INSTRUCTOR’S SOLUTIONS
MANUAL
AN I NTRODUCTION TO
MATHEMATICAL S TATISTICS
AND ITS APPLICATIONS
F IFTH E DITION
Richard J. Larsen
Vanderbilt University
Morris L. Marx
University of West Florida
, Contents
Chapter 2: Probability..................................................................................................................................................1
2.2 Samples Spaces and the Algebra of Sets ..........................................................................................................1
2.3 The Probability Function ..................................................................................................................................5
2.4 Conditional Probability.....................................................................................................................................7
2.5 Independence..................................................................................................................................................13
2.6 Combinatorics.................................................................................................................................................17
2.7 Combinatorial Probability ..............................................................................................................................23
Chapter 3: Random Variables....................................................................................................................................27
3.2 Binomial and Hypergeometric Probabilities ..................................................................................................27
3.3 Discrete Random Variables ............................................................................................................................34
3.4 Continuous Random Variables.......................................................................................................................37
3.5 Expected Values .............................................................................................................................................39
3.6 The Variance...................................................................................................................................................45
3.7 Joint Densities.................................................................................................................................................49
3.8 Transforming and Combining Random Variables..........................................................................................58
3.9 Further Properties of the Mean and Variance.................................................................................................60
3.10 Order Statistics ............................................................................................................................................64
3.11 Conditional Densities ..................................................................................................................................67
3.12 Moment-Generating Functions....................................................................................................................71
Chapter 4: Special Distributions ................................................................................................................................75
4.2 The Poisson Distribution ................................................................................................................................75
4.3 The Normal Distribution ................................................................................................................................80
4.4 The Geometric Distribution............................................................................................................................87
4.5 The Negative Binomial Distribution ..............................................................................................................89
4.6 The Gamma Distribution ................................................................................................................................91
Chapter 5: Estimation ................................................................................................................................................93
5.2 Estimating Parameters: The Method of Maximum Likelihood and Method of Moments .............................93
5.3 Interval Estimation .........................................................................................................................................98
5.4 Properties of Estimators................................................................................................................................102
5.5 Minimum-Variance Estimators: The Cramér-Rao Lower Bound ................................................................105
5.6 Sufficient Estimators ....................................................................................................................................107
5.7 Consistency...................................................................................................................................................109
5.8 Bayesian Estimation .....................................................................................................................................111
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
,ii Contents
Chapter 6: Hypothesis Testing.................................................................................................................................113
6.2 The Decision Rule ........................................................................................................................................113
6.3 Testing Binomial Data - H0: p = po.......................................................................................................................................................................... 114
6.4 Type I and Type II Errors .............................................................................................................................115
6.5 A Notion of Optimality: The Generalized Likelihood Ratio........................................................................119
Chapter 7: Inferences Based on the Normal Distribution ........................................................................................121
7.3 Deriving the Distribution of Y −µ ...............................................................................................................121
S/ n
7.4 Drawing Inferences about µ .........................................................................................................................123
2
7.5 Drawing Inferences about σ ........................................................................................................................127
Chapter 8: Types of Data: A Brief Overview ..........................................................................................................131
8.2 Classifying Data ...........................................................................................................................................131
Chapter 9: Two-Sample Inference ...........................................................................................................................133
9.2 Testing H 0 : µ X = µY .......................................................................................................................................133
9.3 Testing H 0 : σ X2 = σ Y2 —The F Test .................................................................................................................136
9.4 Binomial Data: Testing H 0 : p X = pY ...........................................................................................................138
9.5 Confidence Intervals for the Two-Sample Problem .....................................................................................140
Chapter 10: Goodness-of-Fit Tests ..........................................................................................................................143
10.2 The Multinomial Distribution ...................................................................................................................143
10.3 Goodness-of-Fit Tests: All Parameters Known.........................................................................................145
10.4 Goodness-of-Fit Tests: Parameters Unknown...........................................................................................148
10.5 Contingency Tables...................................................................................................................................154
Chapter 11: Regression............................................................................................................................................159
11.2 The Method of Least Squares....................................................................................................................159
11.3 The Linear Model......................................................................................................................................169
11.4 Covariance and Correlation.......................................................................................................................174
11.5 The Bivariate Normal Distribution............................................................................................................178
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
, Contents iii
Chapter 12: The Analysis of Variance.....................................................................................................................181
12.2 The F test...................................................................................................................................................181
12.3 Multiple Comparisons: Tukey’s Method ..................................................................................................184
12.4 Testing Subhypotheses with Constrasts ....................................................................................................186
12.5 Data Transformations ................................................................................................................................188
Appendix 12.A.3 The Distribution of SSTR / (k − 1) When H1 Is True................................................................188
SSE / (n − k)
Chapter 13: Randomized Block Designs .................................................................................................................191
13.2 The F Test for a Randomized Block Design .............................................................................................191
13.3 The Paired t Test .......................................................................................................................................195
Chapter 14: Nonparametric Statistics ......................................................................................................................199
14.2 The Sign Test ............................................................................................................................................199
14.3 Wilcoxon Tests..........................................................................................................................................202
14.4 The Kruskal-Wallis Test ...........................................................................................................................206
14.5 The Friedman Test ....................................................................................................................................210
14.6 Testing for Randomness............................................................................................................................212
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
MANUAL
AN I NTRODUCTION TO
MATHEMATICAL S TATISTICS
AND ITS APPLICATIONS
F IFTH E DITION
Richard J. Larsen
Vanderbilt University
Morris L. Marx
University of West Florida
, Contents
Chapter 2: Probability..................................................................................................................................................1
2.2 Samples Spaces and the Algebra of Sets ..........................................................................................................1
2.3 The Probability Function ..................................................................................................................................5
2.4 Conditional Probability.....................................................................................................................................7
2.5 Independence..................................................................................................................................................13
2.6 Combinatorics.................................................................................................................................................17
2.7 Combinatorial Probability ..............................................................................................................................23
Chapter 3: Random Variables....................................................................................................................................27
3.2 Binomial and Hypergeometric Probabilities ..................................................................................................27
3.3 Discrete Random Variables ............................................................................................................................34
3.4 Continuous Random Variables.......................................................................................................................37
3.5 Expected Values .............................................................................................................................................39
3.6 The Variance...................................................................................................................................................45
3.7 Joint Densities.................................................................................................................................................49
3.8 Transforming and Combining Random Variables..........................................................................................58
3.9 Further Properties of the Mean and Variance.................................................................................................60
3.10 Order Statistics ............................................................................................................................................64
3.11 Conditional Densities ..................................................................................................................................67
3.12 Moment-Generating Functions....................................................................................................................71
Chapter 4: Special Distributions ................................................................................................................................75
4.2 The Poisson Distribution ................................................................................................................................75
4.3 The Normal Distribution ................................................................................................................................80
4.4 The Geometric Distribution............................................................................................................................87
4.5 The Negative Binomial Distribution ..............................................................................................................89
4.6 The Gamma Distribution ................................................................................................................................91
Chapter 5: Estimation ................................................................................................................................................93
5.2 Estimating Parameters: The Method of Maximum Likelihood and Method of Moments .............................93
5.3 Interval Estimation .........................................................................................................................................98
5.4 Properties of Estimators................................................................................................................................102
5.5 Minimum-Variance Estimators: The Cramér-Rao Lower Bound ................................................................105
5.6 Sufficient Estimators ....................................................................................................................................107
5.7 Consistency...................................................................................................................................................109
5.8 Bayesian Estimation .....................................................................................................................................111
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
,ii Contents
Chapter 6: Hypothesis Testing.................................................................................................................................113
6.2 The Decision Rule ........................................................................................................................................113
6.3 Testing Binomial Data - H0: p = po.......................................................................................................................................................................... 114
6.4 Type I and Type II Errors .............................................................................................................................115
6.5 A Notion of Optimality: The Generalized Likelihood Ratio........................................................................119
Chapter 7: Inferences Based on the Normal Distribution ........................................................................................121
7.3 Deriving the Distribution of Y −µ ...............................................................................................................121
S/ n
7.4 Drawing Inferences about µ .........................................................................................................................123
2
7.5 Drawing Inferences about σ ........................................................................................................................127
Chapter 8: Types of Data: A Brief Overview ..........................................................................................................131
8.2 Classifying Data ...........................................................................................................................................131
Chapter 9: Two-Sample Inference ...........................................................................................................................133
9.2 Testing H 0 : µ X = µY .......................................................................................................................................133
9.3 Testing H 0 : σ X2 = σ Y2 —The F Test .................................................................................................................136
9.4 Binomial Data: Testing H 0 : p X = pY ...........................................................................................................138
9.5 Confidence Intervals for the Two-Sample Problem .....................................................................................140
Chapter 10: Goodness-of-Fit Tests ..........................................................................................................................143
10.2 The Multinomial Distribution ...................................................................................................................143
10.3 Goodness-of-Fit Tests: All Parameters Known.........................................................................................145
10.4 Goodness-of-Fit Tests: Parameters Unknown...........................................................................................148
10.5 Contingency Tables...................................................................................................................................154
Chapter 11: Regression............................................................................................................................................159
11.2 The Method of Least Squares....................................................................................................................159
11.3 The Linear Model......................................................................................................................................169
11.4 Covariance and Correlation.......................................................................................................................174
11.5 The Bivariate Normal Distribution............................................................................................................178
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
, Contents iii
Chapter 12: The Analysis of Variance.....................................................................................................................181
12.2 The F test...................................................................................................................................................181
12.3 Multiple Comparisons: Tukey’s Method ..................................................................................................184
12.4 Testing Subhypotheses with Constrasts ....................................................................................................186
12.5 Data Transformations ................................................................................................................................188
Appendix 12.A.3 The Distribution of SSTR / (k − 1) When H1 Is True................................................................188
SSE / (n − k)
Chapter 13: Randomized Block Designs .................................................................................................................191
13.2 The F Test for a Randomized Block Design .............................................................................................191
13.3 The Paired t Test .......................................................................................................................................195
Chapter 14: Nonparametric Statistics ......................................................................................................................199
14.2 The Sign Test ............................................................................................................................................199
14.3 Wilcoxon Tests..........................................................................................................................................202
14.4 The Kruskal-Wallis Test ...........................................................................................................................206
14.5 The Friedman Test ....................................................................................................................................210
14.6 Testing for Randomness............................................................................................................................212
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
