Instructor’s Solutions
Manual
Probability and
Statistical Inference
Eighth Edition
Robert V. Hogg
University of Iowa
Elliot A. Tanis
Hope College
,The author and publisher of this book have used their best efforts in preparing this book. These efforts
include the development, research, and testing of the theories and programs to determine their effectiveness.
The author and publisher make no warranty of any kind, expresses or implied, with regard to these
programs or the documentation contained in this book. The author and publisher shall not be liable in
any event for incidental or consequential damages in connection with, or arising out of, the furnishing,
performance, or use of these programs.
Reproduced by Pearson Prentice Hall from electronic iles supplied by the author.
Copyright ©2010 Pearson Education, Inc.
Publishing as Pearson Prentice Hall, Upper Saddle River, NJ 07458.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-58476-2
ISBN-10: 0-321-58476-7
,Contents
Preface v
1 Probability 1
1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Properties of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Methods of Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Bayes’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Discrete Distributions 11
2.1 Random Variables of the Discrete Type . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Mathematical Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 The Mean, Variance, and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Bernoulli Trials and the Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . 19
2.5 The Moment-Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Continuous Distributions 27
3.1 Continuous-Type Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Random Variables of the Continuous Type . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 The Uniform and Exponential Distributions . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 The Gamma and Chi-Square Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.7 Additional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Bivariate Distributions 57
4.1 Distributions of Two Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 The Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 The Bivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Distributions of Functions of Random Variables 69
5.1 Functions of One Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Transformations of Two Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Several Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 The Moment-Generating Function Technique . . . . . . . . . . . . . . . . . . . . . . . 79
5.5 Random Functions Associated with Normal Distributions . . . . . . . . . . . . . . . . 81
5.6 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.7 Approximations for Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 86
iii
, iv
6 Estimation 91
6.1 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Confidence Intervals for Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3 Confidence Intervals for the Difference of Two Means . . . . . . . . . . . . . . . . . . . 95
6.4 Confidence Intervals for Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.5 Confidence Intervals for Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.6 Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.7 A Simple Regression Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.8 More Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7 Tests of Statistical Hypotheses 115
7.1 Tests about Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Tests about One Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.3 Tests of the Equality of Two Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.4 Tests for Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.5 One-Factor Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.6 Two-Factor Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.7 Tests Concerning Regression and Correlation . . . . . . . . . . . . . . . . . . . . . . . 128
8 Nonparametric Methods 131
8.1 Chi-Square Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.2 Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.3 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.4 Distribution-Free Confidence Intervals for Percentiles . . . . . . . . . . . . . . . . . . . 138
8.5 The Wilcoxon Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.6 Run Test and Test for Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.7 Kolmogorov-Smirnov Goodness of Fit Test . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.8 Resampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9 Bayesian Methods 157
9.1 Subjective Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9.2 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
9.3 More Bayesian Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10 Some Theory 161
10.1 Sufficient Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.2 Power of a Statistical Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.3 Best Critical Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.4 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
10.5 Chebyshev’s Inequality and Convergence in Probability . . . . . . . . . . . . . . . . . 169
10.6 Limiting Moment-Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.7 Asymptotic Distributions of Maximum
Likelihood Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
11 Quality Improvement Through Statistical Methods 173
11.1 Time Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
11.2 Statistical Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
11.3 General Factorial and 2k Factorial Designs . . . . . . . . . . . . . . . . . . . . . . . . . 179
Manual
Probability and
Statistical Inference
Eighth Edition
Robert V. Hogg
University of Iowa
Elliot A. Tanis
Hope College
,The author and publisher of this book have used their best efforts in preparing this book. These efforts
include the development, research, and testing of the theories and programs to determine their effectiveness.
The author and publisher make no warranty of any kind, expresses or implied, with regard to these
programs or the documentation contained in this book. The author and publisher shall not be liable in
any event for incidental or consequential damages in connection with, or arising out of, the furnishing,
performance, or use of these programs.
Reproduced by Pearson Prentice Hall from electronic iles supplied by the author.
Copyright ©2010 Pearson Education, Inc.
Publishing as Pearson Prentice Hall, Upper Saddle River, NJ 07458.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-58476-2
ISBN-10: 0-321-58476-7
,Contents
Preface v
1 Probability 1
1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Properties of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Methods of Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Bayes’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Discrete Distributions 11
2.1 Random Variables of the Discrete Type . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Mathematical Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 The Mean, Variance, and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Bernoulli Trials and the Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . 19
2.5 The Moment-Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Continuous Distributions 27
3.1 Continuous-Type Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Random Variables of the Continuous Type . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 The Uniform and Exponential Distributions . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 The Gamma and Chi-Square Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.7 Additional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Bivariate Distributions 57
4.1 Distributions of Two Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 The Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 The Bivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Distributions of Functions of Random Variables 69
5.1 Functions of One Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Transformations of Two Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Several Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 The Moment-Generating Function Technique . . . . . . . . . . . . . . . . . . . . . . . 79
5.5 Random Functions Associated with Normal Distributions . . . . . . . . . . . . . . . . 81
5.6 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.7 Approximations for Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 86
iii
, iv
6 Estimation 91
6.1 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Confidence Intervals for Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3 Confidence Intervals for the Difference of Two Means . . . . . . . . . . . . . . . . . . . 95
6.4 Confidence Intervals for Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.5 Confidence Intervals for Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.6 Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.7 A Simple Regression Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.8 More Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7 Tests of Statistical Hypotheses 115
7.1 Tests about Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Tests about One Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.3 Tests of the Equality of Two Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.4 Tests for Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.5 One-Factor Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.6 Two-Factor Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.7 Tests Concerning Regression and Correlation . . . . . . . . . . . . . . . . . . . . . . . 128
8 Nonparametric Methods 131
8.1 Chi-Square Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.2 Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.3 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.4 Distribution-Free Confidence Intervals for Percentiles . . . . . . . . . . . . . . . . . . . 138
8.5 The Wilcoxon Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.6 Run Test and Test for Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.7 Kolmogorov-Smirnov Goodness of Fit Test . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.8 Resampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9 Bayesian Methods 157
9.1 Subjective Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9.2 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
9.3 More Bayesian Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10 Some Theory 161
10.1 Sufficient Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.2 Power of a Statistical Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.3 Best Critical Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.4 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
10.5 Chebyshev’s Inequality and Convergence in Probability . . . . . . . . . . . . . . . . . 169
10.6 Limiting Moment-Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.7 Asymptotic Distributions of Maximum
Likelihood Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
11 Quality Improvement Through Statistical Methods 173
11.1 Time Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
11.2 Statistical Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
11.3 General Factorial and 2k Factorial Designs . . . . . . . . . . . . . . . . . . . . . . . . . 179
