SOLUTION MANUAL for
Modern Mathematical Statistics with Applications
3rd Edition by Jay L. Devore
,Contents
1 Overview and Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Language of Statistics . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Graphical Methods in Descriptive Statistics . . . . . . . . . . 9
1.3 Measures of Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Measures of Variability . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. 49
2.1 Sample Spaces and Events . . . . . . . . . . ............. 49
2.2 Axioms, Interpretations, and Properties
of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3 Counting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.4 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.6 Simulation of Random Events . . . . . . . . . . . . . . . . . . . . . 94
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3 Discrete Random Variables and Probability
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 111
3.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . .... 111
3.2 Probability Distributions for Discrete Random
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.3 Expected Values of Discrete Random Variables . . . . . . . 126
3.4 Moments and Moment Generating Functions . . . . . . . . . 137
3.5 The Binomial Probability Distribution . . . . . . . . . . . . . . . 144
3.6 The Poisson Probability Distribution . . . . . . . . . . . . . . . . 156
3.7 Other Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . 164
3.8 Simulation of Discrete Random Variables . . . . . . . . . . . . 173
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4 Continuous Random Variables and Probability
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.1 Probability Density Functions and Cumulative
Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.2 Expected Values and Moment Generating Functions . . . 203
4.3 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 213
4.4 The Gamma Distribution and Its Relatives . . . . . . . . . . . 230
4.5 Other Continuous Distributions . . . . . . . . . . . . . . . . . . . . 239
ix
,x Contents
4.6 Probability Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
4.7 Transformations of a Random Variable . . . . . . . . . . . . . . 258
4.8 Simulation of Continuous Random Variables . . . . . . . . . 263
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
5 Joint Probability Distributions and Their Applications . . . . . 277
5.1 Jointly Distributed Random Variables . . . . . . . . . . . . . . . 277
5.2 Expected Values, Covariance, and Correlation . . . . . . . . 294
5.3 Linear Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
5.4 Conditional Distributions and Conditional
Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5.5 The Bivariate Normal Distribution . . . . . . . . . . . . . . . . . 330
5.6 Transformations of Multiple Random Variables . . . . . . . 336
5.7 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
6 Statistics and Sampling Distributions . . . . . . . . . . . . . . . . .... 357
6.1 Statistics and Their Distributions . . . . . . . . . . . . . . . .... 357
6.2 The Distribution of Sample Totals, Means,
and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
6.3 The v2, t, and F Distributions . . . . . . . . . . . . . . . . . . . . . 380
6.4 Distributions Based on Normal Random Samples . . . . . . 388
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Appendix: Proof of the Central Limit Theorem . . . . . . . . . . . . . . 395
7 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... 397
7.1 Concepts and Criteria for Point Estimation . . ......... 397
7.2 The Methods of Moments and Maximum
Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
7.3 Sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
7.4 Information and Efficiency . . . . . . . . . . . . . . . . . . . . . . . 436
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
8 Statistical Intervals Based on a Single Sample . . . . . . . . . . . . 451
8.1 Basic Properties of Confidence Intervals . . . . . . . . . . . . . 452
8.2 The One-Sample t Interval and Its Relatives . . . . . . . . . . 463
8.3 Intervals for a Population Proportion . . . . . . . . . . . . . . . 475
8.4 Confidence Intervals for the Population Variance
and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . .... 481
8.5 Bootstrap Confidence Intervals . . . . . . . . . . . . . . . . .... 484
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 494
9 Tests of Hypotheses Based on a Single Sample . . . . . . . . . . . . 501
9.1 Hypotheses and Test Procedures . . . . . . . . . . . . . . . . . . . 501
9.2 Tests About a Population Mean . . . . . . . . . . . . . . . . . . . 512
9.3 Tests About a Population Proportion. . . . . . . . . . . . . . . . 526
9.4 P-Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
9.5 The Neyman–Pearson Lemma and Likelihood Ratio
Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
, Contents xi
9.6 Further Aspects of Hypothesis Testing . . . . . . . . . . . . . . 553
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
10 Inferences Based on Two Samples . . . . . . . . . . . . . . . . . . . . . . 565
10.1 The Two-Sample z Confidence Interval and Test . . . . . . 565
10.2 The Two-Sample t Confidence Interval and Test . . . . . . 575
10.3 Analysis of Paired Data . . . . . . . . . . . . . . . . . . . . . . . . . 591
10.4 Inferences About Two Population Proportions . . . . . . . . 602
10.5 Inferences About Two Population Variances . . . . . . . . . . 611
10.6 Inferences Using the Bootstrap and Permutation
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 617
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 630
11 The Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
11.1 Single-Factor ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . 640
11.2 Multiple Comparisons in ANOVA . . . . . . . . . . . . . . . . . 653
11.3 More on Single-Factor ANOVA . . . . . . . . . . . . . . . . . . . 662
11.4 Two-Factor ANOVA without Replication . . . . . . . . . . . . 672
11.5 Two-Factor ANOVA with Replication . . . . . . . . . . . . . . 687
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
12 Regression and Correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
12.1 The Simple Linear Regression Model . . . . . . . . . . . . . . . 704
12.2 Estimating Model Parameters . . . . . . . . . . . . . . . . . . . . . 713
12.3 Inferences About the Regression Coefficient b1 . . . . . . . 727
12.4 Inferences for the (Mean) Response . . . . . . . . . . . . . . . . 737
12.5 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745
12.6 Investigating Model Adequacy: Residual Analysis . . . . . 757
12.7 Multiple Regression Analysis . . . . . . . . . . . . . . . . . . . . . 767
12.8 Quadratic, Interaction, and Indicator Terms. . . . . . . . . . . 783
12.9 Regression with Matrices . . . . . . . . . . . . . . . . . . . . . . . . 795
12.10 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817
13 Chi-Squared Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
13.1 Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
13.2 Two-Way Contingency Tables . . . . . . . . . . . . . . . . . . . . 840
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851
14 Nonparametric Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855
14.1 Exact Inference for Population Quantiles . . . . . . . . . . . . 855
14.2 One-Sample Rank-Based Inference . . . . . . . . . . . . . . . . . 861
14.3 Two-Sample Rank-Based Inference . . . . . . . . . . . . . . . . . 871
14.4 Nonparametric ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . 879
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886
15 Introduction to Bayesian Estimation . . . . . . . . . . . . . . . . . . . . 889
15.1 Prior and Posterior Distributions . . . . . . . . . . . . . . . . . . . 889
15.2 Bayesian Point and Interval Estimation . . . . . . . . . . . . . . 896
Modern Mathematical Statistics with Applications
3rd Edition by Jay L. Devore
,Contents
1 Overview and Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Language of Statistics . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Graphical Methods in Descriptive Statistics . . . . . . . . . . 9
1.3 Measures of Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Measures of Variability . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. 49
2.1 Sample Spaces and Events . . . . . . . . . . ............. 49
2.2 Axioms, Interpretations, and Properties
of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3 Counting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.4 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.6 Simulation of Random Events . . . . . . . . . . . . . . . . . . . . . 94
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3 Discrete Random Variables and Probability
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 111
3.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . .... 111
3.2 Probability Distributions for Discrete Random
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.3 Expected Values of Discrete Random Variables . . . . . . . 126
3.4 Moments and Moment Generating Functions . . . . . . . . . 137
3.5 The Binomial Probability Distribution . . . . . . . . . . . . . . . 144
3.6 The Poisson Probability Distribution . . . . . . . . . . . . . . . . 156
3.7 Other Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . 164
3.8 Simulation of Discrete Random Variables . . . . . . . . . . . . 173
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4 Continuous Random Variables and Probability
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.1 Probability Density Functions and Cumulative
Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.2 Expected Values and Moment Generating Functions . . . 203
4.3 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 213
4.4 The Gamma Distribution and Its Relatives . . . . . . . . . . . 230
4.5 Other Continuous Distributions . . . . . . . . . . . . . . . . . . . . 239
ix
,x Contents
4.6 Probability Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
4.7 Transformations of a Random Variable . . . . . . . . . . . . . . 258
4.8 Simulation of Continuous Random Variables . . . . . . . . . 263
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
5 Joint Probability Distributions and Their Applications . . . . . 277
5.1 Jointly Distributed Random Variables . . . . . . . . . . . . . . . 277
5.2 Expected Values, Covariance, and Correlation . . . . . . . . 294
5.3 Linear Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
5.4 Conditional Distributions and Conditional
Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5.5 The Bivariate Normal Distribution . . . . . . . . . . . . . . . . . 330
5.6 Transformations of Multiple Random Variables . . . . . . . 336
5.7 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
6 Statistics and Sampling Distributions . . . . . . . . . . . . . . . . .... 357
6.1 Statistics and Their Distributions . . . . . . . . . . . . . . . .... 357
6.2 The Distribution of Sample Totals, Means,
and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
6.3 The v2, t, and F Distributions . . . . . . . . . . . . . . . . . . . . . 380
6.4 Distributions Based on Normal Random Samples . . . . . . 388
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Appendix: Proof of the Central Limit Theorem . . . . . . . . . . . . . . 395
7 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... 397
7.1 Concepts and Criteria for Point Estimation . . ......... 397
7.2 The Methods of Moments and Maximum
Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
7.3 Sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
7.4 Information and Efficiency . . . . . . . . . . . . . . . . . . . . . . . 436
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
8 Statistical Intervals Based on a Single Sample . . . . . . . . . . . . 451
8.1 Basic Properties of Confidence Intervals . . . . . . . . . . . . . 452
8.2 The One-Sample t Interval and Its Relatives . . . . . . . . . . 463
8.3 Intervals for a Population Proportion . . . . . . . . . . . . . . . 475
8.4 Confidence Intervals for the Population Variance
and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . .... 481
8.5 Bootstrap Confidence Intervals . . . . . . . . . . . . . . . . .... 484
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 494
9 Tests of Hypotheses Based on a Single Sample . . . . . . . . . . . . 501
9.1 Hypotheses and Test Procedures . . . . . . . . . . . . . . . . . . . 501
9.2 Tests About a Population Mean . . . . . . . . . . . . . . . . . . . 512
9.3 Tests About a Population Proportion. . . . . . . . . . . . . . . . 526
9.4 P-Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
9.5 The Neyman–Pearson Lemma and Likelihood Ratio
Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
, Contents xi
9.6 Further Aspects of Hypothesis Testing . . . . . . . . . . . . . . 553
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
10 Inferences Based on Two Samples . . . . . . . . . . . . . . . . . . . . . . 565
10.1 The Two-Sample z Confidence Interval and Test . . . . . . 565
10.2 The Two-Sample t Confidence Interval and Test . . . . . . 575
10.3 Analysis of Paired Data . . . . . . . . . . . . . . . . . . . . . . . . . 591
10.4 Inferences About Two Population Proportions . . . . . . . . 602
10.5 Inferences About Two Population Variances . . . . . . . . . . 611
10.6 Inferences Using the Bootstrap and Permutation
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 617
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 630
11 The Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
11.1 Single-Factor ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . 640
11.2 Multiple Comparisons in ANOVA . . . . . . . . . . . . . . . . . 653
11.3 More on Single-Factor ANOVA . . . . . . . . . . . . . . . . . . . 662
11.4 Two-Factor ANOVA without Replication . . . . . . . . . . . . 672
11.5 Two-Factor ANOVA with Replication . . . . . . . . . . . . . . 687
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
12 Regression and Correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
12.1 The Simple Linear Regression Model . . . . . . . . . . . . . . . 704
12.2 Estimating Model Parameters . . . . . . . . . . . . . . . . . . . . . 713
12.3 Inferences About the Regression Coefficient b1 . . . . . . . 727
12.4 Inferences for the (Mean) Response . . . . . . . . . . . . . . . . 737
12.5 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745
12.6 Investigating Model Adequacy: Residual Analysis . . . . . 757
12.7 Multiple Regression Analysis . . . . . . . . . . . . . . . . . . . . . 767
12.8 Quadratic, Interaction, and Indicator Terms. . . . . . . . . . . 783
12.9 Regression with Matrices . . . . . . . . . . . . . . . . . . . . . . . . 795
12.10 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817
13 Chi-Squared Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
13.1 Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
13.2 Two-Way Contingency Tables . . . . . . . . . . . . . . . . . . . . 840
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851
14 Nonparametric Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855
14.1 Exact Inference for Population Quantiles . . . . . . . . . . . . 855
14.2 One-Sample Rank-Based Inference . . . . . . . . . . . . . . . . . 861
14.3 Two-Sample Rank-Based Inference . . . . . . . . . . . . . . . . . 871
14.4 Nonparametric ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . 879
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886
15 Introduction to Bayesian Estimation . . . . . . . . . . . . . . . . . . . . 889
15.1 Prior and Posterior Distributions . . . . . . . . . . . . . . . . . . . 889
15.2 Bayesian Point and Interval Estimation . . . . . . . . . . . . . . 896
