INSTRUCTOR'S SOLUTIONS MANUAL AN INTRODUCTION TO MATHEMATICAL STATISTICS AND ITS APPLICATIONS FIFTH EDITION PDF

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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 This should be only distributed free of cost. If you have paid for this from an online solution manual ven...

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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

This should be only distributed free of cost. If you have paid for this from an online solution manual vendor, you have been cheated. Copyright © 2012, 2006, 2001 Pearson Education, Inc. Publishing as Prentice Hall, 75 Arlington Street, Boston, MA 02116. All rights reserved. This manual may be reproduced for classroom use only. ISBN-13: 978-0-321-69401-0 ISBN-10: 0-321-69401-5

Contents

Chapter 2: Probability..................................................................................................................................................1 2.2 2.3 2.4 2.5 2.6 2.7

Samples Spaces and the Algebra of Sets ..........................................................................................................1 The Probability Function ..................................................................................................................................5 Conditional Probability.....................................................................................................................................7 Independence..................................................................................................................................................13 Combinatorics.................................................................................................................................................17 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 4.3 4.4 4.5 4.6

The Poisson Distribution ................................................................................................................................75 The Normal Distribution ................................................................................................................................80 The Geometric Distribution............................................................................................................................87 The Negative Binomial Distribution ..............................................................................................................89 The Gamma Distribution ................................................................................................................................91

Chapter 5: Estimation ................................................................................................................................................93 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Estimating Parameters: The Method of Maximum Likelihood and Method of Moments .............................93 Interval Estimation .........................................................................................................................................98 Properties of Estimators................................................................................................................................102 Minimum-Variance Estimators: The Cramér-Rao Lower Bound ................................................................105 Sufficient Estimators ....................................................................................................................................107 Consistency...................................................................................................................................................109 Bayesian Estimation .....................................................................................................................................111

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

ii

Contents

Chapter 6: Hypothesis Testing.................................................................................................................................113 6.2 6.3 6.4 6.5

The Decision Rule ........................................................................................................................................113 Testing Binomial Data - H0: p = po.......................................................................................................................................................................... 114 Type I and Type II Errors .............................................................................................................................115 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 9.3 9.4 9.5

Testing H 0 : µ X = µY .......................................................................................................................................133 Testing H 0 : σ X2 = σ Y2 —The F Test .................................................................................................................136 Binomial Data: Testing H 0 : p X = pY ...........................................................................................................138 Confidence Intervals for the Two-Sample Problem .....................................................................................140

Chapter 10: Goodness-of-Fit Tests ..........................................................................................................................143 10.2 10.3 10.4 10.5

The Multinomial Distribution ...................................................................................................................143 Goodness-of-Fit Tests: All Parameters Known.........................................................................................145 Goodness-of-Fit Tests: Parameters Unknown...........................................................................................148 Contingency Tables...................................................................................................................................154

Chapter 11: Regression............................................................................................................................................159 11.2 11.3 11.4 11.5

The Method of Least Squares....................................................................................................................159 The Linear Model......................................................................................................................................169 Covariance and Correlation.......................................................................................................................174 The Bivariate Normal Distribution............................................................................................................178

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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 14.3 14.4 14.5 14.6

The Sign Test ............................................................................................................................................199 Wilcoxon Tests..........................................................................................................................................202 The Kruskal-Wallis Test ...........................................................................................................................206 The Friedman Test ....................................................................................................................................210 Testing for Randomness............................................................................................................................212

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Chapter 2: Probability Section 2.2: Sample Spaces and the Algebra of Sets 2.2.1

S = {( s, s, s ), ( s, s, f ), ( s, f , s ), ( f , s, s ), ( s, f , f ), ( f , s, f ), ( f , f , s ), ( f , f , f )} A=

{ (s, f , s), ( f , s, s)} ; B = {( f , f , f )}

2.2.2

Let (x, y, z) denote a red x, a blue y, and a green z. Then A = {(2,2,1), (2,1,2), (1, 2,2), (1,1,3), (1,3,1), (3,1,1)}

2.2.3

(1,3,4), (1,3,5), (1,3,6), (2,3,4), (2,3,5), (2,3,6)

2.2.4

There are 16 ways to get an ace and a 7, 16 ways to get a 2 and a 6, 16 ways to get a 3 and a 5, and 6 ways to get two 4’s, giving 54 total.

2.2.5

The outcome sought is (4, 4). It is “harder” to obtain than the set {(5, 3), (3, 5), (6, 2), (2, 6)} of other outcomes making a total of 8.

2.2.6

The set N of five card hands in hearts that are not flushes are called straight flushes. These are five cards whose denominations are consecutive. Each one is characterized by the lowest value in the hand. The choices for the lowest value are A, 2, 3, …, 10. (Notice that an ace can be high or low). Thus, N has 10 elements.

2.2.7

P = {right triangles with sides (5, a, b): a2 + b2 = 25}

2.2.8

A = {SSBBBB, SBSBBB, SBBSBB, SBBBSB, BSSBBB, BSBSBB, BSBBSB, BBSSBB, BBSBSB, BBBSSB}

2.2.9

(a) S = {(0, 0, 0, 0) (0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1), (0, 1, 0, 0), (0, 1, 0, 1), (0, 1, 1, 0), (0, 1, 1, 1), (1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 1, 0), (1, 0, 1, 1, ), (1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0), (1, 1, 1, 1, )} (b) A = {(0, 0, 1, 1), (0, 1, 0, 1), (0, 1, 1, 0), (1, 0, 0, 1), (1, 0, 1, 0), (1, 1, 0, 0, )} (c) 1 + k

2.2.10 (a) S = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (2, 4), (4, 1), (4, 2), (4, 4)} (b) {2, 3, 4, 5, 6, 8} 2.2.11 Let p1 and p2 denote the two perpetrators and i1, i2, and i3, the three in the lineup who are innocent. Then S = {( p1 , i1 ), ( p1 , i2 ), ( p1 , i3 ), ( p2 , i1 ), ( p2 , i2 ), ( p2 , i3 ), ( p1 , p2 ), (i1 , i2 ), (i1 , i3 ), (i2 , i3 )} . The event A contains every outcome in S except (p1, p2). 2.2.12 The quadratic equation will have complex roots—that is, the event A will occur—if b2 − 4ac < 0.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

2

Chapter 2: Probability

2.2.13 In order for the shooter to win with a point of 9, one of the following (countably infinite) sequences of sums must be rolled: (9,9), (9, no 7 or no 9,9), (9, no 7 or no 9, no 7 or no 9,9), … 2.2.14 Let (x, y) denote the strategy of putting x white chips and y black chips in the first urn (which results in 10 − x white chips and 10 − y black chips being in the second urn). Then S = {( x, y ) : x = 0,1,...,10, y = 0,1,...,10, and 1 ≤ x + y ≤ 19} . Intuitively, the optimal strategies are (1, 0) and (9, 10). 2.2.15 Let Ak be the set of chips put in the urn at 1/2k minute until midnight. For example, A1 = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}. Then the set of chips in the urn at midnight is ∞

∪(A

k

− {k + 1}) = ∅ .

k =1

2.2.16

2.2.17 If x2 + 2x ≤ 8, then (x + 4)(x − 2) ≤ 0 and A = {x: −4 ≤ x ≤ 2}. Similarly, if x2 + x ≤ 6, then (x + 3)(x − 2) ≤ 0 and B = {x: −3 ≤ x ≤ 2). Therefore, A ∩ B = {x: −3 ≤ x ≤ 2} and A ∪ B = {x: −4 ≤ x ≤ 2}. 2.2.18 A ∩ B ∩ C = {x: x = 2, 3, 4} 2.2.19 The system fails if either the first pair fails or the second pair fails (or both pairs fail). For either pair to fail, though, both of its components must fail. Therefore, A = (A11 ∩ A21) ∪ (A12 ∩ A22). 2.2.20 (a)

(c)

(b) empty set

_____________________ −∞ ∞

(d)

2.2.21 40 2.2.22 (a) {E1, E2}

(b) {S1, S2, T1, T2}

(c) {A, I}

2.2.23 (a) If s is a member of A ∪ (B ∩ C) then s belongs to A or to B ∩ C. If it is a member of A or of B ∩ C, then it belongs to A ∪ B and to A ∪ C. Thus, it is a member of (A ∪ B) ∩ (A ∪ C). Conversely, choose s in (A ∪ B) ∩ (A ∪ C). If it belongs to A, then it belongs to A ∪ (B ∩ C). If it does not belong to A, then it must be a member of B ∩ C. In that case it also is a member of A ∪ (B ∩ C).

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Section 2.2: Sample Spaces and the Algebra of Sets

3

(b) If s is a member of A ∩ (B ∪ C) then s belongs to A and to B ∪ C. If it is a member of B, then it belongs to A ∩ B and, hence, (A ∩ B) ∪ (A ∩ C). Similarly, if it belongs to C, it is a member of (A ∩ B) ∪ (A ∩ C). Conversely, choose s in (A ∩ B) ∪ (A ∩ C). Then it belongs to A. If it is a member of A ∩ B then it belongs to A ∩ (B ∪ C). Similarly, if it belongs to A ∩ C, then it must be a member of A ∩ (B ∪ C). 2.2.24 Let B = A1 ∪ A2 ∪ … ∪ Ak. Then A1C ∩ A2C ∩ ... ∩ AkC = (A1 ∪ A2 ∪ …∪ Ak)C = BC. Then the expression is simply B ∪ BC = S. 2.2.25 (a) Let s be a member of A ∪ (B ∪ C). Then s belongs to either A or B ∪ C (or both). If s belongs to A, it necessarily belongs to (A ∪ B) ∪ C. If s belongs to B ∪ C, it belongs to B or C or both, so it must belong to (A ∪ B) ∪ C. Now, suppose s belongs to (A ∪ B) ∪ C. Then it belongs to either A ∪ B or C or both. If it belongs to C, it must belong to A ∪ (B ∪ C). If it belongs to A ∪ B, it must belong to either A or B or both, so it must belong to A ∪ (B ∪ C). (b) Suppose s belongs to A ∩ (B ∩ C), so it is a member of A and also B ∩ C. Then it is a member of A and of B and C. That makes it a member of (A ∩ B) ∩ C. Conversely, if s is a member of (A ∩ B) ∩ C, a similar argument shows it belongs to A ∩ (B ∩ C). 2.2.26 (a) (b) (c) (d) (e)

AC ∩ BC ∩ CC A∩B∩C A ∩ BC ∩ CC (A ∩ BC ∩ CC) ∪ (AC ∩ B ∩ CC) ∪ (AC ∩ BC ∩ C) (A ∩ B ∩ CC) ∪ (A ∩ BC ∩ C) ∪ (AC ∩ B ∩ C)

2.2.27 A is a subset of B. 2.2.28 (a) {0} ∪ {x: 5 ≤ x ≤ 10} (d) {x: 0 < x < 3} 2.2.29 (a) B and C

(b) {x: 3 ≤ x < 5} (e) {0} ∪ {x : 3 ≤ x ≤ 10}

(c) {x: 0 < x ≤ 7} (f) {0} ∪ {x : 7 < x ≤ 10}

(b) B is a subset of A.

2.2.30 (a) A1 ∩ A2 ∩ A3 (b) A1 ∪ A2 ∪ A3 The second protocol would be better if speed of approval matters. For very important issues, the first protocol is superior. 2.2.31 Let A and B denote the students who saw the movie the first time and the second time, respectively. Then N(a) = 850, N(b) = 690, and N [( A ∪ B)C ] = 4700 (implying that N(A ∪ B) = 1300). Therefore, N(A ∩ B) = number who saw movie twice = 850 + 690 − 1300 = 240. 2.2.32 (a)

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4

Chapter 2: Probability

(b)

2.2.33 (a)

(b)

2.2.34 (a)

A ∪ (B ∪ C)

(A ∪ B) ∪ C

(b)

A ∩ (B ∩ C)

(A ∩ B) ∩ C

...