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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 files 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 1 Probability 1 1 Basic Concepts 1 2 Properties of Probability 1 3 Methods of Enumeration 1 4 Conditional Probability 1 5 Independent Events 1 6 Bayes s Theorem v 1 1 2 3 4 6 7 2 Discrete Distributions 2 1 Random Variables of the Discrete Type 2 2 Mathematical Expectation 2 3 The Mean Variance and Standard Deviation 2 4 Bernoulli Trials and the Binomial Distribution 2 5 The Moment Generating Function 2 6 The Poisson Distribution 11 11 15 16 19 22 24 3 Continuous Distributions 3 1 Continuous Type Data 3 2 Exploratory Data Analysis 3 3 Random Variables of the Continuous Type 3 4 The Uniform and Exponential Distributions 3 5 The Gamma and Chi Square Distributions 3 6 The Normal Distribution 3 7 Additional Models 27 27 30 37 45 48 50 54 4 Bivariate Distributions 4 1 Distributions of Two Random Variables 4 2 The Correlation Coefficient 4 3 Conditional Distributions 4 4 The Bivariate Normal Distribution 57 57 59 61 66 69 69 73 76 79 81 84 86 5 Distributions of Functions of Random Variables 5 1 Functions of One Random Variable 5 2 Transformations of Two Random Variables 5 3 Several Independent Random Variables 5 4 The Moment Generating Function Technique 5 5 Random Functions Associated with Normal Distributions 5 6 The Central Limit Theorem 5 7 Approximations for Discrete Distributions iii iv 6 Estimation 6 1 Point Estimation 6 2 Confidence Intervals for Means 6 3 Confidence Intervals for the Difference of Two 6 4 Confidence Intervals for Variances 6 5 Confidence Intervals for Proportions 6 6 Sample Size 6 7 A Simple Regression Problem 6 8 More Regression Means 91 91 94 95 97 99 100 101 107 7 Tests of Statistical Hypotheses 7 1 Tests about Proportions 7 2 Tests about One Mean 7 3 Tests of the Equality of Two Means 7 4 Tests for Variances 7 5 One Factor Analysis of Variance 7 6 Two Factor Analysis of Variance 7 7 Tests Concerning Regression and Correlation 115 115 117 120 123 124 127 128 8 Nonparametric Methods 8 1 Chi Square Goodness of Fit Tests 8 2 Contingency Tables 8 3 Order Statistics 8 4 Distribution Free Confidence Intervals for Percentiles 8 5 The Wilcoxon Tests 8 6 Run Test and Test for Randomness 8 7 Kolmogorov Smirnov Goodness of Fit Test 8 8 Resampling Methods 131 131 135 136 138 140 144 147 149 9 Bayesian Methods 157 9 1 Subjective Probability 157 9 2 Bayesian Estimation 158 9 3 More Bayesian Concepts 159 10 Some Theory 10 1 Sufficient Statistics 10 2 Power of a Statistical Test 10 3 Best Critical Regions 10 4 Likelihood Ratio Tests 10 5 Chebyshev s Inequality and Convergence 10 6 Limiting Moment Generating Functions 10 7 Asymptotic Distributions of Maximum Likelihood Estimators in Probability 161 161 162 166 168 169 170 171 11 Quality Improvement Through Statistical Methods 11 1 Time Sequences 11 2 Statistical Quality Control 11 3 General Factorial and 2k Factorial Designs 173 173 176 179 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 in the Maple Card that is 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 REMARK Note that Probability and Statistics Explorations with MAPLE second edition written by Zaven Karian and Elliot Tanis is available for download from Pearson Education s online catalog It has been slightly revised and now contains references to several of the exercises in the 8th edition of Probability and Statistical Inference 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 tanis hope edu and he will post corrections on his web page http www math hope edu tanis R V H E A T v vi Chapter 1 Probability 1 1 Basic Concepts 1 1 2 a S bbb gbb bgb bbg bgg gbg ggb ggg b S female male c S 000 001 002 003 999 1 1 4 a Clutch size Frequency b h x 4 3 5 5 6 7 7 27 8 26 9 37 10 8 11 2 12 0 13 1 14 1 0 30 0 25 0 20 0 15 0 10 0 05 2 4 6 8 10 12 14 Figure 1 1 4 Clutch sizes for the common gallinule c 9 1 x 2 Section 1 2 Properties of Probability 1 1 6 a No Boxes Frequency b h x 0 20 4 10 5 19 6 13 7 8 8 13 9 7 10 9 11 5 12 2 13 4 14 4 15 2 16 2 19 1 24 1 0 18 0 16 0 14 0 12 0 10 0 08 0 06 0 04 0 02 2 4 6 8 10 12 14 16 18 20 22 24 x Figure 1 1 6 Number of boxes of cereal 1 1 8 a f 1 2 3 …
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