SOLUTIONS
MANUAL
P ROBABILITY
AND S TATISTICAL I NFERENCE
TENTH EDITION
Robert V. Hogg
Elliot A. Tanis
Dale L. Zimmerman
, iii
Contents
Preface v
1 Probability 1
1.1 Properties of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Methods of Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Discrete Distributions 7
2.1 Random Variables of the Discrete Type . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Mathematical Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Special Mathematical Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 The Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 The Negative Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Continuous Distributions 19
3.1 Random Variables of the Continuous Type . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 The Exponential, Gamma, and Chi-Square Distributions . . . . . . . . . . . . . . . . . 26
3.3 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Additional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Bivariate Distributions 33
4.1 Bivariate Distributions of the Discrete Type . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 The Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 Bivariate Distributions of the Continuous Type . . . . . . . . . . . . . . . . . . . . . . 37
4.5 The Bivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 Distributions of Functions of Random Variables 45
5.1 Functions of One Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Transformations of Two Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Several Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4 The Moment-Generating Function Technique . . . . . . . . . . . . . . . . . . . . . . . 53
5.5 Random Functions Associated with Normal Distributions . . . . . . . . . . . . . . . . 55
5.6 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.7 Approximations for Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.8 Chebyshev’s Inequality and Convergence in Probability . . . . . . . . . . . . . . . . . 61
5.9 Limiting Moment-Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 62
c 2020 Pearson Education, Inc.
Copyrightt °
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,iv Contents
6 Point Estimation 63
6.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.3 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.4 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.5 A Simple Regression Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.6 Asymptotic Distributions of Maximum
Likelihood Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.7 Sufficient Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.8 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7 Interval Estimation 87
7.1 Confidence Intervals for Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Confidence Intervals for the Difference of Two Means . . . . . . . . . . . . . . . . . . . 88
7.3 Confidence Intervals For Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.4 Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.5 Distribution-Free Confidence Intervals for Percentiles . . . . . . . . . . . . . . . . . . . 92
7.6 More Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.7 Resampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8 Tests of Statistical Hypotheses 107
8.1 Tests About One Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.2 Tests of the Equality of Two Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.3 Tests for Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.4 Tests about Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.5 Some Distribution-Free Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.6 Power of a Statistical Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.7 Best Critical Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.8 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9 More Tests 127
9.1 Chi-Square Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.2 Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.3 One-Factor Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.4 Two-Way Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
9.5 General Factorial and 2 k Factorial Designs . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.6 Tests Concerning Regression and Correlation . . . . . . . . . . . . . . . . . . . . . . . 136
9.7 Statistical Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Copyrigh
Copyright c 2020 Pearson
Copyrightt ° Pearson Education, Inc.
, Preface v
Preface
This solutions manual provides answers for the even-numbered exercises in Probability and Statistical
Inference, tenth edition, by Robert V. Hogg, Elliot A. Tanis, and Dale L. Zimmerman. Complete
solutions are given for most of these exercises. You, the instructor, may decide how many of these
solutions and answers you want to make available to your students. Note that the answers for the
odd-numbered exercises are given in the textbook. Our hope is that this solutions manual will be
helpful to each of you in your teaching.
All of the figures in this manual were generated using Maple, a computer algebra system. Most
of the figures were generated and many of the solutions, especially those involving data, were solved
using procedures that were written by Zaven Karian from Denison University. We thank him for
providing these. These procedures are available free of charge for your use. They are available for
down load at http://www.math.hope.edu/tanis/. Short descriptions of these procedures are provided
on the “Maple Card.” Complete descriptions of these procedures are given in Probability and Statistics:
Explorations with MAPLE, second edition, 1999, written by Zaven Karian and Elliot Tanis, published
by Prentice Hall (ISBN 0-13-021536-8). You can download a slightly revised edition of this manual
at http://www.math.hope.edu/tanis/MapleManual.pdf.
We also want to acknowledge the many suggestions/corrections that were made by our accuracy
checker, Kyle Siegrist.
If you find an error or wish to make a suggestion, please send them to .
These errata will be posted on http://homepage.divms.uiowa.edu/∼dzimmer/.
E.A.T.
D.L.Z.
Copyrigh
Copyrigh c 2020 Pearson Education, Inc.
yrightt °
MANUAL
P ROBABILITY
AND S TATISTICAL I NFERENCE
TENTH EDITION
Robert V. Hogg
Elliot A. Tanis
Dale L. Zimmerman
, iii
Contents
Preface v
1 Probability 1
1.1 Properties of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Methods of Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Discrete Distributions 7
2.1 Random Variables of the Discrete Type . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Mathematical Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Special Mathematical Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 The Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 The Negative Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Continuous Distributions 19
3.1 Random Variables of the Continuous Type . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 The Exponential, Gamma, and Chi-Square Distributions . . . . . . . . . . . . . . . . . 26
3.3 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Additional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Bivariate Distributions 33
4.1 Bivariate Distributions of the Discrete Type . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 The Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 Bivariate Distributions of the Continuous Type . . . . . . . . . . . . . . . . . . . . . . 37
4.5 The Bivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 Distributions of Functions of Random Variables 45
5.1 Functions of One Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Transformations of Two Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Several Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4 The Moment-Generating Function Technique . . . . . . . . . . . . . . . . . . . . . . . 53
5.5 Random Functions Associated with Normal Distributions . . . . . . . . . . . . . . . . 55
5.6 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.7 Approximations for Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.8 Chebyshev’s Inequality and Convergence in Probability . . . . . . . . . . . . . . . . . 61
5.9 Limiting Moment-Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 62
c 2020 Pearson Education, Inc.
Copyrightt °
Copyrigh
,iv Contents
6 Point Estimation 63
6.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.3 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.4 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.5 A Simple Regression Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.6 Asymptotic Distributions of Maximum
Likelihood Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.7 Sufficient Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.8 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7 Interval Estimation 87
7.1 Confidence Intervals for Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Confidence Intervals for the Difference of Two Means . . . . . . . . . . . . . . . . . . . 88
7.3 Confidence Intervals For Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.4 Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.5 Distribution-Free Confidence Intervals for Percentiles . . . . . . . . . . . . . . . . . . . 92
7.6 More Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.7 Resampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8 Tests of Statistical Hypotheses 107
8.1 Tests About One Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.2 Tests of the Equality of Two Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.3 Tests for Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.4 Tests about Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.5 Some Distribution-Free Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.6 Power of a Statistical Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.7 Best Critical Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.8 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9 More Tests 127
9.1 Chi-Square Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.2 Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.3 One-Factor Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.4 Two-Way Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
9.5 General Factorial and 2 k Factorial Designs . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.6 Tests Concerning Regression and Correlation . . . . . . . . . . . . . . . . . . . . . . . 136
9.7 Statistical Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Copyrigh
Copyright c 2020 Pearson
Copyrightt ° Pearson Education, Inc.
, Preface v
Preface
This solutions manual provides answers for the even-numbered exercises in Probability and Statistical
Inference, tenth edition, by Robert V. Hogg, Elliot A. Tanis, and Dale L. Zimmerman. Complete
solutions are given for most of these exercises. You, the instructor, may decide how many of these
solutions and answers you want to make available to your students. Note that the answers for the
odd-numbered exercises are given in the textbook. Our hope is that this solutions manual will be
helpful to each of you in your teaching.
All of the figures in this manual were generated using Maple, a computer algebra system. Most
of the figures were generated and many of the solutions, especially those involving data, were solved
using procedures that were written by Zaven Karian from Denison University. We thank him for
providing these. These procedures are available free of charge for your use. They are available for
down load at http://www.math.hope.edu/tanis/. Short descriptions of these procedures are provided
on the “Maple Card.” Complete descriptions of these procedures are given in Probability and Statistics:
Explorations with MAPLE, second edition, 1999, written by Zaven Karian and Elliot Tanis, published
by Prentice Hall (ISBN 0-13-021536-8). You can download a slightly revised edition of this manual
at http://www.math.hope.edu/tanis/MapleManual.pdf.
We also want to acknowledge the many suggestions/corrections that were made by our accuracy
checker, Kyle Siegrist.
If you find an error or wish to make a suggestion, please send them to .
These errata will be posted on http://homepage.divms.uiowa.edu/∼dzimmer/.
E.A.T.
D.L.Z.
Copyrigh
Copyrigh c 2020 Pearson Education, Inc.
yrightt °
