AN INTRODUCTION TO - AND ITS APPLICATIONS S IXTH EDITION Richard Larsen

SOLUTIONS

MANUAL

AN INTRODUCTION TO

M

ATHEMATICAL STATISTICS

AND ITS APPLICATIONS

S

IXTH EDITION

Richard Larsen

Mo rris Marx No Questions For Solution in Chapter 1 Chapter 9 (Solution Not Available) 1 / 4

Contents Chapter 2: Probability 1..............................................................................................................

2.2 Samples Spaces and the Algebra of Sets 1.............................................................................

2.3 The Probability Function 6.....................................................................................................

2.4 Conditional Probability 7.......................................................................................................

2.5 Independence13.....................................................................................................................

2.6 Combinatorics 17...................................................................................................................

2.7 Combinatorial Probability 24.................................................................................................Chapter 3: Random Variables 27................................................................................................

3.2 Binomial and Hypergeometric Probabilities 27.....................................................................

3.3 Discrete Random Variables 36...............................................................................................

3.4 Continuous Random Variables 41..........................................................................................

3.5 Expected Values 44................................................................................................................

3.6 The Variance 52.....................................................................................................................

3.7 Joint Densities 57...................................................................................................................

3.8 Transforming and Combining Random Variables69.............................................................

3.9 Further Properties of the Mean and Variance 73....................................................................

3.10 Order Statistics 79..................................................................................................................

3.11 Conditional Densities 82........................................................................................................

3.12 Moment-Generating Functions 88..........................................................................................Chapter 4: Special Distributions 93............................................................................................

4.2 The Poisson Distribution 93...................................................................................................

4.3 The Normal Distribution 98...................................................................................................

4.4 The Geometric Distribution 105.............................................................................................

4.5 The Negative Binomial Distribution 107...............................................................................

4.6 The Gamma Distribution 109.................................................................................................Chapter 5: Estimation 111...........................................................................................................

5.2 Estimating Parameters: The Method of Maximum Likelihood

and Method of Moments 111.................................................................................................

5.3 Interval Estimation 118 5.4 Properties of Estimators 123

5.5 Minimum-Variance Estimators:

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5.65.6 Sufficient Estima

tors 130 .......................................................................................................

5.7 Consistency 133 .....................................................................................................................

5.8 Bayesian Estimation 135 ........................................................................................................ 2 / 4

Chapter 6: Hypothesis Testing137.............................................................................................

6.2 The Decision Rule 137...........................................................................................................

6.3 Testing Binomial Data138 ................................................................................................................................................................

6.4 Type I and Type II Errors 139................................................................................................

6.5 A Notion of Optimality: The Generalized Likelihood Ratio 144...........................................Chapter 7: Inferences Based on the Normal Distribution 147....................................................

7.3 Deriving the Distribution of

Y 

S / n

.................................................................................... 147

7.4 Drawing Inferences About

 ......................................................................................... 150

7.5 Drawing Inferences About  2

........................................................................................ 156

Chapter 8: Types of Data: A Brief Overview 161......................................................................

8.2 Classifying Data 161..............................................................................................................Chapter 9: Two-Sample Inference 163.......................................................................................

9.2 Testing H

: 

X



Y ....................................................................................................... 163 9.3 Testing H

:

X 2



Y 2 —The F Test ................................................................................. 166

9.4 Binomial Data: Testing H

: p

X  p Y

............................................................................. 168

9.5 Confidence Intervals for the Two-Sample Problem170........................................................Chapter 10: Goodness-of-Fit Tests 173......................................................................................

10.2 The Multinomial Distribution 173.........................................................................................

10.3 Goodness-of-Fit Tests: All Parameters Known175...............................................................

10.4 Goodness-of-Fit Tests: Parameters Unknown 178.................................................................

10.5 Contingency Tables 185.........................................................................................................Chapter 11: Regression 189........................................................................................................

11.2 The Method of Least Squares 189..........................................................................................

11.3 The Linear Model199............................................................................................................

11.4 Covariance and Correlation204.............................................................................................

11.5 The Bivariate Normal Distribution 208..................................................................................Chapter 12: The Analysis of Variance211.................................................................................

12.2 The F test211.........................................................................................................................

12.3 Multiple Comparisons: Tukey’s Method 214.........................................................................Copyright © 2018 Pearson Education, Inc.iv 3 / 4

12.4 Testing Subhypotheses with Constrasts 216..........................................................................

12.5 Data Transformations 218......................................................................................................Appendix 12.A.2 The Distribution of SSTR /(k 1) )  SSE /(nk When H 1 ............................................ Is True 218 Chapter 13: Randomized Block Designs 221.............................................................................

13.2 The F Test for a Randomized Block Design 221...................................................................

13.3 The Paired t Test224..............................................................................................................Chapter 14: Nonparametric Statistics229...................................................................................

14.2 The Sign Test 229..................................................................................................................

14.3 Wilcoxon Tests 232................................................................................................................

14.4 The Kruskal-Wallis Test 236.................................................................................................

14.5 The Friedman Test239...........................................................................................................

14.6 Testing for Randomness 241..................................................................................................v Copyright © 2018 Pearson Education, Inc.

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