Solutions Manual Probability and Statistical Inference 10th Edition by Robert Hogg; Elliot Tanis and Dale Zimmerman
Solutions Manual Probability and Statistical Inference 10th Edition by
Robert Hogg; Elliot Tanis and Dale Zimmerman
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
,Solutions Manual Probability and Statistical Inference 10th Edition by Robert Hogg; Elliot Tanis and Dale Zimmerman
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 Bivariate Distributions .................................................................................................................. 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 Distributions of Functions of a Random Variable ..................................................................... 69
5.2 Transformations of Two Random Variables................................................................................. 71
5.3 Several Independent Random Variables ....................................................................................... 74
5.4 The Moment-Generating Function Technique ........................................................................... 77
5.5 Random Functions Associated with Normal Distributions ....................................................... 79
5.6 The Central Limit Theorem........................................................................................................... 82
5.7 Approximations for Discrete Distributions.................................................................................... 84
iii
iv
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
,Solutions Manual Probability and Statistical Inference 10th Edition by Robert Hogg; Elliot Tanis and Dale Zimmerman
6 Estimation 89
6.1 Point Estimation ............................................................................................................................ 89
6.2 Confidence Intervals for Means ..................................................................................................... 92
6.3 Confidence Intervals For Difference of Two Means..................................................................... 93
6.4 Confidence Intervals For Variances .............................................................................................. 95
6.5 Confidence Intervals For Proportions ........................................................................................... 97
6.6 Sample Size ..................................................................................................................................... 98
6.7 A Simple Regression Problem ....................................................................................................... 99
6.8 More Regression ............................................................................................................................ 105
7 Tests of Statistical Hypotheses 113
7.1 Tests about Proportions .............................................................................................................. 113
7.2 Tests about One Mean and One Variance ................................................................................ 115
7.3 Tests of the Equality of Two Means.......................................................................................... 118
7.4 Tests for Variances ....................................................................................................................... 121
7.5 One-Factor Analysis of Variance ................................................................................................ 122
7.6 Two-Factor Analysis of Variance ............................................................................................... 125
7.7 Tests Concerning Regression and Correlation ........................................................................... 126
8 Nonparametric Methods 129
8.1 Chi-Square Goodness of Fit Tests .............................................................................................. 129
8.2 Contingency Tables ...................................................................................................................... 133
8.3 Order Statistics ............................................................................................................................ 134
8.4 Distribution-Free Confidence Intervals for Percentiles ............................................................. 136
8.5 The Wilcoxon Tests...................................................................................................................... 138
8.6 Run Test and Test for Randomness ........................................................................................... 142
8.7 Kolmogorov-Smirnov Goodness of Fit Test .............................................................................. 145
8.8 Resampling ..................................................................................................................................... 147
9 Bayesian Methods 155
9.1 Subjective Probability .................................................................................................................. 155
9.2 Bayesian Estimation ..................................................................................................................... 156
9.3 More Bayesian Concepts.............................................................................................................. 157
10 Some Theory 159
10.1 Sufficient Statistics ........................................................................................................................ 159
10.2 Power of a Statistical Test .......................................................................................................... 160
10.3 Best Critical Regions ................................................................................................................... 164
10.4 Likelihood Ratio Tests ................................................................................................................. 166
10.5 Chebyshev’s Inequality and Convergence in Probability .......................................................... 167
10.6 Limiting Moment-Generating Functions .................................................................................... 168
10.7 Asymptotic Distributions of Maximum
Likelihood Estimators ................................................................................................................................................. 168
11 Quality Improvement Through Statistical Methods 171
11.1 Time Sequences ............................................................................................................................. 171
11.2 Statistical Quality Control .......................................................................................................... 174
11.3 General Factorial and 2k Factorial Designs .............................................................................. 177
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
, Solutions Manual Probability and Statistical Inference 10th Edition by Robert Hogg; Elliot Tanis and Dale Zimmerman
Preface
This solutions manual provides answers for the even-numbered exercises in Probability and Statistical Inference, 8th
edition, by Robert V. Hogg and Elliot A. Tanis. Complete solutions are given for most of these exercises. You,
the instructor, may decide how many of these answers you want to make available to your students. Note that
the answers for the odd-numbered exercises are given in the textbook.
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 on the CD-ROM in the textbook. Short descriptions of these
procedures are provided on the “Maple Card” on the CD-ROM. 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).
Our hope is that this solutions manual will be helpful to each of you in your teaching.
If you find an error or wish to make a suggestion, send these to Elliot Tanis at
and he will post corrections on his web page, http://www.math.hope.edu/tanis/.
R.V.H.
E.A.T.
v
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solutions Manual Probability and Statistical Inference 10th Edition by
Robert Hogg; Elliot Tanis and Dale Zimmerman
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
,Solutions Manual Probability and Statistical Inference 10th Edition by Robert Hogg; Elliot Tanis and Dale Zimmerman
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 Bivariate Distributions .................................................................................................................. 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 Distributions of Functions of a Random Variable ..................................................................... 69
5.2 Transformations of Two Random Variables................................................................................. 71
5.3 Several Independent Random Variables ....................................................................................... 74
5.4 The Moment-Generating Function Technique ........................................................................... 77
5.5 Random Functions Associated with Normal Distributions ....................................................... 79
5.6 The Central Limit Theorem........................................................................................................... 82
5.7 Approximations for Discrete Distributions.................................................................................... 84
iii
iv
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
,Solutions Manual Probability and Statistical Inference 10th Edition by Robert Hogg; Elliot Tanis and Dale Zimmerman
6 Estimation 89
6.1 Point Estimation ............................................................................................................................ 89
6.2 Confidence Intervals for Means ..................................................................................................... 92
6.3 Confidence Intervals For Difference of Two Means..................................................................... 93
6.4 Confidence Intervals For Variances .............................................................................................. 95
6.5 Confidence Intervals For Proportions ........................................................................................... 97
6.6 Sample Size ..................................................................................................................................... 98
6.7 A Simple Regression Problem ....................................................................................................... 99
6.8 More Regression ............................................................................................................................ 105
7 Tests of Statistical Hypotheses 113
7.1 Tests about Proportions .............................................................................................................. 113
7.2 Tests about One Mean and One Variance ................................................................................ 115
7.3 Tests of the Equality of Two Means.......................................................................................... 118
7.4 Tests for Variances ....................................................................................................................... 121
7.5 One-Factor Analysis of Variance ................................................................................................ 122
7.6 Two-Factor Analysis of Variance ............................................................................................... 125
7.7 Tests Concerning Regression and Correlation ........................................................................... 126
8 Nonparametric Methods 129
8.1 Chi-Square Goodness of Fit Tests .............................................................................................. 129
8.2 Contingency Tables ...................................................................................................................... 133
8.3 Order Statistics ............................................................................................................................ 134
8.4 Distribution-Free Confidence Intervals for Percentiles ............................................................. 136
8.5 The Wilcoxon Tests...................................................................................................................... 138
8.6 Run Test and Test for Randomness ........................................................................................... 142
8.7 Kolmogorov-Smirnov Goodness of Fit Test .............................................................................. 145
8.8 Resampling ..................................................................................................................................... 147
9 Bayesian Methods 155
9.1 Subjective Probability .................................................................................................................. 155
9.2 Bayesian Estimation ..................................................................................................................... 156
9.3 More Bayesian Concepts.............................................................................................................. 157
10 Some Theory 159
10.1 Sufficient Statistics ........................................................................................................................ 159
10.2 Power of a Statistical Test .......................................................................................................... 160
10.3 Best Critical Regions ................................................................................................................... 164
10.4 Likelihood Ratio Tests ................................................................................................................. 166
10.5 Chebyshev’s Inequality and Convergence in Probability .......................................................... 167
10.6 Limiting Moment-Generating Functions .................................................................................... 168
10.7 Asymptotic Distributions of Maximum
Likelihood Estimators ................................................................................................................................................. 168
11 Quality Improvement Through Statistical Methods 171
11.1 Time Sequences ............................................................................................................................. 171
11.2 Statistical Quality Control .......................................................................................................... 174
11.3 General Factorial and 2k Factorial Designs .............................................................................. 177
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
, Solutions Manual Probability and Statistical Inference 10th Edition by Robert Hogg; Elliot Tanis and Dale Zimmerman
Preface
This solutions manual provides answers for the even-numbered exercises in Probability and Statistical Inference, 8th
edition, by Robert V. Hogg and Elliot A. Tanis. Complete solutions are given for most of these exercises. You,
the instructor, may decide how many of these answers you want to make available to your students. Note that
the answers for the odd-numbered exercises are given in the textbook.
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 on the CD-ROM in the textbook. Short descriptions of these
procedures are provided on the “Maple Card” on the CD-ROM. 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).
Our hope is that this solutions manual will be helpful to each of you in your teaching.
If you find an error or wish to make a suggestion, send these to Elliot Tanis at
and he will post corrections on his web page, http://www.math.hope.edu/tanis/.
R.V.H.
E.A.T.
v
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
