AND STATISTICAL INFERENCE

SOLUTIONS

MANUAL

PROBABILITY

AND STATISTICAL INFERENCE

TENTH EDITION

Robert V. Hogg Elliot A. Tanis Dale L. Zimmerman 1 / 4

CopCopyrighyrighttcc°° iii Contents Preface v

• Probability 1

1.1 Properties of Probability 1. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Methods of En umeration 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Conditional Probability

3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Independent Events 4 . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5 Bayes’ Theorem 5.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• Discrete Distributions 7

2.1 Random Variables of the Discrete Type 7. . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Mathematical Expectation

9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Special Mathematical Exp ectations 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• Continuous Distributions 19

3.1 Random Variables of the Continuous Type 19. . . . . . . . . . . . . . . . . . . . . . . . .

3.2 The Exponen tial, Gamma, and Chi-Square Distributions 26. . . . . . . . . . . . . . . . .

3.3 The Normal Distribution

28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4 Additional Models

30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• Bivariate Distributions 33

4.1 Bivariate Distributions of the Discrete Type 33. . . . . . . . . . . . . . . . . . . . . . . .

4.2 The Correlation Co efficient 34. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 Conditional Distributions 36.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 Bivariate Distributions of the Continuous Type 37. . . . . . . . . . . . . . . . . . . . . .

4.5 The Bivariate Normal Distribution 41. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• Distributions of F

unctions of Random Variables 45 5.1 Functions of One Random Variable 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2 Transformations of Tw

• Random Variables 47. . . . . . . . . . . . . . . . . . . . . . . .

5.3 Several Random V ariables 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

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CopCopyrighyrighttcc°° iv Contents

• Point Estimation 63

6.1 Descriptive Statistics 63. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2 Exploratory Data Analysis 65. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3 Order Statistics

70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4 Maximum Lik elihood 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• Interval Estimation

87 7.1 Confidence In tervals for Means 87. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2 Confidence In tervals for the Difference of Two Means 88. . . . . . . . . . . . . . . . . . .

7.3 Confidence In tervals 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 Metho ds 99. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• Tests of

Statistical Hypotheses 107 8.1 Tests Ab out One Mean 107. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.2 Tests of the Equality of Two Means 109. . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3 Tests for Variances 111. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.4 Tests ab out 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 Likelihoo d Ratio Tests 124. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• More T

ests 127 9.1 Chi-Square Go odness-of-Fit Tests 127. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2 Contingency T ables 130. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.3 One-Factor Analysis of Variance 131. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4 Two-Wa y 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 Qualit y Control 137. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Copyright °c 2020 Pearson Education, Inc. 3 / 4

v Preface 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.p

df.We also w 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 [email protected].

These errata will be posted on http://homepage.divms.uiowa.edu/∼dzimmer/.

E.A.T.

D.L.Z.

CopCopCopyrighyrighyrighttt °°°ccc 202020202020 PPPearsonearsonearson Education,Education,Education, Inc.Inc.Inc.

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