of Y Given X......Page 706
16.9 Decomposition of the Sample Variation of Y......Page 710
16.10 The Correlation Model......Page 714
16.11 Estimating the Population Correlation Coefficient .......Page 716
16.12 Inferences about the Population Correlation Coefficient .......Page 717
16.13 Exercises......Page 724
Appendix A......Page 736
Solutions to Selected Exercises......Page 786
References and Suggested Reading......Page 804
Index......Page 808