INSTRUCTOR’S SOLUTIONS
MANUAL
AN INTRODUCTION TO
MATHEMATICAL STATISTICS
AND ITS A PPLICATIONS
FIFTH EDITION
Richard J. Larsen
Vanderbilt University
Morris L. Marx
University Of West Florida
Contents
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
,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
7.5 Drawing Inferences About Σ2...................................................................................................................................................................127
Chapter 8: Types Of Data: A Brief Overview .......................................................................................................... 131
8.2 Classifying Data ............................................................................................................................................ 131
Chapter 9: Two-Sample Inference ............................................................................................................................ 133
9.2 Testing H0 : µx = µy.................................................................................................................................................................................................................................................. 133
9.3 Testing H0 :Σ 2X = Σ 2Y —The F Test .................................................................................................................. 136
9.4 Binomial Data: Testing H0 : Px = 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 − When H1 Is True ................................................................. 188
1)
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 INTRODUCTION TO
MATHEMATICAL STATISTICS
AND ITS A PPLICATIONS
FIFTH EDITION
Richard J. Larsen
Vanderbilt University
Morris L. Marx
University Of West Florida
Contents
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
,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
7.5 Drawing Inferences About Σ2...................................................................................................................................................................127
Chapter 8: Types Of Data: A Brief Overview .......................................................................................................... 131
8.2 Classifying Data ............................................................................................................................................ 131
Chapter 9: Two-Sample Inference ............................................................................................................................ 133
9.2 Testing H0 : µx = µy.................................................................................................................................................................................................................................................. 133
9.3 Testing H0 :Σ 2X = Σ 2Y —The F Test .................................................................................................................. 136
9.4 Binomial Data: Testing H0 : Px = 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 − When H1 Is True ................................................................. 188
1)
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.
