SOLUTIONS MANAUA for Probability and Statistical Inference 10th Edition Robert V. Hogg; Elliot A. Ta by StudyGuide

INSTRUCTOR’S SOLUTIONS MANUAL P ROBABILITY AND S TATISTICAL I NFERENCE TENTH EDITION

Robert V. Hogg Elliot A. Tanis Dale L. Zimmerman

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ISBN-13: 978-0-13-518948-1 ISBN-10: 0-13-518948-9

iii

Contents Preface

1 Probability 1.1 Properties of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Methods of Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 4 5

2 Discrete Distributions 2.1 Random Variables of the Discrete Type . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mathematical Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Special Mathematical Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Negative Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 9 11 14 16 17 18

3 Continuous Distributions 3.1 Random Variables of the Continuous Type . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Exponential, Gamma, and Chi-Square Distributions . . . . . . . . . . . . . . . . . 3.3 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Additional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 26 28 30

4 Bivariate Distributions 4.1 Bivariate Distributions of the Discrete Type . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Bivariate Distributions of the Continuous Type . . . . . . . . . . . . . . . . . . . . . . 4.5 The Bivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 34 36 37 41

5 Distributions of Functions of Random Variables 45 5.1 Functions of One Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Transformations of Two Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3 Several Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.4 The Moment-Generating Function Technique . . . . . . . . . . . . . . . . . . . . . . . 53 5.5 Random Functions Associated with Normal Distributions . . . . . . . . . . . . . . . . 55 5.6 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.7 Approximations for Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.8 Chebyshev’s Inequality and Convergence in Probability . . . . . . . . . . . . . . . . . 61 5.9 Limiting Moment-Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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Contents

6 Point Estimation 6.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 A Simple Regression Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Asymptotic Distributions of Maximum Likelihood Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Sufficient Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 65 70 73 76

7 Interval Estimation 7.1 Confidence Intervals for Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Confidence Intervals for the Difference of Two Means . . . . . . . . . . . . . . . . . . . 7.3 Confidence Intervals For Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Distribution-Free Confidence Intervals for Percentiles . . . . . . . . . . . . . . . . . . . 7.6 More Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Resampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 88 90 91 92 93 99

81 81 84

8 Tests of Statistical Hypotheses 107 8.1 Tests About One Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.2 Tests of the Equality of Two Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 8.3 Tests for Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.4 Tests about Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.5 Some Distribution-Free Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8.6 Power of a Statistical Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 8.7 Best Critical Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.8 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 9 More Tests 127 9.1 Chi-Square Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 9.2 Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 9.3 One-Factor Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.4 Two-Way Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 9.5 General Factorial and 2k Factorial Designs . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.6 Tests Concerning Regression and Correlation . . . . . . . . . . . . . . . . . . . . . . . 136 9.7 Statistical Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

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Preface

Preface This solutions manual provides answers for the even-numbered exercises in Probability and Statistical Inference, tenth edition, by Robert V. Hogg, Elliot A. Tanis, and Dale L. Zimmerman. Complete solutions are given for most of these exercises. You, the instructor, may decide how many of these solutions and answers you want to make available to your students. Note that the answers for the odd-numbered exercises are given in the textbook. Our hope is that this solutions manual will be helpful to each of you in your teaching. 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 for down load at http://www.math.hope.edu/tanis/. Short descriptions of these procedures are provided on the “Maple Card.” 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). You can download a slightly revised edition of this manual at http://www.math.hope.edu/tanis/MapleManual.pdf. We also want to acknowledge the many suggestions/corrections that were made by our accuracy checker, Kyle Siegrist. If you find an error or wish to make a suggestion, please send them to dale-zimmerman@uiowa.edu. These errata will be posted on http://homepage.divms.uiowa.edu/∼dzimmer/. E.A.T. D.L.Z.

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Chapter 1 Probability

Chapter 1

Probability 1.1

Properties of Probability

1.1-2 Sketch a figure and fill in the probabilities of each of the disjoint sets. Let A = {insure more than one car}, P (A) = 0.85. Let B = {insure a sports car}, P (B) = 0.23.

Let C = {insure exactly one car}, P (C) = 0.15.

It is also given that P (A ∩ B) = 0.17. Since A ∩ C = φ, P (A ∩ C) = 0. It follows that

P (A ∩ B ∩ C 0 ) = 0.17. Thus P (A0 ∩ B ∩ C 0 ) = 0.06 and P (B 0 ∩ C) = 0.09.

1.1-4 (a) S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTTH, TTHH, HTHT, THTH, THHT, HTTT, THTT, TTHT, TTTH, TTTT}; (b) (i) 5/16, (ii) 0, (iii) 11/16, (iv) 4/16, (v) 4/16, (vi) 9/16, (vii) 4/16. 1.1-6 (a) P (A ∪ B) = 0.5 + 0.6 − 0.4 = 0.7; (b)

A = (A ∩ B 0 ) ∪ (A ∩ B) P (A) = P (A ∩ B 0 ) + P (A ∩ B) 0.5 = P (A ∩ B 0 ) + 0.4 0 P (A ∩ B ) = 0.1;

(c) P (A0 ∪ B 0 ) = P [(A ∩ B)0 ] = 1 − P (A ∩ B) = 1 − 0.4 = 0.6. 1.1-8 Let A ={lab work done}, B ={referral to a specialist}, P (A) = 0.41, P (B) = 0.53, P ([A ∪ B]0 ) = 0.21. P (A ∪ B) = P (A) + P (B) − P (A ∩ B) 0.79 = 0.41 + 0.53 − P (A ∩ B) P (A ∩ B) = 0.41 + 0.53 − 0.79 = 0.15.

1.1-10

A∪B∪C = A ∪ (B ∪ C) P (A ∪ B ∪ C) = P (A) + P (B ∪ C) − P [A ∩ (B ∪ C)] = P (A) + P (B) + P (C) − P (B ∩ C) − P [(A ∩ B) ∪ (A ∩ C)] = P (A) + P (B) + P (C) − P (B ∩ C) − P (A ∩ B) − P (A ∩ C) + P (A ∩ B ∩ C).

1.1-12 (a) 1/3; (b) 2/3; (c) 0; (d) 1/2.

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Section 1.2 Methods of Enumeration √ √ 2[r − r( 3/2)] 3 =1− . 1.1-14 P (A) = 2r 2 1.1-16 Note that the respective probabilities are p0 , p1 = p0 /4, p2 = p0 /42 , · · ·. ∞ X p0 = 1 4k k=0

p0 1 − 1/4 p0

3 4 15 1 = . 16 16

1 − p 0 − p1 = 1 −

1.2

Methods of Enumeration

1.2-2 (a) (4)(5)(2) = 40; (b) (2)(2)(2) = 8. µ ¶ 6 1.2-4 (a) 4 = 80; 3 (b) 4(26 ) = 256; (c)

(4 − 1 + 3)! = 20. (4 − 1)!3!

1.2-6 S ={ DDD, DDFD, DFDD, FDDD, DDFFD, DFDFD, FDDFD, DFFDD, FDFDD, FFDDD, FFF, FFDF, FDFF, DFFF FFDDF, FDFDF, DFFDF, FDDFF, DFDFF, DDFFF } so there are 20 possibilities. Note that the winning player (2 choices) must win the last set and two of the previous sets, so the number of outcomes is ·µ ¶ µ ¶ µ ¶¸ 2 3 4 2 + + = 20. 2 2 2 1.2-8 3 · 3 · 212 = 36,864. ¶ ¶ µ µ n−1 n−1 + 1.2-10 r−1 r

1.2-12

n! (n − r)(n − 1)! + r(n − 1)! = = = r!(n − r)! r!(n − r)! µ ¶ n n µ ¶ X X n n . (−1)r (−1)r (1)n−r = (1 − 1)n = r r r=0 r=0 n

(1 + 1) =

n µ ¶ X n r=0

(n − 1)! (n − 1)! + r!(n − 1 − r)! (r − 1)!(n − r)!

(1) (1)

n−r

¶ 5 − 1 + 29 33! 1.2-14 = = 40,920. 29 29!4! ¶ µ ¶µ 19 52 − 19 102,486 6 3 µ ¶ = 0.2917; = 1.2-16 (a) 52 351,325 9

n µ ¶ X n r=0

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µ ¶ n . r

Chapter 1 Probability µ

¶µ ¶µ ¶µ ¶µ ¶µ ¶µ ¶ 10 7 3 5 2 6 7,695 2 1 0 1 0 2 µ ¶ = 0.00622. = (b) 52 1,236,664 9 P5 1.2-18 (a) P (A) = n=1 (1/2)n = 1 − (1/2)5 ; P10 (b) P (B) = n=1 (1/2)n = 1 − (1/2)10 ; 19 3

(d) P (A ∩ B) = P (A) = 1 − (1/2)5 ;

(e) P (C) = P (B) − P (A) = (1/2)5 − (1/2)10 ; (f ) P (B 0 ) = 1 − P (B) = (1/2)10 .

1.3

Conditional Probability

1.3-2 (a)

1041 ; 1456

(b)

392 ; 633

(c)

649 . 823

(d) The proportion of women who favor a gun law is greater than the proportion of men who favor a gun law. 1.3-4 (a) P (HH) =

1 13 12 · = ; 52 51 17

(b) P (HC) =

13 13 13 · = ; 52 51 204

12 4 1 3 51 1 · + · = = . 52 51 52 51 52 · 51 52

1.3-6 Let H ={died from heart disease}; P ={at least one parent had heart disease}. P (H | P 0 ) =

N (H ∩ P 0 ) 110 = . 0 N (P ) 648

3 2 1 1 · · = ; 20 19 18 1140 µ ¶µ ¶ 3 17 1 1 1 2 µ ¶ · (b) = ; 20 17 380 3 ¶ µ ¶µ 17 3 9 X 35 1 2 2k − 2 µ ¶ = = 0.4605; · (c) 20 20 − 2k 76 k=1 2k

1.3-8 (a)

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Section 1.4 Independent Events (d) Draw second. The probability of winning is 1 − 0.4605 = 0.5395. 52 51 50 49 48 47 8,808,975 · · · · · = = 0.74141; 52 52 52 52 52 52 11,881,376 (b) P (A0 ) = 1 − P (A) = 0.25859.

1.3-10 (a) P (A) =

1.3-12 (a) It doesn’t matter because P (B1 ) = (b) P (B) =

1 2 = on each draw. 18 9

1 1 1 , P (B5 ) = , P (B18 ) = ; 18 18 18

1.3-14 (a) 5 · 4 · 3 = 60; (b) 5 · 5 · 5 = 125. 1.3-16

1.4

3 5 2 4 23 · + · = . 5 8 5 8 40

Independent Events P (A ∩ B) = P (A)P (B) = (0.3)(0.6) = 0.18; P (A ∪ B) = P (A) + P (B) − P (A ∩ B) = 0.3 + 0.6 − 0.18 = 0.72; P (A ∩ B) 0 (b) P (A|B) = = = 0. P (B) 0.6

1.4-2 (a)

1.4-4 Proof of (b): P (A0 ∩ B)

Proof of (c): P (A0 ∩ B 0 )

1.4-6

P [A ∩ (B ∩ C)]

= P (B)P (A0 |B) = P (B)[1 − P (A|B)] = P (B)[1 − P (A)] = P (B)P (A0 ). = P [(A ∪ B)0 ] = 1 − P (A ∪ B) = 1 − P (A) − P (B) + P (A ∩ B) = 1 − P (A) − P (B) + P (A)P (B) = [1 − P (A)][1 − P (B)] = P (A0 )P (B 0 ).

= P [A ∩ B ∩ C] = P (A)P (B)P (C) = P (A)P (B ∩ C).

P [A ∩ (B ∪ C)]

P [A0 ∩ (B ∩ C 0 )]

= P [(A ∩ B) ∪ (A ∩ C)] = P (A ∩ B) + P (A ∩ C) − P (A ∩ B ∩ C) = P (A)P (B) + P (A)P (C) − P (A)P (B)P (C) = P (A)[P (B) + P (C) − P (B ∩ C)] = P (A)P (B ∪ C). = P (A0 ∩ C 0 ∩ B) = P (B)[P (A0 ∩ C 0 ) | B] = P (B)[1 − P (A ∪ C | B)] = P (B)[1 − P (A ∪ C)] = P (B)P [(A ∪ C)0 ] = P (B)P (A0 ∩ C 0 ) = P (B)P (A0 )P (C 0 ) = P (A0 )P (B)P (C 0 ) = P (A0 )P (B ∩ C 0 ).

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Chapter 1 Probability

P [A0 ∩ B 0 ∩ C 0 ]

1.4-8

= P [(A ∪ B ∪ C)0 ] = 1 − P (A ∪ B ∪ C) = 1 − P (A) − P (B) − P (C) + P (A)P (B) + P (A)P (C)+ P (B)P (C) − P A)P (B)P (C) = [1 − P (A)][1 − P (B)][1 − P (C)] = P (A0 )P (B 0 )P (C 0 ).

1 2 3 1 4 3 5 2 3 2 · · + · · + · · = . 6 6 6 6 6 6 6 6 6 9

9 3 3 · = ; 4 4 16 9 1 3 3 2 ; (b) · + · = 4 4 4 4 16 2 1 2 4 10 (c) · + · = . 4 4 4 4 16 µ ¶3 µ ¶2 1 1 ; 1.4-12 (a) 2 2 µ ¶3 µ ¶2 1 1 (b) ; 2 2 µ ¶3 µ ¶2 1 1 ; (c) 2 2 µ ¶3 µ ¶2 5! 1 1 (d) . 3! 2! 2 2

1.4-10 (a)

1.4-14 (a) 1 − (0.4)3 = 1 − 0.064 = 0.936;

(b) 1 − (0.4)8 = 1 − 0.00065536 = 0.99934464.

1.4-16 (a)

µ ¶2k ∞ X 1 4

k=0

(b)

5 5

5 ; 9

3 1 4 3 1 4 3 2 1 1 + · · + · · · · = . 5 5 4 3 5 4 3 2 1 5

1.4-18 (a) 7; (b) (1/2)7 ; (c) 63; (d) No! (1/2)63 = 1/9,223,372,036,854,775,808. 1.4-20 No. The equations that must hold are (1 − p1 )(1 − p2 ) = p1 (1 − p2 ) + p2 (1 − p1 ) = p1 p2 . There are no real solutions.

1.5

Bayes’ Theorem

1.5-2 (a)

(b)

P (G)

= P (A ∩ G) + P (B ∩ G) = P (A)P (G | A) + P (B)P (G | B) = (0.40)(0.85) + (0.60)(0.75) = 0.79;

P (A | G)

P (A ∩ G) P (G)

(0.40)(0.85) = 0.43. 0.79

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Section 1.5 Bayes’ Theorem 1.5-4 Let event B denote an accident and let A1 be the event that age of the driver is 16–25. Then (0.1)(0.05) P (A1 | B) = (0.1)(0.05) + (0.55)(0.02) + (0.20)(0.03) + (0.15)(0.04) =

50 50 = = 0.179. 50 + 110 + 60 + 60 280

1.5-6 Let B be the event that the policyholder dies. Let A1 , A2 , A3 be the events that the deceased is standard, preferred and ultra-preferred, respectively. Then (0.60)(0.01) P (A1 | B) = (0.60)(0.01) + (0.30)(0.008) + (0.10)(0.007) 60 60 = = 0.659; = 60 + 24 + 7 91 24 P (A2 | B) = = 0.264; 91 7 = 0.077. P (A3 | B) = 91 1.5-8 Let A be the event that the tablet is under warranty. (0.40)(0.10) P (B1 | A) = (0.40)(0.10) + (0.30)(0.05) + (0.20)(0.03) + (0.10)(0.02) 40 40 = = 0.635; = 40 + 15 + 6 + 2 63 15 P (B2 | A) = = 0.238; 63 6 P (B3 | A) = = 0.095; 63 2 = 0.032. P (B4 | A) = 63 1.5-10 (a) P (D + ) = (0.02)(0.92) + (0.98)(0.05) = 0.0184 + 0.0490 = 0.0674; 0.0490 0.0184 (b) P (A− | D+ ) = = 0.727; P (A+ | D+ ) = = 0.273; 0.0674 0.0674 9310 (0.98)(0.95) (c) P (A− | D− ) = = = 0.998; (0.02)(0.08) + (0.98)(0.95) 16 + 9310 P (A+ | D− ) = 0.002; (d) Yes, particularly those in part (b).

1.5-12 Let D = {defective roll}. Then P (I ∩ D) P (I | D) = P (D) = = =

P (I) · P (D | I) P (I) · P (D | I) + P (II) · P (D | II) (0.60)(0.03) (0.60)(0.03) + (0.40)(0.01) 0.018 0.018 = = 0.818. 0.018 + 0.004 0.022

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Chapter 2 Discrete Distributions

Chapter 2

Discrete Distributions 2.1

Random Variables of the Discrete Type

2.1-2 (a)

(b)

  0.6, 0.3, f (x) =  0.1,

x = 1, x = 5, x = 10;

f(x)

0.6 0.5 0.4 0.3 0.2 0.1 1

Figure 2.1–2: A probability histogram

2.1-4 (a)

9 X

x=1

log10

x+1 x

x=1

= =

(b)

  0, log10 n, F (x) =  1,

9 X

[log10 (x + 1) − log10 x]

log10 2 − log10 1 + log10 3 − log10 2 + · · · + log10 10 − log10 9

log10 10 = 1;

x < 1, n − 1 ≤ x < n, 9 ≤ x.

n = 2, 3, . . . , 9,

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Section 2.1 Random Variables of the Discrete Type 2.1-6 (a) f (x) = (b)

1 , 10

x = 0, 1, 2, · · · , 9;

N ({0})/150 = 11/150 = 0.073;

N ({5})/150 = 13/150 = 0.087;

N ({2})/150 = 13/150 = 0.087;

N ({7})/150 = 16/150 = 0.107;

N ({1})/150 = 14/150 = 0.093;

N ({6})/150 = 22/150 = 0.147;

N ({3})/150 = 12/150 = 0.080;

N ({8})/150 = 18/150 = 0.120;

N ({4})/150 = 16/150 = 0.107; (c)

N ({9})/150 = 15/150 = 0.100.

f(x), h(x) 0.14 0.12 0.10 0.08 0.06 0.04 0.02 1

Figure 2.1–6: Michigan daily lottery digits

2.1-8 (a) f (x) = (b)

6 − |7 − x| , x = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12; 36 f(x)

0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 1

10 11 12

Figure 2.1–8: Probability histogram for the sum of a pair of dice

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Chapter 2 Discrete Distributions

2.1-10 (a) The space of W is S = {0, 1, 2, 3, 4, 5, 6, 7}. 1 1 1 · = , assuming independence. 2 4 8 1 1 1 P (W = 1) = P (X = 0, Y = 1) = · = , 2 4 8 1 1 1 P (W = 2) = P (X = 2, Y = 0) = · = , 2 4 8 1 1 1 P (W = 3) = P (X = 2, Y = 1) = · = , 2 4 8 1 1 1 P (W = 4) = P (X = 0, Y = 4) = · = , 2 4 8 1 1 1 P (W = 5) = P (X = 0, Y = 5) = · = , 2 4 8 1 1 1 P (W = 6) = P (X = 2, Y = 4) = · = , 2 4 8 1 1 1 P (W = 7) = P (X = 2, Y = 5) = · = . 2 4 8 1 That is, f (w) = P (W = w) = , w ∈ S. 8 P (W = 0) = P (X = 0, Y = 0) =

(b)

f (w) 0.12 0.10 0.08 0.06 0.04 0.02 1

Figure 2.1–10: Probability histogram of sum of two special dice 2.1-12 Let x equal the number of orange balls and 144 − x the number of blue balls. Then x 144 − x 144 − x x x x − 1 144 − x 143 − x · + · = · + · 144 143 144 143 144 143 144 143 x2 − x + 144 · 143 − 144x − 143x + x2 = 2 · 144x − 2 · x2 x2 − 144x + 5,148 = 0 (x − 78)(x − 66) = 0 Thus there are 78 orange balls and 66 blue balls.

2.2

Mathematical Expectation µ ¶ µ ¶ µ ¶ 4 1 4 + (0) + (1) = 0; 9 9 9 µ ¶ µ ¶ µ ¶ 4 1 4 8 E(X 2 ) = (−1)2 + (0)2 + (1)2 = ; 9 9 9 9

2.2-2 E(X) = (−1)

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Section 2.3 Special Mathematical Expectations E(3X 2 − 2X + 4) = 3

2.2-4

µ ¶ 20 8 − 2(0) + 4 = . 9 3

6 X

f (x) =

x=0

E(Payment)

2.2-6 Note that

E(X) =

¶ µ 9 1 1 1 1 1 1 +c + + + + + 10 1 2 3 4 5 6

2 ; 49 µ ¶ 1 71 2 1 1 1 1 1· +2· +3· +4· +5· = units. 49 2 3 4 5 6 490

∞ 6 6 X 1 6 π2 = = 2 = 1, so this is a pdf. 2 2 2 2 π x π x=1 x π 6 x=1 ∞ X

∞ X

x=1

∞

6 X 1 6 = 2 = +∞ 2 2 π x π x=1 x

and it is well known that the sum of this harmonic series is not finite.

2.2-8 E(|X − c|) = When c = 5,

1X |x − c|, where S = {1, 2, 3, 5, 15, 25, 50}. 7

E(|X − 5|) =

x∈S

1 [(5 − 1) + (5 − 2) + (5 − 3) + (5 − 5) + (15 − 5) + (25 − 5) + (50 − 5)] . 7

If c is either increased or decreased by 1, this expectation is increased by 1/7. Thus c = 5, the median, minimizes this expectation while b = E(X) = 101/7 minimizes E[(X − b) 2 ].

2.2-10 (1) ·

21 −6 −1 15 + (−1) · = = ; 36 36 36 6

(1) ·

15 21 −6 −1 + (−1) · = = ; 36 36 36 6

(4) ·

6 30 −6 −1 + (−1) · = = . 36 36 36 6

2.2-12 (a) The average class size is (b)

  0.4, 0.3, f (x) =  0.3,

(16)(25) + (3)(100) + (1)(300) = 50; 20

x = 25, x = 100, x = 300;

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Chapter 2 Discrete Distributions

2.3

Special Mathematical Expectations

2.3-2 (a)

E(X) 3 X x =

= = E[X(X − 1)]

= = =

σ2

= = =

(b)

µ ¶x µ ¶3−x 3! 1 3 x! (3 − x)! 4 4 x=1 µ ¶X µ ¶k µ ¶2−k 2 1 1 3 2! 3 4 k! (2 − k)! 4 4 k=0 ¶2 µ ¶µ 1 3 3 1 + = ; 3 4 4 4 4 µ ¶x µ ¶3−x 3 X 3! 1 3 x(x − 1) x! (3 − x)! 4 4 x=2 µ ¶2 µ ¶3 1 3 1 2(3) +6 4 4 4 µ ¶2 µ ¶µ ¶ 3 1 1 ; 6 =2 4 4 4 E[X(X − 1)] + E(X) − µ2 µ ¶µ ¶ µ ¶ µ ¶2 3 1 3 3 (2) + − 4 4 4 4 µ ¶µ ¶ µ ¶µ ¶ µ ¶µ ¶ 3 1 3 1 1 3 (2) + =3 ; 4 4 4 4 4 4

E(X) 4 X x =

= =

µ ¶x µ ¶4−x 4! 1 1 x! (4 − x)! 2 2 x=1 µ ¶k µ ¶3−k µ ¶X 3 1 1 1 3! 4 2 k! (3 − k)! 2 2 k=0 µ ¶µ ¶3 1 1 1 4 + = 2; 2 2 2

E[X(X − 1)]

= = =

σ2

µ ¶x µ ¶4−x 1 1 4! x! (4 − x)! 2 2 x=2 µ ¶4 µ ¶4 µ ¶4 1 1 1 + (6)(4) + (12) 2(6) 2 2 2 µ ¶4 µ ¶2 1 1 48 = 12 ; 2 2 µ ¶2 µ ¶2 1 4 4 (12) + − = 1. 2 2 2 4 X

x(x − 1)

2.3-4 E[(X − µ)/σ] = (1/σ)[E(X) − µ] = (1/σ)(µ − µ) = 0; E{[(X − µ)/σ]2 } = (1/σ 2 )E[(X − µ)2 ] = (1/σ 2 )(σ 2 ) = 1. 2.3-6 f (1) =

2 3 3 , f (2) = , f (3) = , 8 8 8

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Section 2.3 Special Mathematical Expectations 3 2 3 + 2 · + 3 · = 2, 8 8 8 3 2 3 3 σ 2 = 12 · + 22 · + 32 · − 22 = . 8 8 8 4 µ=1·

2.3-8

E(X)

4 X

x·

4 X

x2 ·

x=1

50 = 3.125; 16

= E(X 2 )

x=1

2x − 1 16

85 ; 8

= Var(X)

2x − 1 16

µ ¶2 85 25 55 − = 0.8594; = 8 8 64 √ 55 = 0.9270. 8

2.3-10 µ = E(X) = (−5)(1/16) + (−1)(5/8) + (3)(5/16) = 0 so E[(X −µ)3 ] = E(X 3 ) = (−5)3 (1/16)+(−1)3 (5/8)+(33 )(5/16) = (−125−10+135)/16 = 0. Thus the index of skewness likewise equals 0. The distribution is not symmetric, however.

2.3-12 (a) f (x) = (b)

(c)

σ2

364 365

¶x−1 µ

1 365

x = 1, 2, 3, . . . ;

1 = 365, 1 365 364 365 = 132,860, µ ¶2 1 365 364.500;

P (X > 400)

P (X < 300)

¶400 364 = 0.3337, 365 µ ¶299 364 1− = 0.5597. 365 µ

2.3-14 P (X ≥ 100) = P (X > 99) = (0.99)99 = 0.3697. 2.3-16 (a) f (x) = (1/2)x−1 , x = 2, 3, 4, . . .; (b)

M (t)

E[e ] = ∞ X

∞ X

etx (1/2)x−1

x=2

(et /2)x

e2t 2(et /2)2 = , 1 − et /2 2 − et

x=2

t < ln 2;

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Chapter 2 Discrete Distributions

(c)

M 0 (t)

M 00 (t)

σ2

4e2t − e3t (2 − et )2 M 0 (0) = 3; (2 − et )2 (8e2t − 3e3t ) − (4e2t − e3t )2 ∗ (2 − et )(−et ) (2 − et )4 00 2 M (0) − µ = 11 − 9 = 2;

(d) (i) P (X ≤ 3) = 3/4; (ii) P (X ≥ 5) = 1/8; (iii) P (X = 3) = 1/4. 2.3-18

P (X > k + j | X > k)

P (X > k + j) P (X > k)

q k+j = q j = P (X > j). qk

2.3-20 (a) Construct a square in the first quadrant with lengths of sides equal to one. Within this square construct squares with one vertex at the origin and side lengths equal to 1 − 1/2x ; ¶2 µ ¶2 µ 1 (2x−1 − 1)2 (2x − 1)2 1 = − (b) f (x) = 1 − x − 1 − x−1 2 2 22x 22x−2 =

(2x − 1)2 22 (2x−1 − 1)2 (2x − 1)2 (2x − 2)2 − = − 22x 22x 22x 22x

2x+1 − 3 22x − 2 · 2x + 1 − 22x + 22 · 2x − 4 = , x = 1, 2, 3, . . . ; 2x 2 22x ¸ X ¸ ∞ · x+1 ∞ · X 3 3 3/4 4/4 2 2 · 2x (c) − − − = 2 − 1 = 1; = = 2x 2x 2x x 2 2 2 4 1 − 1/2 1 − 1/4 x=1 x=1 =

(d) M (t) = E(e

∞ X

x=1

(e)

M 0 (t)

µ (f )

M 00 (t)

σ2

¸ ∞ · 3(etx) 2et 3et 2x+1 − 3 X 2(2et )x = − = − ; · 22x 4x 4x 2 − et 4 − et x=1

4et − 2e2t + 2e2t 12et − 3e2t + 3e2t − t 2 (2 − e ) (4 − et )2

12et 4et − t 2 (2 − e ) (4 − et )2

M 0 (0) = 4 − =

12 8 = ; 9 3

(2 − et )2 · 4et − 4et − 4et · 2(2 − et )(−et ) − (2 − et )4 (4 − et )2 · 12et − 12et − 12et · 2(4 − et )(−et ) (4 − et )4 µ ¶2 8 4 + 8 108 + 72 8 00 2 − − M (0) − µ = = . 1 81 3 3

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Section 2.4 The Binomial Distribution

2.4

The Binomial Distribution 11 7 , f (1) = ; 18 18 11 7 4 µ = (−1) + (1) =− ; 18 18 18 ¶2 µ ¶ µ ¶2 µ ¶ µ 11 7 4 77 4 . + 1+ = σ 2 = −1 + 18 18 18 18 81

2.4-2 f (−1) =

The variance can also be computed by noting that X 2 = 1 so µ ¶2 4 77 2 2 σ =1−µ =1− − . = 18 81

2.4-4 (a) X is b(7, 0.15); (b)

(i)

P (X ≥ 2)

1 − P (X ≤ 1) = 1 − 0.7166 = 0.2834;

(ii)

P (X = 1)

P (X ≤ 1) − P (X ≤ 0) = 0.7166 − 0.3206 = 0.3960;

(iii)

P (X ≤ 3)

0.9879.

2.4-6 (a) X is b(15, 0.75);

15 − X is b(15, 0.25);

(b) P (X ≥ 10) = P (15 − X ≤ 5) = 0.8516;

(d) P (X = 10) = P (X ≥ 10) − P (X ≥ 11) = P (15 − X ≤ 5) − P (15 − X ≤ 4) = 0.8516 − 0.6865 = 0.1651; √ (e) µ = (15)(0.75) = 11.25, σ 2 = (15)(0.75)(0.25) = 2.8125; σ = 2.8125 = 1.67705. 2.4-8 (a) 1 − 0.014 = 0.99999999;

(b) 0.994 = 0.960596.

2.4-10 (a) X is b(8, 0.90); (b)

(i)

P (X = 8)

P (8 − X = 0) = 0.4305;

(ii)

P (X ≤ 6)

P (8 − X ≥ 2)

1 − P (8 − X ≤ 1) = 1 − 0.8131 = 0.1869;

P (8 − X ≤ 2) = 0.9619.

(iii) P (X ≥ 6) 2.4-12 (a)

f (x) =

(b)

µ σ2 σ

 125/216,        75/216,       

x = −1, x = 1,

15/216,

x = 2,

1/216,

x = 3;

125 75 15 1 17 + (1) · + (2) · + (3) · = − ; 216 216 216 216 216 µ ¶2 269 17 = E(X 2 ) − µ2 = − − = 1.2392; 216 216 = 1.11;

(−1) ·

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Chapter 2 Discrete Distributions

0.5 0.4 0.3 0.2 0.1 –1

Figure 2.4–12: Losses in chuck-a-luck 2.4-14 Let X equal the number of winning tickets when n tickets are purchased. Then P (X ≥ 1)

(a)

1 − P (X = 0) µ ¶n 9 = 1− . 10

1 − (0.9)n

0.50

(0.9)n

0.50

n ln 0.9

ln 0.5

ln 0.5 = 6.58 ln 0.9

1 − (0.9)n

0.95

(0.9)n

0.05

ln 0.05 = 28.43 ln 0.9

so n = 7; (b)

so n = 29.

2.4-16 It is given that X is b(10, 0.10). We are to find M so that P (1000X ≤ M ) ≥ 0.99) or P (X ≤ M/1000) ≥ 0.99. From Appendix Table II, P (X ≤ 4) = 0.9984 > 0.99. Thus M/1000 = 4 or M = 4000 dollars.

2.4-18 Let X equal the number of infected individuals. X is b(5, 0.05). The expected number of tests is 1 P (X = 0) + 6 P (X > 0) = 1 (0.7738) + 6 (1 − 0.7738) = 2.131. 2.4-20 (a) (i) b(5, 0.7); (ii) µ = 3.5, σ 2 = 1.05; (iii) 0.1607; (b) (i) geometric, p = 0.3; (ii) µ = 10/3, σ 2 = 70/9; (iii) 0.51; (c) (i) Bernoulli, p = 0.55; (ii) µ = 0.55, σ 2 = 0.2475; (iii) 0.55;

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Section 2.5 The Hypergeometric Distribution (d) (ii) µ = 2.1, σ 2 = 0.89; (iii) 0.7; (e) (i) discrete uniform on 1, 2, . . . , 10; (ii) 5.5, 8.25; (iii) 0.2.

2.5

The Hypergeometric Distribution

µ ¶µ ¶ 4 4 1 4 0 µ ¶ = 2.5-2 (a) ; 8 70 4 µ ¶µ ¶ 4 4 2 4 0 (b) 2 µ ¶ = . 8 70 4

µ ¶µ ¶ 3 17 91 137 0 5 2.5-4 P (X ≥ 1) = 1 − P (X = 0) = 1 − µ ¶ = 1 − = = 0.60. 20 228 228 5 2.5-6 We have N = N1 + N2 . Thus E[X(X − 1)]

n X

x=0

n X

x=2

x(x − 1)f (x) x(x − 1)

N1 ! N2 ! · x! (N1 − x)! (n − x)! (N2 − n + x)! µ ¶ N n

n X

(N1 − 2)! N2 ! · (x − 2)! (N − x)! (n − x)! (N 1 2 − n + x)! µ ¶ . N1 (N1 − 1) x=2 N n

In the summation, let k = x − 2, and in the denominator, note that ¶ µ ¶ µ N! N (N − 1) N − 2 N = = . n n!(N − n)! n(n − 1) n − 2 Thus

E[X(X − 1)]

n−2 X

N1 (N1 − 1) N (N − 1) k=0 n(n − 1)

¶µ ¶ N2 N1 − 2 k n−2−k µ ¶ N −2 n−2

N1 (N1 − 1)(n)(n − 1) . N (N − 1)

µ ¶µ ¶ 5 65 1 1 0 5 µ ¶ · = ; 2.5-8 (a) 70 25 302,575,350 5

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Chapter 2 Discrete Distributions

µ ¶µ ¶ 5 65 24 24 5 0 µ ¶ · = ; (b) 70 25 302,575,350 5 µ ¶µ ¶ 5 65 1 325 1 4 µ ¶ · (c) = ; 70 25 302,575,350 5

µ ¶µ ¶ 5 65 7,800 24 1 4 µ ¶ · (d) = ; 70 25 302,575,350 5 µ ¶µ ¶ 5 65 8,259,888 1 0 5 µ ¶ · = . (e) 70 25 302,575,350 5

2.5-10 (a) P (2, 1, 6, 10) means that 2 is in position 1 so 1 cannot be selected. Thus µ ¶µ ¶µ ¶ 1 1 8 4 56 0 1 5 µ ¶ P (2, 1, 6, 10) = = ; = 10 210 15 6 ¶ ¶µ ¶µ µ i−1 1 n−i r−1 1 k−r µ ¶ (b) P (i, r, k, n) = . n k

2.6

The Negative Binomial Distribution

2.6-2

10 − 1 5−1

¶µ ¶5 µ ¶5 1 63 126 1 = . = 2 2 1024 512

2.6-4 Let “being missed” be a success and let X equal the number of trials until the first success. Then p = 0.01. P (X ≤ 50) = 1 − 0.9950 = 1 − 0.605 = 0.395. 2.6-6 (a)

(b)

= ln(1 − p + pet ), ¸ · pet R0 (t) = = p, 1 − p + pet t=0 · ¸ (1 − p + pet )(pet ) − (pet )(pet ) 00 R (t) = = p(1 − p); (1 − p + pet )2 t=0 R(t)

= n ln(1 − p + pet ), ¸ · npet 0 R (t) = = np, 1 − p + pet t=0 ¸ · (1 − p + pet )(pet ) − (pet )(pet ) 00 = np(1 − p); R (t) = n (1 − p + pet )2 t=0 R(t)

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The Poisson Distribution (c)

R(t) R0 (t)

ln p + t − ln[1 − (1 − p)et ], · ¸ 1−p (1 − p)et 1 = 1+ = 1+ = , 1 − (1 − p)et t=0 p p

R00 (t)

R(t)

(d)

R0 (t) R00 (t)

¤ 1−p ; (−1){1 − (1 − p)et }2 {−(1 − p)et } t=0 = p2

r [ln p + t − ln{1 − (1 − p)et }] , ¸ · r 1 = , = r 1 − (1 − p)et t=0 p =

£ ¤ r(1 − p) . r (−1){1 − (1 − p)et }−2 {−(1 − p)et } t=0 = p2

2.6-8 (0.7)(0.7)(0.3) = 0.147.

2.6-10 (a) Let X equal the number of boxes that must be purchased. Then 10 X 1 = 29.29; E(X) = (11 − x)/10 x=1 (b)

2.7

100 · E(X) ≈ 8 years. 365

The Poisson Distribution

2.7-2 λ = µ = σ 2 = 3 so P (X = 2) = 0.423 − 0.199 = 0.224. 2.7-4

λ1 e−λ 1! −λ e λ(λ − 6) 3

λ2 e−λ 2! 0

= =

λ = 6 Thus P (X = 4) = 0.285 − 0.151 = 0.134. 2.7-6

M (t)

eλ(e −1) ,

M 0 (t)

eλ(e −1) λet ,

M 00 (t)

eλ(e −1) λ2 e2t + eλ(e −1) λet ,

M 000 (t)

E(X 3 )

eλ(e −1) λ3 e3t + eλ(e −1) λ2 · 2e2t + eλ(e −1) λ2 e2t + eλ(e −1) λet ,

M 000 (0) = λ3 + 3λ2 + λ,

√ [λ3 + 3λ2 + λ − λ3 − 3λ(λ + λ2 ) + 3λ2 · λ]/λ3/2 = 1/ λ.

Thus the index of skewness is positive. As λ increases, the index of skewness deceases to zero. 2.7-8 np = 1000(0.005) = 5, (a) P (X ≤ 1) ≈ 0.040;

(b) P (X = 4, 5, 6) = P (X ≤ 6) − P (X ≤ 3) ≈ 0.762 − 0.265 = 0.497. √ 2.7-10 σ = 9 = 3, P (3 < X < 15) = P (X ≤ 14) − P (X ≤ 3) = 0.959 − 0.021 = 0.938. 2.7-12 Since E(X) = 0.2, the expected loss is (0.2)($100,000) = $2,000.

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Chapter 3 Continuous Distributions

Chapter 3

Continuous Distributions 3.1

Random Variables of the Continuous Type R1

c ⇒ c = 6; 6 £ ¤0.6 R 0.6 (b) P (0.3 < X < 0.6) = 0.3 6(x − x2 ) dx = 3x2 − 2x3 0.3 = 0.432; Rπ (c) 0 0.50 6(x − x2 ) dx = 1/2 ⇒ π0.50 = 1/2.

3.1–2 (a)

c · (x − x2 ) dx =

3.1–4 X is U (4, 5);

(b) σ 2 = 1/12; (c) 0.5. Z 200 Z n 1 1 dx + [n − 5(x − n)] dx 3.1–6 E(profit) = [x − 0.5(n − x)] 200 200 n 0 · ¸n · ¸200 1 1 x2 (n − x)2 5x2 + = + 6nx − 200 2 4 200 2 n 0 ¤ 1 £ = −3.25n2 + 1200n − 100000 200 1 derivative = [−6.5n + 1200] = 0 200 1200 n = ≈ 185. 6.5 Z c 3.1–8 (a) (i) x3 /4 dx = 1 (a) µ = 9/2;

c4 /16

c (ii) F (x)

= 2; Z x f (t) dt −∞

Z x

t3 /4 dt

=  0,    x4 /16, F (x) =    1,

x4 /16, −∞ < x < 0, 0 ≤ x < 2, 2 ≤ x < ∞;

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Section 3.1 Random Variables of the Continuous Type

(iii) F(x)

f(x) 2.0

1.0

1.5

0.75

1.0

0.5

0.25

0.4

0.8

1.2

1.6

2.0

0.4

0.8

1.2

1.6

2.0

Figure 3.1–8: (a) Continuous distribution pdf and cdf

(iv)

Z 2

(x)(x3 /4) dx =

8 ; 5

(x2 )(x3 /4) dx =

8 ; 3

E(X 2 )

Z 2 0

(b) (i)

Z c

σ2

µ ¶2 8 8 8 − ; = 3 5 75 R2 3 3 √ √ x x /4 dx − 3 · (8/5) · (8/75) − (8/5)3 3 8 75 0 = − = − 1.04978; 70 (8/75)3/2

(3/16)x2 dx =

c3 /8

c Z x

−c

(ii) F (x)

f (t) dt

−∞

Z x

(3/16)t2 dt

−2

· 3 ¸x t 16 −2

x3 1 + , 16 2

 0,      x3 1 + , F (x) =  16 2     1,

−∞ < x < −2, −2 ≤ x < 2, 2 ≤ x < ∞;

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Chapter 3 Continuous Distributions

(iii)

–2

f(x) 1.0

F(x) 1.0

0.8

0.6

0.4

0.2

–1

–2

–1

Figure 3.1–8: (b) Continuous distribution pdf and cdf

(iv)

µ σ2

= =

Z 2

(x)(3/16)(x2 ) dx = 0;

−2 Z 2 −2

−2

(x2 )(3/16)(x2 ) dx =

x3 (3/16)x2 dx − 3 · (0) · (12/5) − 03

Z 1

c √ dx = 1 x 2c = 1

12 ; 5

(12/5)3/2

= 0;

1/2.

The pdf in part (c) is unbounded. Z x f (t) dt (ii) F (x) = −∞

Z x

1 √ dt 0 2 t h√ ix √ t = x,

=  0,    √ x, F (x) =    1,

−∞ < x < 0,

0 ≤ x < 1, 1 ≤ x < ∞;

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Section 3.1 Random Variables of the Continuous Type (iii)

f(x) 2.0

–0.2

F(x) 1.0

1.5

0.75

1.0

0.5

0.25

0.2

0.4

0.6

0.8

x 1.2

0.2

0.4

0.6

0.8

1.0

Figure 3.1–8: (c) Continuous distribution pdf and cdf (iv)

3.1–10 (a)

E(X 2 )

σ2

Z ∞

c dx 2 x 1 · ¸∞ −c x 1

(b) E(X) =

Z 1

√ 1 (x)(1/2)/ x dx = ; 3 0 Z 1 √ 1 (x2 )(1/2)/ x dx = ; 5 0 µ ¶2 1 1 4 − = ; 5 3 45 R1 3 √ √ x /(2 x ) dx − 3 · (1/3) · (4/45) − (1/3)3 2 45 0 = = 0.63888. 21 (4/45)3/2

Z ∞

x ∞ dx = [ln x]1 , which is unbounded. x2

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Chapter 3 Continuous Distributions

 0,    (x3 + 1)/2, F (x) =    1,

3.1–12 (a)

−∞ < x < −1, −1 ≤ x < 1, 1 ≤ x < ∞; F(x)

f(x)

1.0

1.4 1.2

0.75

1.0 0.8

0.5

0.6 0.4

0.25

0.2 –1.0

–0.6

–0.2

0.2

0.6

1.0

–1.0 –0.75–0.50–0.25

0.25 0.5 0.75 1.0

Figure 3.1–12: (a) f (x) = (3/2)x2 and F (x) = (x3 + 1)/2

 0,    (x + 1)/2, F (x) =    1,

(b)

–1.0

–0.6

−∞ < x < −1, −1 ≤ x < 1, 1 ≤ x < ∞;

f(x) 1.0

F(x) 1.0

0.8

0.6

0.4

0.2

–0.2

0.2

0.6

1.0

–1.0

–0.6

–0.2

Figure 3.1–12: (b) f (x) = 1/2 and F (x) = (x + 1)/2

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0.2

0.6

1.0

Section 3.1 Random Variables of the Continuous Type (c)

F (x) =

f(x) 1.0

–1.0

–0.6

 0,       (x + 1)2 /2,

−∞ < x < −1, −1 ≤ x < 0,

  1 − (1 − x) /2,     1, 2

0 ≤ x < 1, 1 ≤ x < ∞; F(x) 1.0

0.8

0.6

0.4

0.2

–0.2

0.2

0.6

1.0

–1.0

–0.6

–0.2

Figure 3.1–12: (c) f (x) and F (x) for Exercise 3.1-12(c) 3.1–14 (a) σ 2 = (b) (c)

x2 (3/2)x2 dx = 3/5 = 0.6; −1 R1 σ 2 = −1 x2 (1/2) dx = 1/3 = 0.333; R0 R1 σ 2 = −1 x2 (x + 1) dx + 0 x2 (1 − x) dx = 1/6 = 0.1667.

3.1–16 F (x) = (x + 1)2 /4,

−1 < x < 1.

(a) F (π0.64 ) = (π0.64 + 1)2 /4 π0.64 + 1

0.64 √ 2.56

π0.64

0.6;

(b) (π0.25 + 1) /4

π0.25 + 1

0.25 √ 1.00

π0.25

π0.81 + 1

0.81 √ 3.24

π0.81

0.8.

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0.2

0.6

1.0

Chapter 3 Continuous Distributions

3.1–18 (b)

   0,    x   ,    2   1 F (x) = ,  2     x 1   −   2 2    1,

−∞ < x ≤ 0, 0 < x ≤ 1, 1 < x ≤ 2, 2 ≤ x < 3, 3 ≤ x < ∞;

f(x) 1.0

F(x) 1.0

0.8

0.6

0.4

0.2

0.2 0.5

1.0

1.5

2.0

2.5

3.0

0.5

1.0

1.5

Figure 3.1–18: f (x) and F (x) for Exercise 3.1-18(a) and (b) (c)

q1 2 q1

0.25

0.5;

(d) 1 ≤ m ≤ 2, not unique; (e)

3.1–20 (a)

q3 1 − 2 2 q3 2

0.75

5 4 5 . q3 = 2 Z 1 Z ∞ c x dx + dx = 1 x3 0 1 · 2 ¸1 h x c i∞ = 1 − 2 0 2x2 1 1 c + = 1 2 2 =

c (b) E(X) =

Z 1 0

x2 dx +

Z ∞ 1

1 4 dx = ; 2 x 3

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2.0

2.5

3.0

Section 3.2 The Exponential, Gamma, and Chi-Square Distributions

(d) P (1/2 ≤ X ≤ 2) =

Z 1

x dx +

1/2

Z 2 1

1 3 dx = . x3 4

(e) Since the median of this distribution is 1, π.82 must be greater than 1. Thus solve £ ¤π.82 Rπ 2 0.82 − 0.5 = 1 .82 (1/x3 ) dx = −(1/2x2 ) 1 = [1 − (1/π.82 )]/2 2 ⇒ 1 − (1/π.82 )

3.2

2 0.64 ⇒ 1/π.82 = 0.36 ⇒ π.82 = 1/(0.6) = 5/3.

The Exponential, Gamma, and Chi-Square Distributions

µ ¶ 2 −2x/3 3.2–2 (a) f (x) = e , 0 ≤ x < ∞; 3 Z ∞ h i∞ 2 −2x/3 (b) P (X > 2) = e dx = −e−2x/3 = e−4/3 . 3 2 2 3.2–4 Let F (x) = P (X ≤ x). Then P (X > x + y | X > x)

P (X > y)

1 − F (x + y) = 1 − F (y). 1 − F (x) That is, with g(x) = 1 − F (x), g(x + y) = g(x)g(y). This functional equation implies that 1 − F (x) = g(x) = acx = e(cx) ln a = ebx where b = c ln a. That is, F (x) = 1 − ebx . Since F (∞) = 1, b must be negative, say b = −λ with λ > 0. Thus F (x) = 1 − e−λx , 0 ≤ x, the distribution function of an exponential distribution. Z ∞ 3 −3x/100 e dx 3.2–6 (a) P (X > 40) = 100 40 £ ¤∞ = −e−3x/100 40 = e−1.2 ; (b) Flaws occur randomly so we are observing a Poisson process. 3.2–8 Either use integration by parts or F (x)

P (X ≤ x)

1−

α−1 X k=0

(λx)k e−λx . k!

Thus, with λ = 1/θ = 1/4 and α = 2, µ ¶ 5 −5/4 −5/4 e P (X < 5) = 1 − e − 4 = 0.35536. 3.2–10 The moment generating function of X is M (t) = (1 − θt)−α , t < 1/θ. Thus M 0 (t)

αθ(1 − θt)−α−1

M 00 (t)

α(α + 1)θ 2 (1 − θt)−α−2 .

The mean and variance are

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Chapter 3 Continuous Distributions

= M 0 (0) = αθ

σ2

= M 00 (0) − (αθ)2 = α(α + 1)θ 2 − (αθ)2 = αθ 2 .

3.2–12 (a) W has a gamma distribution with α = 7, θ = 1/16. (b) Using Table III in the Appendix, P (W ≤ 0.5)

= =

1−

6 X 8k e−8

k=0

1 − 0.313 = 0.687,

because here λw = (16)(0.5) = 8. 3.2–14 a = 5.226, b = 21.03. 3.2–16 Note that λ = 5/10 = 1/2 is the mean number of arrivals per minute. Thus θ = 2 and the pdf of the waiting time before the eighth toll is 1 f (x) = x8−1 e−x/2 Γ(8)28 1 µ ¶ x16/2−1 e−x/2 , 0 < x < ∞, = 16 16/2 Γ 2 2 the pdf of a chi-square distribution with r = 16 degrees of freedom. Using Table IV, P (X > 26.30) = 0.05. 3.2–18 Note that X = 80 + Y where Y has a gamma distribution with shape parameter 2 and scale parameter 50. (a) E(X) = 80 + E(Y ) = 80 + 2 · 50 = 180 and Var(X) = Var(Y ) = 2 · 502 = 5000;

(b) The same calculus argument shows that the median of the gamma distribution in general is (α − 1)θ. So the median is X is 80 + (1)50 = 130; (c) P (X < 200) = P (Y < 120) = 0.6916 using tables or computer software or calculus: · ¸200 Z 200 x − 80 −(x−80)/50 x − 80 −(x−80)/50 −(x−80)/50 e dx = − e −e 502 50 80 80

3.2–20

E[v(T )]

= = = =

Z 3

−120 −120/50 e − e−120/50 + 1 50

1−

100(23−t − 1)e−t/5 /5dt

−20e−t/5 dt + 100

Z 3

17 −12/5 e = 0.6916 = 69.16%. 5

e(3−t) ln 2 e−t/5 /5dt

−100(1 − e−0.6 ) + 100e3 ln 2

Z 3

e−t ln 2 e−t/5 /5dt

−100(1 − e−0.6 ) + 100e3 ln 2 −

e−(ln 2+0.2)t ln 2 + 0.2

¸3 0

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= $121.734.

Section 3.3 The Normal Distribution M 0 (t)

M 00 (t)

M 000 (t)

3.2–22

r , (1 − 2t)(r/2)+1 r(r + 2) , (1 − 2t)(r/2)+2 r(r + 2)(r + 4) ⇒ E(X 3 ) = M 000 (0) = r(r + 2)(r + 4). (1 − 2t)(r/2)+3

Thus E[(X − µ)3 ] = E(X 3 ) − 3µσ 2 − µ3 = r(r + 2)(r + 4) − 3(r)(2r) − r 3 = 8r, so 8r γ= = (2r)3/2

8 , r

which decreases as r increases.

3.3

The Normal Distribution

3.3–2 (a) 0.2823:

(b) 0.4931;

(d) 0.0536;

(e) 0.1038;

(f ) 0.3174;

(g) 0.0456;

(h) 0.0026.

3.3–4 (a) 1.282;

(b)−1.645;

(d) −1.82.

3.3–6 M (t) = e166t+400t /2 , − ∞ < t < ∞, so (a) µ = 166;

(b) σ 2 = 400;

(c) P (170 < X < 200) = P (0.2 < Z < 1.7) = 0.3761; (d) P (148 ≤ X ≤ 172) = P (−0.9 ≤ Z ≤ 0.3) = 0.4338. 3.3–8 We must solve f 00 (x) = 0. We have √ ln f (x) = − ln( 2π σ) − (x − µ)2 /2σ 2 , f 0 (x) f (x)

f (x)f 00 (x) − [f 0 (x)]2 [f (x)]2 f 00 (x) (x − µ)2 σ4 x−µ

−2(x − µ) 2σ 2

−1 σ2 ½ · 0 ¸¾ −1 f (x) + = 0 = f (x) σ2 f (x) 1 = σ2 = ±σ or x = µ ± σ. =

Here is an alternative solution: 2 1 φ(z) = √ e−z /2 2π φ0 (z) = −zφ(z) φ00 (z)

φ (z)

−φ(z) + z 2 φ(z)

0 ⇒ −1 + z 2 = 0 ⇒ z = ±1,

the inflection points. Now let

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Chapter 3 Continuous Distributions

f (x)

f 0 (x)

f 00 (x)

¸ ¶ · µ 1 1 (x − µ)2 x−µ √ exp − = φ 2σ 2 σ σ σ 2π ¶ µ 1 0 x−µ φ σ2 σ µ ¶ 1 00 x − µ φ σ3 σ µ ¶2 x−µ x−µ 0 ⇒ −1 + =0 ⇒ = ±1. σ σ

Thus the points of inflection are at x = µ ± σ. 3.3–10

G(y)

P (Y ≤ y) = P (aX + b ≤ y) µ ¶ y−b = P X≤ if a > 0 a Z (y−b)/a 2 2 1 √ e−(x−µ) /2σ dx = σ 2π −∞

Let w = ax + b so dw = a dx. Then Z y 2 2 2 1 √ e−(w−b−aµ) /2a σ dw G(y) = −∞ aσ 2π which is the distribution function of the normal distribution N (b + aµ, a2 σ 2 ). The case when a < 0 can be handled similarly. 3.3–12 X is N (500, 2500); so [(X − 500)/50]2 is χ2 (1) and " # µ ¶2 X − 500 P 2.706 ≤ ≤ 5.024 = 0.975 − 0.900 = 0.075. 50 3.3–14

G(x)

g(x)

P (X ≤ x)

P (eY ≤ x)

P (Y ≤ ln x)

Z ln x

2 1 1 G0 (x) = √ e−(ln x−10) /2 , x 2π

2 1 √ e−(y−10) /2 dy = Φ(ln x − 10) 2π −∞

P (10,000 < X < 20,000)

0 < x < ∞.

P (ln 10,000 < Y < ln 20,000)

Φ(ln 20,000 − 10) − Φ(ln 10,000 − 10)

0.461557 − 0.214863 = 0.246694.

3.3–16 (a) N (0, 1); (b) N (−1, 1); (c) N (2, 1). Note the slopes of these graphs at 0.

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Section 3.4 Additional Models

3.4

Additional Models

3.4–2 With b = ln 1.1, G(w)

h a w ln 1.1 a i 1 − exp − e + ln 1.1 ln 1.1

G(64) − G(63)

0.01

0.00002646 =

P (W ≤ 71 | 70 < W )

P (70 < W ≤ 71) P (70 < W )

0.0217.

3.4–4

λ(w) H(w)

using Maple

aebw + c Rw = 0 (aebt + c) dt

¢ a ¡ bw e − 1 + cw b h a¡ i ¢ 1 − exp − ebw − 1 − cw , b

3.4–6

1 37792.19477

G(w)

g(w)

0<w<∞

a − (ebw − 1) − cw (aebw + c)e b ,

(a) 1/4 − 1/8 = 1/8;

(b) 1/4 − 1/4 = 0;

(d) 1 − 1/2 = 1/2;

(e) 3/4 − 3/4 = 0;

(f ) 1 − 3/4 = 1/4.

0 < w < ∞.

3.4–8 There is a discrete point of probability at x = 0, P (X = 0) = 1/3, and F 0 (x) = (2/3)e−x for 0 < x. Thus Z ∞ µ = E(X) = (0)(1/3) + x(2/3)e−x dx 0

= 2

E(X )

(2/3) [−xe

−x

(0) (1/3) +

∞

− e−x ]0

Z ∞

= 2/3,

x2 (2/3)e−x dx

= so σ2

3.4–10

T =

= 4/3, (2/3)[−x2 e−x − 2xe−x − 2e−x ]∞ 0

Var(X) = 4/3 − (2/3)2 = 8/9. (

X, 4,

X ≤ 4,

4 < X;

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Chapter 3 Continuous Distributions

E(T )

Z 4 µ ¶ Z ∞ µ ¶ 1 −x/5 1 −x/5 x e dx + 4 e dx 5 5 0 4 £ ¤4 £ ¤∞ −xe−x/5 − 5e−x/5 0 + 4 −e−x/5 4

= = =

5 − 4e−4/5 − 5e−4/5 + 4e−4/5

5 − 5e−4/5 ≈ 2.753.

3.4–12 (a)

(b)

t =

ln x

x =

dx dt

g(t)

f (et )

α + β ln w

dt dw

β w

h(w)

eα+β ln w e−e

βwβ−1 eα e−w e ,

3.4–14 (a) (0.03)

Z 1

2/30

t dx = et e−e , dt

α+β ln w

µ ¶ β w

β α

0 < w < ∞.

6(1 − x)5 dx = 0.0198;

(b) E(X) = (0.97)(0) + 0.03

Z 1 0

3.4–16

−∞ < t < ∞;

x6(1 − x)5 dx = 0.0042857;

The expected payment is E(X) · [$ 30,000] = $ 128.57. Z 2 Z 1 1 −x/10 1 −x/10 e dx + (m/2) e dx = 200 m 10 10 1 0 m[1 − e−1/10 ] + 0.5m[e−1/10 − e−2/10 ]

200

m − 0.5me−1/10 − 0.5me−2/10

200

= =

200 1 − 0.5e−1/10 − 0.5e−2/10 1447.

Z ∞ µ ¶3 4 4 t 3.4–18 P (X > x) = e−(t/4) dt = e−(x/4) ; 4 x

P (X > 5) e−625/256 = = e−369/256 . P (X > 4) e−1 Z 60 h i 2 60 2x −(x/50)2 dx = −e−(x/50) = e−16/25 − e−36/25 ; 3.4–20 (a) e 2 50 40 40 i h 2 ∞ (b) P (X > 80) = −e−(x/50) = e−64/25 . P (X > 5 | X > 4) =

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Chapter 4 Bivariate Distributions

Chapter 4

Bivariate Distributions 4.1

Bivariate Distributions of the Discrete Type

4.1–2

4 16

1 • 16

4 16

1 • 16

4 16

1 • 16

4 16

1 • 16

4 16

(e) Independent, because fX (x)fY (y) = f (x, y). 4.1–4

1 25

1 • 25

1 25

1 • 25

2 25

1 • 25

2 25

1 • 25

3 25

1 • 25

2 25

1 • 25

3 25

1 • 25

2 25

1 • 25

3 25

1 • 25

2 25

1 • 25

2 25

1 • 25

1 25

1 • 25

1 25

1 • 25

0 1 5

2 1 5

4 1 5

1 5

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8 1 5

Section 4.2 The Correlation Coefficient (c) Not independent, because fX (x)fY (y) 6= f (x, y) and also because the support is not rectangular. 4.1–6

25! (0.30)7 (0.40)8 (0.20)6 (0.10)4 = 0.00405. 7!8!6!4!

4.1–8 (a) x = 0, 1, . . . , 6, y = max(0, x − 2), . . . , min(x, 4);   1/15, 2/15, fX (x) =  3/15, fY (y) = 3/15,

x = 0, 6, x = 1, 5, x = 2, 3, 4; y = 0, 1, 2, 3, 4;

(b) x = 0, 1, 2, 3, 4, y = max(0, x − 2), . . . , x;   1/12, 2/12, fX (x) =  3/12,   3/12, 2/12, fY (y) =  1/12,

x = 0, x = 1, x = 2, 3, 4; y = 0, 1, 2, y = 3, y = 4;

(c) x = 0, 1, 2, 3, 4, y = max(2 − x, x − 2), . . . , min(2 + x, 6 − x);   1/13, 3/13, fX (x) =  5/13,   1/13, 3/13, fY (y) =  5/13,

4.1–10 (a) f (x, y) =

y = 0, 4, y = 1, 3, y = 2.

7! (0.78)x (0.01)y (0.21)7−x−y , x!y!(7 − x − y)!

(b) X is b(7, 0.78),

4.2

x = 0, 4, x = 1, 3, x = 2;

0 ≤ x + y ≤ 7;

x = 0, 1, . . . , 7.

The Correlation Coefficient

4.2–2 (c)

µX

0.5(0) + 0.5(1) = 0.5,

µY

0.2(0) + 0.6(1) + 0.2(2) = 1,

2 σX

(0 − 0.5)2 (0.5) + (1 − 0.5)2 (0.5) = 0.25,

σY2

(0 − 1)2 (0.2) + (1 − 1)2 (0.6) + (2 − 1)2 (0.2) = 0.4,

Cov(X, Y )

(0)(0)(0.2) + (1)(2)(0.2) + (0)(1)(0.3) +

(1)(1)(0.3) − (0.5)(1) = 0.2, √ 0.2 √ ρ = √ = 0.4; 0.25 0.4 Ã√ ! √ 0.4 (d) y = 1 + 0.4 √ (x − 0.5) = 0.6 + 0.8x. 0.25 4.2–4 Note that X is b(3, 1/6), Y is b(3, 1/2) so

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Chapter 4 Bivariate Distributions

(a) E(X) = 3(1/6) = 1/2; (b) E(Y ) = 3(1/2) = 3/2; (c) Var(X) = 3(1/6)(5/6) = 5/12; (d) Var(Y ) = 3(1/2)(1/2) = 3/4; (e) Cov(X, Y )

0 + (1)f (1, 1) + 2f (1, 2) + 2f (2, 1) − (1/2)(3/2)

(1)(1/6) + 2(1/8) + 2(1/24) − 3/4

−1/4;

−1/4 −1 (f ) ρ = r =√ . 5 5 3 · 12 4 4.2–6 (b)

1 6

• 16

2 6

• 16

3 6

• 16

• 61

3 6

2 6

1 6

µ ¶ µ ¶µ ¶ 2 2 1 4 −5 1 − = − = ; (c) Cov(X, Y ) = (1)(1) 6 3 3 6 9 18 µ ¶2 2 2 4 5 2 + − = σY2 , (d) σX = = 6 6 3 9 −5/18 1 ρ = p = − ; 2 (5/9)(5/9) s µ ¶ 2 1 5/9 2 (e) y = − x− 3 2 5/9 3 1 y = 1 − x. 2 2 4.2–8 (a) µX = 3, µY = 2, σX = 8/3, σY2 = 2, Cov(X, Y ) = 2, ρ =

(b) µX =

√ 3/2 = 0.886;

227 2 227 19 2 181 181 29 , µY = ,σ = ,σ = , Cov(X, Y ) = ,ρ= = 0.7974; 12 12 X 144 Y 144 144 227

2 = 14/13, σY2 = 14/13, Cov(X, Y ) = 0, ρ = 0. (c) µX = 2, µY = 2, σX

4.2–10 (a) fX (1) = 0.15, fX (2) = 0.25, fX (3) = 0.45, fX (4) = 0.15; fY (1) = 0.35, fY (2) = 0.65; µX = 2.60; µY = 1.65; 2 σX = 0.8400; σY2 = 0.2275;

(b) Cov(X, Y ) = −0.0900; ρ = −0.2059;

= = =

E{[(X − µX ) + v(Y − µY )]2 }

E[(X − µX )2 + 2E[(X − µX )(Y − µY ) + E[(Y − µY )2 ]

2 σX + 2Cov(X, Y )v + σY2 v 2 ≥ 0.

Thus the discriminant of this quadratic must be less than or equal to 0. So we have

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Section 4.3 Conditional Distributions 2 2 [Cov(X, Y )]2 − σX σY ρ2 −1

4.3

≤ 0 ≤ 1 ≤ ρ ≤ 1.

Conditional Distributions

4.3–2 2

1 4

3 4

g(x | 2)

3 4

1 4

g(x | 1)

equivalently, g(x | y) =

3 − 2|x − y| , 4

x = 1, 2, for y = 1 or 2;

h(y | 1)

h(y | 2)

1 4

3 4

1 4

equivalently, h(y | x) =

3 − 2|x − y| , 4

y = 1, 2, for x = 1 or 2;

µX |1 = 5/4, µX |2 = 7/4, µY |1 = 5/4, µY |2 = 7/4; 2 2 2 2 σX |1 = σX |2 = σY |1 = σY |2 = 3/16.

4.3–4 (a) X is b(400, 0.75); (b) E(X) = 300, Var(X) = 75; (c) b(300, 2/3); (d) E(Y | X = 300) = 200, Var(Y | X = 300) = 200/3. 4.3–6 (a) P (X = 500) = 0.40, P (Y = 500) = 0.35, P (Y = 500 | X = 500) = 0.50, P (Y = 100 | X = 500) = 0.25;

2 (b) µX = 485, µY = 510, σX = 118,275, σY2 = 130,900;

(c) µX |Y =100 = 2400/7, µY |X =500 = 525; (d) Cov(X, Y ) = 49650; (e) ρ = 0.399. 4.3–8 (a) X and Y have a trinomial distribution with n = 30, pX = 1/6, pY = 1/6; (b) The conditional pmf of X, given Y = y, is µ

pX b n − y, 1 − pY

= b(30 − y, 1/5).

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Chapter 4 Bivariate Distributions

(c) Since E(X) = 5 and Var(X) = 25/6, E(X 2 ) = Var(X) + [E(X)]2 = 25/6 + 25 = 175/6. Similarly, E(Y ) = 5, Var(Y ) = 25/6, E(Y 2 ) = 175/6. The correlation coefficient is ρ=− so E(XY ) = −1/5 Thus

(1/6)(1/6) = −1/5 (5/6)(5/6)

(25/6)(25/6) + (5)(5) = 145/6.

µ ¶ µ ¶ 145 175 120 175 −4 +3 = = 20. E(X − 4XY + 3Y ) = 6 6 6 6 2

4.3–10 (a) f (x, y) = 1/[10(10 − x)], (b) fY (y) =

y X

x = 0, 1, · · · , 9, y = x, x + 1, · · · , 9;

1 , 10(10 − x) x=0

4.4

y = 0, 1, . . . , 9; x = 0, 1, . . . , 9.

Bivariate Distributions of the Continuous Type

4.4–2 (a) fX (x)

Z 1

(x + y) dy ¸1 · 1 1 0 ≤ x ≤ 1; = xy + y 2 = x + , 2 2 0 Z 1 1 fY (y) = 0 ≤ y ≤ 1; (x + y) dx = y + , 0 ¶µ 2 ¶ µ 1 1 y+ = fX (x)fY (y); f (x, y) = x + y 6= x + 2 2 ¶ · ¸1 Z 1 µ 1 1 3 1 2 7 (b) (i) µX = x x+ dx = x + x ; = 2 3 4 12 0 0 ¶ Z 1 µ 7 1 ; dy = (ii) µY = y y+ 2 12 0 ¸1 ¶ · Z 1 µ 5 1 1 4 1 3 2 2 = (iii) E(X ) = x x+ x + x , dx = 2 4 6 12 0 0 µ ¶2 5 7 11 2 = E(X 2 ) − µ2X = σX − ; = 12 12 144 0

(iv)

4.4–4 (a)

Similarly, σY2 =

0 ≤ X ≤ 12

11 . 144

Z 21 Z 1 0

Z 12 0

3 dy dx x2 2

3 11 (1 − x2 ) dx = ; 2 16

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Section 4.4 Bivariate Distributions of the Continuous Type

(b)

(c)

¡1

2 ≤Y

≤1

Z 1 Z √y 1 2

3 dx dy 2

µ ¶3/2 1 3√ ; y dy = 1 − = 1 2 2 2 Z 1 Z √y ¡ ¢ 3 1 1 dx dy P X ≥ 2, Y ≥ 2 = 1 1 2 2 2 ¶ Z 1 µ 3 √ 1 = y− dy 1 2 2 2 µ ¶3/2 5 1 ; = − 8 2 Z 1

(d) X and Y are dependent. 4.4–6 (a) The variances are · 5 ¸1 3 2 3 3x x · x dx = = σX = E(X ) − 0 = 2 10 −1 5 −1 2

Z 1

and σY2 = E(Y 2 ) − 02

−1 Z 0

y 2 (1 − |y|) dy 2

y (1 + y) dy +

−1

Z 1 0

y 2 (1 − y) dy

1 1 1 1 1 − + − = ; 3 4 3 4 6 Z 1Z 1 Z 0Z 1 3 2 3 2 x (1 − y) dxdy + x (1 + y) dxdy −y 2 0 −1 −y 2 =

(b)

P (−X ≤ Y )

1 1 1 1 1 1 1 1 1 − + − + − − + = . 2 4 8 10 2 4 8 10 2 Z 1 Z 0 15 2 15 2 x (1 − x2 ) dx + x· x dx x· 7 7 0 −1 · ¸0 · ¸1 15 x4 x6 15 x4 − + 7 4 6 −1 7 4 0 · ¸ · ¸ 15 −1 1 15 1 5 + = + , 7 4 6 7 4 14 Z 1 Z 0 15 2 2 2 15 2 x (1 − x ) dx + x2 · x dx x · 7 7 0 −1 · ¸0 · ¸1 x7 15 x5 15 x5 − + 7 5 7 −1 7 5 0 · ¸ · ¸ 15 1 27 15 1 1 − , + = 7 5 7 7 5 49 µ ¶2 27 5 83 − ; = 49 14 196 =

4.4–8

µX = E(X)

= = =

E(X 2 )

= = 2 σX

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Chapter 4 Bivariate Distributions

µY = E(Y )

= = =

E(Y 2 )

= = =

σY2

Z 1

10 (y + y 4 ) dy 7 0 · ¸1 y6 10 y 3 + 7 3 6 0 · ¸ 5 10 1 1 = + 7 3 6 7 Z 1 10 (y + y 4 ) dy y2 · 7 0 · ¸1 y7 10 y 4 + 7 4 7 0 · ¸ 10 1 1 55 + = 7 4 7 98 µ ¶2 5 5 55 = − . 98 7 98 y·

4.4–10 The area of the space is Z 6 Z 14−2t2 2

dt1 dt2 =

Z 6 2

(13 − 2t2 ) dt2 = 20;

Thus P (T1 + T2 > 10)

Z 4 Z 14−2t2

10−t2

1 dt1 dt2 20

Z 4

4 − t2 dt2 20 2 ¸4 · 1 (4 − t2 )2 = . − 40 10 2

= =

4.4–12 E[a1 u1 (X1 , X2 ) + a2 u2 (X1 , X2 )] X X = [a1 u1 (x1 , x2 ) + a2 u2 (x1 , x2 )]f (x1 , x2 ) (x1 ,x2 ) ∈R

X X

u1 (x1 , x2 )f (x1 , x2 ) + a2

(x1 , x2 ) ∈R

4.4–14 (a)

fX (x)

a1 E[u1 (X1 , X2 )] + a2 E[u2 (X1 , X2 )].

Z 1 x

fY (y)

Z y 0

8xy dy = 4x(1 − x2 ), 8xy dx = 4y 3 ,

X X

0 ≤ x ≤ 1,

0 ≤ y ≤ 1;

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u2 (x1 , x2 )f (x1 , x2 )

Section 4.4 Bivariate Distributions of the Continuous Type

(b)

µX

Z 1

µY

Z 1

2 σX

Z 1

σY2

Z 1

Cov(X, Y )

(x − 8/15)2 4x(1 − x2 ) dx = (y − 4/5)2 ∗ 4y 3 dy = Z 1Z 1 x

20 4x + . 33 11

Z 1 0

11 , 225

2 , 75

(x − 8/15)(y − 4/5)8xy dy dx =

√ 2 66 p = ; 33 (11/225)(2/75)

2 1 1 , µY = , and E(Y 2 ) = ; 3 3 2 1 1 2 , σX = − 6 6

2x2 (1 − x) dx =

Cov(X, Y ) =

Z 1Z 1 0

2xy dy dx −

µ ¶2 µ ¶2 1 1 1 1 2 = = , σY2 = − ; 3 18 2 3 18

µ ¶µ ¶ 1 2 1 2 1 − = , = 3 3 4 9 36

so ρ= p 4.4–18 (b)

 Z x   1/8 dy    0      Z x 1/8 dy fX (x) =  x−2     Z 4     1/8 dy 

x/8,

0 ≤ x ≤ 2,

1/4,

2 < x < 4,

(6 − x)/8,

4 ≤ x ≤ 6;

1/8 dx = 1/4,

(d)

h(y | x) =

      

1/36 1 p = . 2 1/18 1/18

x−2

Z y+2

0 ≤ y ≤ 4;

1/x,

0 ≤ y ≤ x,

0 ≤ x ≤ 2,

1/2,

x − 2 < y < x,

2 < x < 4,

1/(6 − x),

x − 2 ≤ y ≤ 4,

4 ≤ x ≤ 6;

(e) g(x | y) = 1/2,

4 , 225

4/225

4.4–16 From Example 4.4–3, µX =

E(X 2 ) =

4 , 5

y ∗ 4y 3 dy =

8 , 15

x4x(1 − x2 ) dx =

y ≤ x ≤ y + 2,

0 ≤ y ≤ 4;

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Chapter 4 Bivariate Distributions (f )

 Z x µ ¶ 1   y dy   x  0      Z x 1 y · dy E(Y | x) = 2  x−2     Z  4  y    dy x−2 6 − x

(g) E(X | y) = 4

Z y+2 y

x·

= = =

x , 2 · 2 ¸x y = x − 1, 4 x−2 ¸4 · x+2 y2 = , 2(6 − x) x−2 2

· 2 ¸y+2 x 1 dx = = y + 1, 2 4 y 4

E(Y | x) = Thus

Z 1

4 ≤ x < 6;

(i) x = E(X | y)

(x + 1)y x+1 dy =

¸1 x + 1 x+2 x+1 = y , x+2 x+2 0

E(Y | X) =

= E[E(Y | X)] =

Z 2 1

1 < x < 2.

X +1 , X +2

and E(Y )

2 < x < 4,

0 ≤ y ≤ 4.

Figure 4.4–18: (h) y = E(Y | x) 4.4–20

0 ≤ x ≤ 2,

x+1 dx = x+2

Z 2µ 1

1 1− x+2

= [x − ln(x + 2)]1 = 2 − ln 4 − (1 − ln 3) = 1 − ln(4/3) = 0.7123.

4.5

The Bivariate Normal Distribution

4.5–2

q(x, y)

= =

(x − µX )2 [y − µY − ρ(σY /σX )(x − µX )]2 + 2 σY2 (1 − ρ2 ) σX · 2 2ρ(x − µX )(y − µY ) 1 (y − µY ) − 2 2 1−ρ σY σX σY

¸ 2 ρ2 (x − µX )2 2 (x − µX ) + (1 − ρ ) 2 2 σX σX "µ ¶2 µ ¶µ ¶ µ ¶2 # x − µX x − µX y − µY y − µY 1 + − 2ρ 1 − ρ2 σX σX σY σY +

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Section 4.5 The Bivariate Normal Distribution µ ¶ 5 13 (76 − 70) = 83; 13 10 " µ ¶2 # 5 (b) Var(Y | X = 76) = 169 1 − = 144; 13 ¶ µ 86 − 83 = Φ(0.25) = 0.5987. (c) P (Y ≤ 86 | X = 76) = P Z ≤ 12

4.5–4 (a) E(Y | X = 76) = 80 +

4.5–6 (a) P (19.9 < Y < 26.9) = Φ(1.2) − Φ(−0.8) = 0.8849 − 0.2119 = 0.6730; Note that using an electronic device, you may get 0.88493 − 0.21186 = 0.67307; (b) E(Y | x) = 22.7 + 0.78(3.5/4.2)(x − 22.7) = 0.65x + 7.945; (c) Var(Y | x) = 12.25(1 − 0.782 ) = 4.7971; (d) P (19.9 < Y < 26.9 | X = 23) = Φ(1.8286) − Φ(−1.3674) = 0.9663 − 0.0857 = 0.8806; (e) P (19.9 < Y < 26.9 | X = 25) = Φ(1.2350) − Φ(−1.9610) = 0.8914 − 0.0249 = 0.8665; (f ) 0.16 0.12 0.08 0.04 20

26 22 y

Figure 4.5–6: Conditional pdfs of Y , given x = 21, 23, 25

4.5–8 (a) P (13.6 < Y < 17.8) = Φ(0.20) − Φ(−0.85) = 0.3816; (b) E(Y | x) = 17 + 0(4/3)(x − 8) = 17; (c) Var(Y | x) = 16(1 − 02 ) = 16; (d) P (13.6 < Y < 17.8 | X = 9.1) = 0.3816. 4.5–10 (a) P (2.80 ≤ Y ≤ 5.35) = Φ(1.50) − Φ(0) = 0.4332; µ ¶ 1.7 (b) E(Y | X = 82.3) = 2.80 + (−0.57) (82.3 − 72.30) = 1.877; 10.5 Var(Y | X = 82.3) = 2.89[1 − (−0.57)2 ] = 1.9510; P (2.76 ≤ Y ≤ 5.34 | X = 82.3)

= Φ(2.479) − Φ(0.632) = 0.9934 − 0.7363 = 0.2571.

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Chapter 4 Bivariate Distributions

4.5–12

fX (x)

= = =

Z ∞

Z ∞ 2 2 1 −(x2 +y2 )/2 1 −(x2 +y2 )/2 e dy + e xye−(x +y −2)/2 dy 2π 2π −∞ −∞ Z 1 −x2 /2 ∞ 1 −y2 /2 √ e √ e dy + 0 2π 2π −∞ 2 1 √ e−x /2 , −∞ < x < ∞. 2π

Note that the first integrand is the product of two N (0, 1) pdfs and the integral of a pdf is equal to 1. The second integral is an odd function so it is equal to 0. 2 1 Similarly, fY (y) = √ e−y /2 , −∞ < y < ∞. 2π R∞ R∞ R∞ It also follows that −∞ −∞ f (x, y) dx dy = −∞ fX (x) dx = 1.

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Chapter 5 Distributions of Functions of Random Variables

Chapter 5

Distributions of Functions of Random Variables 5.1

Functions of One Random Variable

5.1–2 The linear transformation is one-to-one with inverse transformation X = (Y − b)/a, so the pdf of Y is g(y) =

5.1–4 Here x =

√

F (x) =

ac1 + b < y < ac2 + b.

1 dx = √ and 0 < x < ∞ maps onto 0 < y < ∞. Thus dy 2 y g(y) =

5.1–6 (a)

1 f [(y − b)/a], |a|

     Z x    

√

¯ ¯ ¯ 1 ¯ 1 y e−y/2 ¯¯ √ ¯¯ = e−y/2 , 2 y 2

x < 0,

2t dt = x2 ,

(b) Let y = x2 ; so x =

0 ≤ x < 1, 1 ≤ x;

1, √

0 < y < ∞.

y. Let Y be U (0, 1); then X =

√

Y has the given x-distribution;

−∞ < x < ∞.

Y = F (X) =

1 1 + e−X

F (x) = By Theorem 5.1-2,

is U (0, 1).

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Section 5.1 Functions of One Random Variable

5.1–10

dx dy

f (x)

g(y)

= =

³ y ´10/7

5 µ ¶ 10 ³ y ´3/7 1 7 5 5

e−x ,

0<x<∞ µ ¶µ ¶3/7 1 2 10/7 y 3/7 e−(y/5) 7 5 10/7 3/7 −(y/5)10/7 , y e 510/7

0 < y < ∞.

(The reason for writing the pdf in that form is because Y has a Weibull distribution with α = 10/7 and β = 5.) 5.1–12 X has cdf F given by

F (x) =

 0        x+1 4        1,

x < −1, −1 ≤ x ≤ 3

3 < x.

The cdf G of Y is given by √ √ G(y) = P (Y ≤ y) = P (− y ≤ X ≤ y).

√ √ Taking cases then gives G(y) = y/2 for 0 ≤ y ≤ 1 and G(y) = ( y + 1)/4 for 1 ≤ y ≤ 9. Taking derivatives then gives the pdf

g(y) =

5.1–14

E(X)

= = =

 1   ,   4√ y    

1 √ , 8 y

0 < y < 1, 1 ≤ y < 9.

Z ∞

x dx π(1 + x2 ) −∞ ¸0 ¸b · · 1 1 lim ln(1 + x2 ) + lim ln(1 + x2 ) a→−∞ 2π b→+∞ 2π a · ¸ 0 1 2 2 lim {− ln(1 + a )} + lim ln(1 + b ) . b→+∞ 2π a→−∞

E(X) does not exist because neither of these limits exists. 5.1–16 The cdf of Y = |X| is given by G(y)

= =

P (Y ≤ y) = P (−y ≤ X ≤ y) Φ(y) − Φ(−y) = 2Φ(y) − 1,

0 < y < ∞,

where Φ is the standard normal cdf. Taking derivatives then gives the pdf 2 2 g(y) = 2φ(y) = √ e−y /2 , 2π

0 < y < ∞.

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Chapter 5 Distributions of Functions of Random Variables

5.1–18 (a) First note that the cdf of X is F (x) = (x2 + 2x + 1)/9 , −1 < x < 2. Then √ √ √ √ G(y) = P (X 2 < y) = P (− y < X < y ) = F ( y ) − F (− y ) ( √ √ √ [(y + 2 y + 1)/9] − [y − 2 y + 1)/9] = 4 y/9, 0 < y < 1, = √ √ (y + 2 y + 1)/9 − 0 = (y + 2 y + 1)/9, 1 ≤ y < 4. 0

Thus g(y) = G (y) =

 2    9√ y ,

1 1    + √ , 9 9 y

0 < y < 1, 1 ≤ y < 4;

(b) First note that the cdf of X is F (x) = (x2 + 4x + 4)/9 , −2 < x < 1. Then √ √ √ √ G(y) = P (X 2 < y) = P (− y < X < y ) = F ( y ) − F (− y ) ( √ √ √ [(y + 4 y + 4)/9] − [(y − 4 y + 4)/9] = 8 y/9, 0 < y < 1, = √ 1 − [(y − 4 y + 4)/9], 1 ≤ y < 4. 0

Thus g(y) = G (y) =

5.2

 4    9√ y ,

1 2    − + √ , 9 9 y

0 < y < 1, 1 ≤ y < 4.

Transformations of Two Random Variables

5.2-2 (a) The joint pdf of X1 and X2 is f (x1 , x2 ) =

1 r /2−1 r2 /2−1 −(x1 +x2 )/2 ³r ´ ³r ´ x11 x2 e , 1 2 (r1 +r2 )/2 Γ Γ 2 2 2 0 < x1 < ∞, 0 < x2 < ∞.

Let Y1 = (X1 /r1 )/(X2 /r2 ) and Y2 = X2 . The Jacobian of the transformation is (r1 /r2 )y2 . Thus g(y1 , y2 ) =

1 ³r ´ ³r ´ 2 1 Γ 2(r1 +r2 )/2 Γ 2 2

r 1 y1 y2 r2

¶r1 /2−1

r /2−1 −(y2 /2)(r1 y1 /r2 +1)

y2 2

0 < y1 < ∞, 0 < y2 < ∞; Z ∞ (b) The marginal pdf of Y1 is g1 (y1 ) = g(y1 , y2 ) dy2 . 0

µ ¶ y2 r 1 y1 + 1 . Then Make the change of variables w = 2 r2 µ ¶µ ¶r1 /2 r1 + r 2 r1 r /2−1 y1 1 Γ 2 r2 0 < y1 < ∞. g1 (y1 ) = ¶(r +r )/2 · 1, ³ r ´ ³ r ´µ r 1 y1 1 2 1 2 Γ Γ 1+ 2 2 r2

5.2-4 (a) F0.05 (9, 24) = 2.30; (b) F0.95 (9, 24) =

1 1 = = 0.3448; F0.05 (24, 9) 2.90

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r 1 y2 r2

Section 5.2 Transformations of Two Random Variables ¶ µ ¶ 1 1 1 =P > > 3.61 = 0.025; W 0.277 W P (0.277 ≤ W ≤ 2.70) = P (W ≤ 2.70) − P (W ≤ 0.277) = 0.975 − 0.025 = 0.95. µ ¶ X1 F (w) = P ≤w , 0<w<1 X1 + X 2 Z ∞Z ∞ xα−1 x2β−1 e−(x1 +x2 )/θ 1 = dx2 dx1 Γ(α)Γ(β)θ α+β (1−w)x1 /w 0 µ ¶ Z ∞ −xα−1 [(1 − w)x1 /w]β−1 e−[x1 +(1−w)x1 /w]/θ −1 1 0 x1 dx1 f (w) = F (w) = Γ(α)Γ(β)θ α+β w2 0 Z e−x1 /θw (1 − w)β−1 ∞ xα+β−1 1 1 dx1 = β+1 α+β Γ(α)Γ(β) w θ 0

5.2-6

5.2-8 (a)

E(X)

Γ(α + β) (θw)α+β (1 − w)β−1 Γ(α + β) wβ+1 θα+β

Γ(α + β) α−1 w (1 − w)β−1 , Γ(α)Γ(β)

Z 1

0 < w < 1.

Γ(α + β) α−1 x (1 − x)β−1 dx Γ(α)Γ(β) Z 1 Γ(α + β)Γ(α + 1) Γ(α + 1 + β) α+1−1 . x (1 − x)β−1 dx Γ(α)Γ(α + β + 1) 0 Γ(α + 1)Γ(β) x

(α)Γ(α)Γ(α + β) (α + β)Γ(α + β)Γ(α) α = ; α+β Z Γ(α + β)Γ(α + 2) 1 Γ(α + 2 + β) α+2−1 2 x (1 − x)β−1 dx E(X ) = Γ(α)Γ(α + 2 + β) 0 Γ(α + 2)Γ(β) =

(α + 1)α . (α + β + 1)(α + β)

α(α + 1) α2 αβ − = ; 2 (α + β + 1)(α + β) (α + β) (α + β + 1)(α + β)2

Thus σ2 (b)

f (x)

f 0 (x)

Γ(α + β) α−1 x (1 − x)β−1 Γ(α)Γ(β) ¤ Γ(α + β) £ (α − 1) xα−2 (1 − x)β−1 − (β − 1) xα−1 (1 − x)β−2 . Γ(α)Γ(β)

Set f 0 (x) equal to zero and solve for x:

Γ(α + β) α−2 x (1 − x)β−2 [(α − 1)(1 − x) − (β − 1) x] = 0 Γ(α)Γ(β) α − αx − 1 + x − βx + x = 0 (α + β − 2)x

x =

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α−1

α−1 . α+β−2

Chapter 5 Distributions of Functions of Random Variables

5.2-10 Use integration by parts two times to show ·µ ¶ µ ¶ µ ¶ ¸p Z p 6 4 6! 3 6 5 6 6 2 2 1 0 y (1 − y) dy = y (1 − y) + y (1 − y) + y (1 − y) 4 5 6 0 3!2! 0 =

6 µ ¶ X n y=4

5.2-12 (a) w1 = 2x1 and

py (1 − p)6−y .

dw1 = 2. Thus dx1 f (x1 ) =

2 , π(1 + 4x21 )

−∞ < x1 < ∞;

(b) For x2 = y1 − y2 , x1 = y2 , | J | = 1. Thus

Z ∞

g(y1 , y2 ) = f (y2 )f (y1 − y2 ),

−∞

(d)

g1 (y1 )

= = = = = = =

−∞ < yi < ∞, i = 1, 2;

f (y2 )f (y1 − y2 ) dy2 ;

Z ∞ 2 2 · dy = h(y2 ) dy2 2 2 π[1 + 4(y1 − y2 )2 ] −∞ −∞ π[1 + 4y2 ] Z ∞ 1 4 1 · dy2 π 2 −∞ [1 + 2iy2 ][1 − 2iy2 ] [1 + 2i(y1 − y2 )][1 − 2i(y1 − y2 )] Z 4 ∞ 1 1 1 −1 1 −1 −1 1 · · · · · · · dy2 π 2 −∞ 2i y2 − 2i 2i y2 + 2i 2i y2 − (y1 − 2i ) 2i y2 − (y1 + 2i ) · µ ¶ µ ¶¸ i i 4(2πi) Res h(y2 ); y2 = + Res h(y2 ); y2 = y1 + π2 2 2 · ¸ 1 1 1 1 1 8πi 1 1 · · + · · π 2 16 i i − y1 −y1 y1 y1 + i i · · ¸ ¸ 1 1 1 1 1 1 y1 + i + y 1 − i · + · = 2π y1 y1 − i y1 + i 2π y1 (y1 − i)(y1 + i)

Z ∞

1 . π(1 + y12 )

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Section 5.2 Transformations of Two Random Variables A Maple solution for Exercise 5.2-12: >f := x-> 2/Pi/(1 + 4*x^2); f := x− > 2

1 π (1 + 4 x2 )

>simplify(int(f(y[2])*f(y[1]-y[2]),y[2]=-infinity..infinity)); 1 π (1 + y12 ) A Mathematica solution for Exercise 5.2-12: In[1]:= f[x_] := 2/(Pi*(1 + 4(x)^2)) g[y1_,y2_] := f[y2]*f[y1-y2] In[3]:= Integrate[g[y1,y2], {y2, -Infinity,Infinity}] Out[3]= 1 _____________ 2 Pi + Pi y1 5.2-14 The joint pdf is x −(x+y)/5 e , 53

h(x, y) = z

x , y zw,

0 < x < ∞, 0 < y < ∞;

The Jacobian is J

The joint pdf of Z and W is zw f (z, w) = 3 e−(z+1)w/5 w, 5

¯ ¯ w ¯ ¯ ¯ 0

¯ z ¯¯ ¯ = w. 1 ¯

0 < z < ∞, 0 < w < ∞;

The marginal pdf of Z is fZ (z)

= = =

Z ∞

zw −(z+1)w/5 e w dw 53 0 µ ¶3 Z ∞ 5 Γ(3)z w3−1 e−w/(5/[z+1]) dw 3 5 z+1 Γ(3)(5/[z + 1])3 0

2z , (z + 1)3

0 < z < ∞.

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Chapter 5 Distributions of Functions of Random Variables

5.2-16 Y = 1/W is a one-to-one transformation from SW = {w : w > 0} to SY = {y : y > 0}, with dw 1 inverse W = g −1 (Y ) = 1/Y and derivative (of the inverse transformation) = − 2. dy y Thus, the pdf of Y is ¯ ¯ ¯ dw ¯ ¯ fY (y) = fW [g −1 (y)] ¯¯ dy ¯ ¯ ¯ Γ[(r1 + r2 )/2] (r1 /r2 )r1 /2 (1/y)r1 /2−1 ¯¯ 1 ¯¯ − = Γ(r1 /2)Γ(r2 /2) [1 + (r1 /r2 )(1/y)](r1 +r2 )/2 ¯ y 2 ¯ Ã ! r /2 Γ[(r1 + r2 )/2] (1/y)r1 /2−1 r11 = (1/y 2 ) r /2 (r1 +r2 )/2 Γ(r1 /2)Γ(r2 /2) r 1 [(r + r y)/(r y)] 1 2 2 2 Ã ! ! Ã r /2 r1 /2 r2 /2 (r1 +r2 )/2 r11 r r y Γ[(r1 + r2 )/2] 2 2 y −1−r1 /2 = Γ(r1 /2)Γ(r2 /2) rr1 /2 (r + r2 y)(r1 +r2 )/2 1 2 Ã ! r /2 Γ[(r1 + r2 )/2] r1 /2 (−1−r1 /2)+(r1 +r2 )/2 r22 = r y r /2 r /2 Γ(r1 /2)Γ(r2 /2) 1 [1 + (r2 /r1 )y](r1 +r2 )/2 r 1 r 2 1

r2 /2 r2 /2−1

Γ[(r1 + r2 )/2] (r2 /r1 ) y , Γ(r1 /2)Γ(r2 /2) [1 + (r2 /r1 )y](r1 +r2 )/2

y > 0,

which is the pdf of an F (r2 , r1 ) distribution. You could also note by Theorem 5.2-1 that the F (r1 , r2 ) distribution of W is the distribution of (U/r1 )/(V /r2 ) where U and V are independent chi-square variables with r1 and r2 degrees of freedom, respectively. It follows that 1/W is the distribution of (V /r 2 )/(U/r1 ). By the same theorem, this is the F (r2 , r1 ) distribution. 5.2-17 The pdf of W is f (w) =

(r1 /r2 )r1 /2 Γ[(r1 + r2 )/2]wr1 /2−1 . Γ(r1 /2)Γ(r2 /2)[1 + (r1 w/r2 )](r1 +r2 )/2

Let α = r1 /2 and β = r2 /2. Then the pdf becomes f (w) =

(α/β)α Γ[(α + β]w α−1 . Γ(α)Γ(β)[1 + αw/β](α+β)

We are given z

dw dz

1 1 + αw/β ¶ µ µ ¶ β 1−z β 1 −1 = α z α z −

β 1 α z2

It follows that the pdf of Z is g(z)

µ ¶α · µ ¶¸α−1 µ ¶ β 1 α β 1−z z α+β β α z α z2

Γ(α + β) Γ(α)Γ(β)

Γ(α + β) (1 − z)α−1 z β−1 , Γ(α)Γ(β)

0 < z < 1.

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Section 5.3 Several Random Variables

5.3

Several Random Variables

5.3–2 (a)

P (X1 = 2, X2 = 4)

3! 2!1!

µ ¶2 µ ¶1 #" µ ¶4 µ ¶1 # 5! 1 1 1 1 2 2 4!1! 2 2

15 15 = ; 8 2 256 (b) {X1 + X2 = 7} can occur in the two mutually exclusive ways: {X1 = 3, X2 = 4} and {X1 = 2, X2 = 5}. The sum of the probabilities of the two latter events is =

" µZ 1.0

5.3–4 (a)

µ ¶3 #" µ ¶5 # " µ ¶3 #" µ ¶5 # 5! 1 3! 1 5! 1 3! 1 5+3 1 + = = . 3!0! 2 4!1! 2 2!1! 2 5!0! 2 28 32

2e−2x1 dx1

0.5

¶µZ 1.2

2e−2x2 dx2

0.7

(e−1 − e−2 )(e−1.4 − e−2.4 )

(0.368 − 0.135)(0.247 − 0.091)

(0.233)(0.156) = 0.036;

(b) E(X1 ) = E(X2 ) = 0.5,

5.3–6

E[X1 (X2 − 0.5)2 ] = E(X1 )Var(X2 ) = (0.5)(0.25) = 0.125. · µ ¶ ¸1 Z 1 Z 1 1 3 4 2 3 3 = ; x E(X) = x6x(1 − x )dx = (6x − 6x ) dx = 2x − 2 2 0 0 0 ·µ ¶ µ ¶ ¸1 Z 1 3 4 3 6 5 E(X 2 ) = (6x3 − 6x4 ) dx = = x − x . 2 5 10 0 0 Thus

µX

µY

3 1 1 1 2 = ; σX − = , and 2 10 4 20 1 1 1 1 1 + = 1; σY2 = + = . 2 2 20 20 10

5.3–8 Let Y = max(X1 , X2 ). Then G(y)

= = =

[P (X ≤ y)]2 ·Z y

¸2 4 dx 5 1 x ¸2 · 1 1− 4 , y

1<y<∞

G0 (y) ¶µ ¶ µ 4 1 , 1 < y < ∞; = 2 1− 4 y y5 µ ¶µ ¶ Z ∞ 1 4 E(Y ) = y·2 1− 4 dy y y5 1 Z ∞ = 8 [y −4 − y −8 ] dy g(y)

32 21

(in 1000s of dollars).

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Chapter 5 Distributions of Functions of Random Variables

µ ¶"µ ¶µ ¶2 #µ ¶ 27 3 1 3 3 = 5.3–10 (a) P (X1 = 1)P (X2 = 3)P (X3 = 1) = ; 4 4 4 4 1024 (b) 3P (X1 = 3, X2 = 1, X3 = 1) + 3P (X1 = 2, X2 = 2, X3 = 1) = ¶ µ ¶ µ 162 27 27 +3 = ; 3 1024 1024 1024 ¶3 µ ¶3 µ 15 3 3 1 = . (c) P (Y ≤ 2) = + · 4 4 4 16 µZ ∞ ¶3 −x 3 5.3–12 P (1 < min Xi ) = [P (1 < Xi )] = e dx = e−3 = 0.05. 1

5.3–14

P (Y > 1000)

= P (X1 > 1000)P (X2 > 1000)P (X3 > 1000) = e−1 e−2/3 e−1/2 = e−13/6 = 0.1146.

5.3–16 Denote the three lifetimes by X1 , X2 , X3 and let Y = X1 + X2 + X3 . E(Y ) = E(X1 + X2 + X3 ) = E(X1 ) + E(X2 ) + E(X3 ) = 3 · 2 · 2 = 12. Var(X1 + X2 + X3 ) = Var(X1 ) + Var(X2 ) + Var(X3 ) = 3 · 2 · 22 = 24. 5.3–18

Cov(W, V ) σW σV

E(W V ) − µW µV σW σV

E(X 2 )E(Y ) − E(X)E(Y )E(X) σXY σX

= = = 5.3–20 Var( X )

p p

2 + µ2X )(µY ) − µ2X µY (σX

E(X 2 Y 2 ) − [E(X)E(Y )]2 σX µY σ X

2 + µ2 )(σ 2 + µ2 ) − µ2 µ2 (σX X Y Y X Y µY σ X p . 2 σ 2 + σ 2 µ2 + σ 2 µ2 σX Y X Y Y X

n µ ¶2 X 1 i=1

σ +2

X X µ 1 ¶µ 1 ¶ i<j

σ2 + ρσ = n 2

2 n2

¶µ

n(n − 1) 2

ρσ 2

σ [1 + (n − 1)ρ]. n

Since 1 + (n − 1)ρ > 1, this variance is larger than the variance of X when the variables are uncorrelated.

5.4

The Moment-Generating Function Technique

5.4–2

MY (t)

E[et(X1 +X2 ) ] = E[etX1 ]E[etX2 ]

(q + pet )n1 (q + pet )n2 = (q + pet )n1 +n2 ,

−∞ < t < ∞.

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Section 5.4 The Moment-Generating Function Technique Thus Y is b(n1 + n2 , p). 5.4–4

E[et(X1 +···+Xn ) ]

n Y

E[etXi ] =

n Y

eµi (e −1)

i=1

(µ1 +µ2 +···+µn )(et −1)

= e , −∞ < t < ∞, the moment generating function of a Poisson random variable with mean µ1 + µ 2 + · · · + µ n . 5.4–6 (a)

E[etY ]

= E[et(X1 +X2 +X3 +X4 +X5 ) ] = E[etX1 etX2 etX3 etX4 etX5 ] = E[etX1 ]E[etX2 ]E[etX3 ]E[etX4 ]E[etX5 ] (1/3)et (1/3)et (1/3)et · · · 1 − (2/3)et 1 − (2/3)et 1 − (2/3)et ¸ · 5 (1/3)et = 1 − (2/3)et

[(1/3)et ]5 , [1 − (2/3)et ]5

t < − ln(1 − 1/3);

(b) So Y has a negative binomial distribution with p = 1/3 and r = 5. 5.4–8

E[etW ]

E[et(X1 +X2 +···+Xh ) ] = E[etX1 ]E[etX2 ] · · · E[etXh ]

[1/(1 − θt)]h = 1/(1 − θt)h ,

t < 1/θ,

the moment generating function for the gamma distribution with mean hθ. 5.4–10 (a) E[etX ] = (1/4)(e0t + e1t + e2t + e3t ), tY

(b) E[e ] = (1/4)(e + e + e + e (c)

E[e

]

t(X+Y )

−∞ < t < ∞;

E[e

E[etX ]E[etY ]

(1/16)(e0t + e1t + e2t + e3t )(e0t + e4t + e8t + e12t )

(1/16)(e0t + e1t + e2t + e3t + · · · e15t ),

(d) P (W = w) = 1/16,

]

12t

w = 0, 1, 2, . . . , 15.

−∞ < t < ∞;

5.4–12 First die has 0 on four faces and 2 on four faces; second die has faces numbered 0, 1, 4, 5, 8, 9, 12, 13. 5.4–14 Let X1 , X2 , X3 be the number of accidents in weeks 1, 2, and 3, respectively. Then Y = X1 + X2 + X3 is Poisson with mean λ = 6 and P (Y = 7) = 0.744 − 0.606 = 0.138. 5.4–16 Let X1 , X2 , X3 , X4 be the number of sick days for employee i, i = 1, 2, 3, 4, respectively. Then Y = X1 + X2 + X3 + X4 is Poisson with mean λ = 8 and P (Y > 10) = 1 − P (Y ≤ 10) = 1 − 0.816 = 0.184. 5.4–18 Let Xi equal the number of cracks in mile i, i = 1, 2, . . . , 40. Then Y =

40 X

is Poisson with mean

λ = 20.

i=1

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Chapter 5 Distributions of Functions of Random Variables It follows that P (Y < 15) = P (Y ≤ 14) =

14 X 20y e−20 y=0

= 0.1049.

5.4–20 Y = X1 + X2 + X3 + X4 has a gamma distribution with α = 6 and θ = 10. So Z ∞ 1 y 6−1 e−y/10 dy = 1 − 0.8843 = 0.1157. P (Y > 90) = 6 90 Γ(6)10 5.4–22 (a)

E(etY )

1 (1 − 2t)r/2

E(etX2 )

E(etX1 etX2 )

E(etX1 )E(etX2 )

= =

(b) X2 is χ2 (r−r1 ).

1 E(etX2 ) (1 − 2t)r1 /2 1 ; (1 − 2t)(r−r1 )/2

5.4–24 (a) MX (t) =

1 [1 − 2(t/4)]r/2

¶4

1 , [1 − (1/2)t]2r

t < 2;

(b) Gamma with α = 2r and θ = 1/2.

5.5

Random Functions Associated with Normal Distributions

5.5–2 0.4

0.3 n = 36 0.2 n =9 0.1 n=1 40

Figure 5.5–2: X is N (50, 36), X is N (50, 36/n), n = 9, 36 5.5–4 (a) P (X < 6.0171) = P (Z < −1.645) = 0.05;

(b) Let W equal the number of boxes that weigh less than 6.0171 pounds. Then W is b(9, 0.05) and P (W ≤ 2) = 0.9916; µ ¶ 6.035 − 6.05 (c) P ( X ≤ 6.035) = P Z ≤ 0.02/3 =

P (Z ≤ −2.25) = 0.0122.

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Section 5.5 Random Functions Associated with Normal Distributions 5.5–6 (a) X is N (µ, 4/100) and Y is N (µ, 9/n). Equating 4/100 to 9/n yields n = 225; (b) Y − X is N [µ − µ, (4/100) + (9/225)], i.e, N (0, 0.08). So µ

0.2 P ( Y − X > 0.2) = P Z > √ 0.08

= P (Z > 0.71) = 0.2389.

5.5–8 X − Y is N (184.09 − 171.93, 39.37 + 50.88); µ

X − Y − 12.16 0 − 12.16 √ P (X > Y ) = P > 9.5 90.25

= P (Z > −1.28) = 0.8997.

5.5–10 Let Y = X1 + X2 + · · · + Xn . Then Y is N (800n, 1002 n). Thus

P (Y ≥ 10000) = 0.90 ¶ 10000 − 800n Y − 800n √ √ = 0.90 ≥ 100 n 100 n −1.282

10000 − 800n √ 100 n

√ 800n − 128.2 n − 10000

√ Either use the quadratic formula to solve for n or use Maple to solve for n. We find that √ n = 3.617 or n = 13.08 so use n = 14 bulbs. 5.5–12 (a) The joint pdf of X1 and X2 is 2 1 1 r/2−1 −x2 /2 x e , f (x1 , x2 ) = √ e−x1 /2 r/2 2 Γ(r/2)2 2π p y1 = x1 / x2 /r, y 2 = x2

The Jacobian is J

y2 /r,

¯ p ¯ y2 /r ¯ ¯ ¯ 0

The joint pdf of Y1 and Y2 is

−∞ < x1 < ∞, 0 < x2 < ∞;

−1/2

y1 ( 12 )y2

√ ¯ / r ¯¯ p ¯ = y2 /r. ¯

√ 1 1 −y12 y2 /2r r/2−1 −y2 /2 y2 √ , y e g(y1 , y2 ) = √ e r Γ(r/2)2r/2 2 2π

−∞ < y1 < ∞, 0 < y2 < ∞;

(b) The marginal pdf of Y1 is g1 (y1 )

= =

Z ∞

√ y2 2 1 1 r/2−1 −y2 /2 √ e−y1 y2 /2r √ dy2 y e 2 r/2 r Γ(r/2)2 2π 0 Z ∞ 1 Γ[(r + 1)/2] (r+1)/2−1 −(y2 /2)(1+y12 /r) √ . y2 e (r+1)/2 πr Γ(r/2) 0 Γ[(r + 1)/2]2

Let u = y2 (1 + y12 /r). Then y2 =

dy2 1 u and = . So 2 1 + y1 /r du 1 + y12 /r

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Chapter 5 Distributions of Functions of Random Variables

g1 (y1 )

= =

Z ∞

√

Γ[(r + 1)/2] πr Γ(r/2)(1 + y12 /r)(r+1)/2

√

Γ[(r + 1)/2] , πr Γ(r/2)(1 + y12 /r)(r+1)/2

1 u(r+1)/2−1 e−u/2 Γ[(r + 1)/2]2(r+1)/2 −∞ < y1 < ∞.

5.5–14 Because Z is N (0, 1), E(Z) = 1 and E(Z 2 ) = 1. U is χ2 (r) so it follows that Z ∞ √ 1 1 √ E[1/ U ] = ur/2−1 e−u/2 du r/2 u Γ(r/2)2 0 Z Γ[(r − 1)/2] ∞ 1 √ = u(r−1)/2−1 e−u/2 du 2 Γ(r/2) 0 Γ[(r − 1)/2]2(r−1)/2 Γ[(r − 1)/2] √ . 2 Γ(r/2)

Note that the last integral is equal to one because the integrand is the pdf of a χ 2 (r −1) random variable. To find E(1/U ) we have Z ∞ 1 1 ur/2−1 e−u/2 du E[1/U ] = r/2 u Γ(r/2)2 0 Z Γ[(r − 2)/2] ∞ 1 u(r−2)/2−1 e−u/2 du = 2 Γ(r/2) Γ[(r − 2)/2]2(r−2)/2 0 =

Γ(r/2 − 1) 1 = . 2Γ(r/2 − 1) (r/2 − 1) r−2

Note that the last integral is equal to one because the integrand is the pdf of a χ 2 (r −2) random variable. " # Z E(T ) = E p U/r p = E[Z]E[1/ U/r]

Var(T )

5.5–16 T =

·√

rΓ[(r − 1)/2] √ 2 Γ(r/2)

E(T 2 ) − 02

E[Z 2 ]E[r/U ]

r , r−2

= 0

provided r ≥ 2;

provided r ≥ 3.

X −µ √ is t with r = 9 − 1 = 8 degrees of freedom. S/ 9

(a) t0.025 (8) = 2.306.

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Section 5.6 The Central Limit Theorem

−t0.025

(b)

S −t0.025 √ n S −X − t0.025 √ n S X − t0.025 √ n

5.6

≤

X −µ √ S/ n

≤

t0.025

≤

X −µ

≤

S t0.025 √ n

≤

−µ

≤

S −X + t0.025 √ n S ≤ X + t0.025 √ . n

The Central Limit Theorem

5.6–2 If

(3/2)x2 , −1 < x < 1, Z 1 x(3/2)x2 dx = 0; =

f (x)

E(X)

−1

Var(X)

Z 1

(3/2)x4 dx =

−1

Thus P (−0.3 ≤ Y ≤ 1.5)

5.6–4

P (39.75 ≤ X ≤ 41.25)

5.6–6 (a)

(b)

¸1 3 5 3 x = . 10 5 −1

−0.3 − 0 Y −0 1.5 − 0 p ≤p ≤p 15(3/5) 15(3/5) 15(3/5)

≈

P (−0.10 ≤ Z ≤ 0.50) = 0.2313.

≈

P (−0.50 ≤ Z ≤ 2.50) = 0.6853.

39.75 − 40 X − 40 41.25 − 40 p ≤p ≤ p (8/32) (8/32) (8/32)

¸2 · 2 4 x3 2 x = 2− − = ; 2 6 0 3 3 0 µ ¶2 Z 2 2 σ2 = x2 (1 − x/2) dx − 3 0 ¸ · 3 2 x4 2 x 4 − = ; = − 3 8 0 9 9   µ ¶ 2 2 5 2 2 X− 5 2 3 − 3 6 − 3  P = Pq ≤q 3 ≤q ≤X≤ 3 6 2 2 2 /18 /18 /18 µ

Z 2

x(1 − x/2) dx =

≈

P (0 ≤ Z ≤ 1.5) = 0.4332.

5.6–8 (a) E(X) = µ = 24.43; (b) Var(X) = (c)

σ2 2.20 = = 0.0733; n 30

P (24.17 ≤ X ≤ 24.82)

≈ P

24.82 − 24.43 24.17 − 24.43 √ ≤Z≤ √ 0.0733 0.0733

= P (−0.96 ≤ Z < 1.44) = 0.7566. c 2020 Pearson Education, Inc. Copyright °

Chapter 5 Distributions of Functions of Random Variables 5.6–10

E(X + Y )

30 + 50 = 80;

Var(X + Y )

2 σX + σY2 + 2ρσX σY

52 + 64 + 28 = 144;

25 X

(Xi + Yi ) is approximately N (25 · 80, 25 · 144). ¶ µ Z − 2000 2090 − 2000 1970 − 2000 < < Thus P (1970 < Z < 2090) = P 60 60 60 Z

i=1

≈

Φ(1.5) − Φ(−0.5)

0.9332 − 0.3085 = 0.6247.

5.6–12 Let Xi equal the time between sales of ticket i − P 1 and i, for i = 1, 2, . . . , 10. Each Xi has 10 a gamma distribution with α = 3, θ = 2. Y = i=1 Xi has a gamma distribution with parameters αY = 30, θY = 2. (a) Thus P (Y ≤ 60) =

Z 60 0

1 y 30−1 e−y/2 dy = 0.52428 using Maple; Γ(30)230

(b) The normal approximation is given by µ ¶ Y − 60 60 − 60 P √ ≈ Φ(0) = 0.5000. ≤ √ 120 120 5.6–14 We are given that Y =

P20

i=1 Xi has mean 200 and variance 80. We want to find y so that

P (Y > y) = P

Y − 200 y − 200 √ > √ 80 80

< 0.20.

We have that

5.7

y − 200 √ 80

0.842

207.5 ↑ 208 days.

Approximations for Discrete Distributions

5.7–2 (a) P (2 < X < 9) = 0.9532 − 0.0982 = 0.8550; ! Ã X − 25(0.2) 8.5 − 5 2.5 − 5 ≤p (b) P (2 < X < 9) = P ≤ 2 2 25(0.2)(0.8) ≈

P (−1.25 ≤ Z ≤ 1.75)

= 5.7–4

P (35 ≤ X ≤ 40)

≈ =

0.8543. µ ¶ 34.5 − 36 40.5 − 36 P ≤Z≤ 3 3

P (−0.50 ≤ Z ≤ 1.50) = 0.6247.

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Section 5.7 Approximations for Discrete Distributions 5.7–6 µX = 84(0.7) = 58.8, Var(X) = 84(0.7)(0.3) = 17.64, µ ¶ 52.5 − 58.8 P (X ≤ 52.5) ≈ Φ = Φ(−1.5) = 0.0668. 4.2 µ ¶ X − 21.37 20.857 − 21.37 5.7–8 (a) P (X < 20.857) = P < 0.4 0.4 =

P (Z < −1.282) = 0.10;

(b) The distribution of Y is b(100, 0.10). Thus ! Ã 5.5 − 10 Y − 100(0.10) ≤ ≈ P (Z ≤ −1.50) = 0.0668; P (Y ≤ 5) = P p 3 100(0.10)(0.90) µ ¶ 21.39 − 21.37 21.31 − 21.37 (c) P (21.31 ≤ X ≤ 21.39) ≈ P ≤Z≤ 0.4/10 0.4/10 = P (−1.50 ≤ Z ≤ 0.50) = 0.6247. 5.7–10 The distribution of Y is b(1000, 18/38). Thus Ã ! 500.5 − 1000(18/38) P (Y > 500) ≈ P Z ≥ p = P (Z ≥ 1.698) = 0.0448. 1000(18/38)(20/38)

5.7–12 (a) E(X) = 100(0.1) = 10, Var(X) = 9, µ ¶ µ ¶ 14.5 − 10 11.5 − 10 P (11.5 < X < 14.5) ≈ Φ −Φ 3 3 = Φ(1.5) − Φ(0.5) = 0.9332 − 0.6915 = 0.2417; (b) P (X ≤ 14) − P (X ≤ 11) = 0.917 − 0.697 = 0.220; ¶ 14 µ X 100 (0.1)x (0.9)100−x = 0.2244. (c) x x=12

5.7–14 (a) E(Y ) = 24(3.5) = 84, Var(Y ) = 24(35/12) = 70, µ ¶ 85.5 − 84 √ P (Y ≥ 85.5) ≈ 1 − Φ = 1 − Φ(0.18) = 0.4286; 70 (b) P (Y < 85.5) ≈ 1 − 0.4286 = 0.5714;

5.7–16 (a) See Figure 5.7-16. (b) (i) When p = 0.1,

¶ µ ¶ 1.5 −1.5 P (−1.5 < Y − 10 < 1.5) ≈ Φ −Φ = 0.6915 − 0.3085 = 0.3830; 3 3 (ii) When p = 0.5, µ ¶ µ ¶ 1.5 −1.5 P (−1.5 < Y − 50 < 1.5) ≈ Φ −Φ = 0.6179 − 0.3821 = 0.2358; 5 5 (iii) When p = 0.8, µ ¶ µ ¶ −1.5 1.5 P (−1.5 < Y − 80 < 1.5) ≈ Φ −Φ = 0.6462 − 0.3538 = 0.2924. 4 4

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Chapter 5 Distributions of Functions of Random Variables 0.12

0.10

0.08

0.06

0.04

0.02

2 10

90 100

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 5.7–16: Normal approximations of the pdfs of Y and Y /100, p = 0.1, 0.5, 0.8 5.7–18 X is N (0, 0.52 ). The probability that one item exceeds 0.98 in absolute value is P (|X| > 0.98)

1 − P (−0.98 ≤ X ≤ 0.98) µ ¶ X −0 0.98 − 0 −0.98 − 0 ≤ ≤ = 1−P 0.5 0.5 0.5

1 − P (−1.96 ≤ Z ≤ 1.96) = 1 − 0.95 = 0.05

If we let Y equal the number out of 100 that exceed 0.98 in absolute value, Y is b(100, 0.05). (a) Let λ = 100(0.05) = 5. P (Y ≥ 7) = 1 − P (Y ≤ 6) = 1 − 0.762 = 0.238; (b)

P (Y ≥ 7)

Y −5

6.5 − 5 p ≥ 2.179 100(0.05)(0.95)

≈

P (Z ≥ 0.688)

1 − 0.7543 = 0.2447;

5.8

Chebyshev’s Inequality and Convergence in Probability

5.8–2 Var(X) = 298 − 172 = 9. (a)

P (10 < X < 24)

P (10 − 17 < X − 17 < 24 − 17)

P ( |X − 17| < 7) ≥ 1 −

because k = 7/3;

40 9 = , 49 49

9 = 0.035, because k = 16/3. 162 ¯ ¶ µ¯ ¯ ¯ Y (0.5)(0.5) − 0.5¯¯ < 0.08 ≥ 1 − = 0.609; 5.8–4 (a) P ¯¯ 100 100(0.08)2 p because k = 0.08/ (0.5)(0.5)/100; (b) P ( |X − 17| ≥ 16) ≤

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Section 5.9 Limiting Moment-Generating Functions ¯ ¶ µ¯ ¯ ¯ Y ¯ < 0.08 ≥ 1 − (0.5)(0.5) = 0.922; ¯ − 0.5 ¯ ¯ 500 500(0.08)2 p because k = 0.08/ (0.5)(0.5)/500;

(b) P

¯ µ¯ ¶ ¯ ¯ Y (0.5)(0.5) ¯ ¯ − 0.5¯ < 0.08 ≥ 1 − = 0.961, (c) P ¯ 1000 1000(0.08)2 because k = 0.08/

5.8–6

P (75 < X < 85)

because k = 5/

5.9

(0.5)(0.5)/1000.

P (75 − 80 < X − 80 < 85 − 80)

P ( |X − 80| < 5) ≥ 1 −

60/15 = 0.84, 25

60/15 = 5/2.

Limiting Moment-Generating Functions

5.9–2 Using Table III with λ = np = 400(0.005) = 2, P (X ≤ 2) = 0.677. 5.9–4 Let Y =

n X

Xi , where X1 , X2 , . . . , Xn are mutually independent χ2 (1) random variables.

i=1

Then µ = E(Xi ) = 1 and σ 2 = Var(Xi ) = 2, i = 1, 2, . . . , n. Hence Y − nµ Y −n √ = √ 2 2n nσ has a limiting distribution that is N (0, 1).

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Chapter 6 Point Estimation

Chapter 6

Point Estimation 6.1

Descriptive Statistics

6.1–2 x = 3.58; s = 0.5116. 6.1–4 (a)

Class Interval (303.5, 307.5) (307.5, 311.5) (311.5, 315.5) (315.5, 319.5) (319.5, 323.5) (323.5, 327.5) (327.5, 331.5) (331.5, 335.5)

Class Limits (304, 307) (308, 311) (312, 315) (316, 319) (320, 323) (324, 327) (328, 331) (332, 335)

Frequency fi 1 5 6 10 11 9 7 1

Class Mark,ui 305.5 309.5 313.5 317.5 321.5 325.5 329.5 333.5

(b) x = 320.1, s = 6.7499; (c)

h(x) 0.06 0.05 0.04 0.03 0.02 0.01

303.5 307.5 311.5 315.5 319.5 323.5 327.5 331.5 335.5

Figure 6.1–4: Melting points of metal alloys There are 31 observations within one standard deviation of the mean (62%) and 48 observations within two standard deviations of the mean (96%).

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Section 6.1 Descriptive Statistics 6.1–6 (a) With the class boundaries 0.5, 5.5, 17.5, 38.5, 163.5, 549.5, the respective frequencies are 11, 9, 10, 10, 10; h(x)

(b)

0.04 0.03 0.02 0.01

100

200

300

400

500

Figure 6.1–6: Mobil home losses (c) This is a skewed to the right distribution. 6.1–8 (a) With the class boundaries 3.5005, 3.5505, 3.6005, . . . , 4.1005, the respective class frequencies are 4, 7, 24, 23, 7, 4, 3, 9, 15, 23, 18, 2; (b)

h(x) 3.5 3.0 2.5 2.0 1.5 1.0 0.5 3.6

3.7

3.8

3.9

4.0

4.1

Figure 6.1–8: Weights of mirror parts (c) This is a bimodal histogram. 6.1–10 (a) 50/204 = 0.245; 93/329 = 0.283; (b) 124/355 = 0.349; 21/58 = 0.362; (c) 174/559 = 0.311; 114/387 = 0.295; (d) Although Player J’s batting average is higher than Player H’s on both grass and artificial turf, Player H’s is higher over all. Note the different numbers of at bats on grass and artificial turf and how this affects the batting averages.

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Chapter 6 Point Estimation

6.2

Exploratory Data Analysis

6.2–2 (a) Baby carrots: 1.02, 1.03, 1.04, 1.04, 1.06; Regular-size carrots: 1.00, 1.15, 1.21, 1.26, 1.43; (b)

1.1

1.2

1.3

1.4

Figure 6.2–2: (b) Box-and-whisker diagrams of small and regular-size carrots (c) Regular-size packages tend to be heavier. 6.2–4 (a) The five-number summary is: min = 1, qe1 = 6.75, m e = 32, qe3 = 90.75, max = 527;

0 100 200 300 400 500 Figure 6.2–4: (a) Box-and-whisker diagram of mobile home losses (b) IQR = 90.75 − 6.75 = 84. The inner fence is at 216.75 and the outer fence is at 342.75;

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Section 6.2 Exploratory Data Analysis (c)

0 100 200 300 400 500 Figure 6.2–4: (c) Box-and-whisker diagram of losses with fences and outliers 6.2–6 (a) Stems 0• 1∗ 1• 2∗ 2• 3∗ 3• 4∗ 4• 5∗ 5•

Leaves 55555555555555555556666666666666677777778888888999999 0000001111111222334 5555666677889 0111133444 5 4 5 5 5

(b) The five-number summary is: min = 5, qe1 = 6, m e = 9, qe3 = 15, max = 55.

20 30 50 10 40 Figure 6.2–6: (b) Box-and-whisker diagram of maximum capital

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Freq 53 19 13 10 1 1 1 0 1 0 1

Chapter 6 Point Estimation

10 20 30 40 50 Figure 6.2–6: (d) Box-and-whisker diagram of maximum capital with outliers and fences (e) The 90th percentile is 22.8. 6.2–8 (a) Stems 101 102 103 104 105 106 107 108 109 110

Leaves 7 000

89 133667788 379 8 139 022

Frequency 1 3 0 0 2 9 3 1 3 3

(Multiply numbers by 10−1 .) Table 6.2–8: Ordered stem-and-leaf diagram of weights of indicator housings

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Section 6.2 Exploratory Data Analysis (b) min = 101.7, qe1 = 106.0, m e = 106.7, qe3 = 108.95, max = 110.2;

(c) The interquartile range in IQR = 108.95 − 106.0 = 2.95. The inner fence is located at 106.0 − 1.5(2.95) = 101.575 so there are no suspected outliers.

102

104

106

108

110

Figure 6.2–8: (b) Weights of indicator housings

6.2–10 (a) With the class boundaries 2.85, 3.85, . . . , 16.85 the respective frequencies are 1, 0, 2, 4, 1, 14, 20, 11, 4, 5, 0, 1, 0, 1;

h(x) 0.30 0.25 0.20 0.15 0.10 0.05 3.85

5.85

7.85

9.85

11.85

13.85

Figure 6.2–10: (a) Lead concentrations

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15.85

Chapter 6 Point Estimation

(b) x = 9.422, s = 2.082; (c)

h(x) 0.30 0.25 0.20 0.15 0.10 0.05 3.85

*5.85

*7.85

*9.85

11.85

13.85

15.85

Figure 6.2–10: (c) Lead concentrations showing x, x ± s, x ± 2s There are 48 (48/64 = 75%) within one standard deviation of the mean and 60 (60/64 = 93.75%) within two standard deviations of the mean; (d) 1976 Leaves

Stems

1 92 9 9943200 98875544443221100000 986632210 766554331100 975320 961 2

2 3 4 5 6 7 8 9 10 11 12 13 14 15 17

1977 Leaves 9

07 3568 3 0112223677788899 1123333444455678889999 22345579 0469 0346 8 7

Multiply numbers by 10−1 Table 6.2–10: (d) Back-to-Back Stem-and-Leaf Diagram of Lead Concentrations

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Section 6.3 Order Statistics (e)

1977

1976

Figure 6.2–10: (e) Box-and-whisker diagrams of 1976 and 1977 lead concentrations

6.3

Order Statistics

6.3–2 (a) The location of the median is (0.5)(17 + 1) = 9, thus the median is m e = 5.2.

The location of the first quartile is (0.25)(17 + 1) = 4.5. Thus the first quartile is qe1 = (0.5)(4.3) + (0.5)(4.7) = 4.5.

The location of the third quartile is (0.75)(17 + 1) = 13.5. Thus the third quartile is qe3 = (0.5)(5.6) + (0.5)(5.7) = 5.65;

(b) The location of the 35th percentile is (0.35)(18) = 6.3. Thus π e0.35 = (0.7)(4.8) + (0.3)(4.9) = 4.83.

The location of the 65th percentile is (0.65)(18) = 11.7. Thus

6.3–4

g(y)

5 ½ X

π e0.65 = (0.3)(5.6) + (0.7)(5.6) = 5.6.

6! (k)[F (y)]k−1 f (y)[1 − F (y)]6−k k!(6 − k)! k=3 ¾ 6! [F (y)]k (6 − k)[1 − F (y)]6−k−1 [−f (y)] + 6[F (y)]5 f (y) + k!(6 − k)! 6! 6! [F (y)]2 f (y)[1 − F (y)]3 − [F (y)]3 [1 − F (y)]2 f (y) 2!3! 3!2! 6! 6! [F (y)]3 f (y)[1 − F (y)]2 − [F (y)]4 [1 − F (y)]1 f (y) + 3!2! 4!1! 6! 6! + [F (y)]4 f (y)[1 − F (y)]1 − [F (y)]5 [1 − F (y)]0 f (y) + 6[F (y)]5 f (y) 4!1! 5!0! 6! [F (y)]2 [1 − F (y)]3 f (y), a < y < b. 2!3! c 2020 Pearson Education, Inc. Copyright °

Chapter 6 Point Estimation 6.3–6 (a)

(b)

F (x)

g1 (w)

n[1 − w]n−1 (1),

gn (w)

E(W1 )

0 < w < 1;

n[w]n−1 (1), 0 < w < 1; Z 1 (w)(n)(1 − w)n−1 dw = =

E(Wn )

0 < x < 1. Thus

¸1 1 1 n+1 (1 − w) . −w(1 − w) − = n+1 n + 1 0 · ¸1 Z 1 n n (w)(n)w n−1 dw = wn+1 = ; n + 1 n + 1 0 0 n

n! [wr ]r−1 [1 − wr ]n−r · 1 (r − 1)!(n − r)!

Γ(r + n − r + 1) r−1 w (1 − wr )n−r+1−1 . Γ(r)Γ(n − r + 1) r

Thus Wr has a beta distribution with α = r, β = n − r + 1. Z 1 n! 2 w2 wr−1 (1 − w)n−r dw 6.3–8 (a) E(Wr ) = (r − 1)!(n − r)! 0 Z 1 r(r + 1) (n + 2)! wr+1 (1 − w)n−r dw = (n + 2)(n + 1) 0 (r + 1)!(n − r)! =

r(r + 1) (n + 2)(n + 1)

since the integrand is like that of a pdf of the (r + 2)th order statistic of a sample of size n + 2 and hence the integral must equal one; (b) Var(Wr ) =

r(r + 1) r2 r(n − r + 1) − = . (n + 2)(n + 1) (n + 1)2 (n + 2)(n + 1)2

6.3–10 g(yi , yj )dyi dyj is the probability that i − 1 of the sample variables are less than yi , one variable is in the infinitesimal interval of length dyi about yi , j − i − 1 variables are between yi and yj , one variable is in the infinitesimal interval of length dyj about yj , and n − j variables are greater than yj . By independence and combinatorics, the probability is n! [F (yi )]i−1 f (yi )dyi [F (yj )−F (yi )]j−1+1 f (yj )dyj [1−F (yj )]n−j . (i − 1)!1!(j − i − 1)!1!(n − j)! Thus, for −∞ < yi < yj < ∞, g(yi , yj ) =

n! [F (yi )]i−1 [F (yj )−F (yi )]j−1+1 [1−F (yj )]n−j f (yi )f (yj ). (i − 1)!(j − i − 1)!(n − j)!

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Section 6.3 Order Statistics 6.3–12

Strengths

p = k/10

z1−p

Strengths

p = k/10

z1−p

1 2 3 4 5

7.2 8.9 9.7 10.5 10.9

0.10 0.20 0.30 0.40 0.50

−1.282 −0.842 −0.524 −0.253 0.000

6 7 8 9

11.7 12.9 13.9 15.3

0.60 0.70 0.80 0.90

0.253 0.524 0.842 1.282

–1

–2

Figure 6.3–12: q-q plot of N (0, 1) quantiles versus data quantiles It seems to be an excellent fit. 6.3–14 (a)

133 134 135 136 137 138 139 Figure 6.3–14: q-q plot of N (0, 1) quantiles versus data quantiles (b) It looks like an excellent fit.

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Chapter 6 Point Estimation

6.4

Maximum Likelihood Estimation

6.4–2 The likelihood function is " n # · ¸n/2 X 1 2 L(θ) = exp − (xi − µ) /2θ , 2πθ i=1

0 < θ < ∞.

The logarithm of the likelihood function is n n 1 X n (xi − µ)2 . ln L(θ) = − (ln 2π) − (ln θ) − 2 2 2θ i=1

Setting the first derivative equal to zero and solving for θ yields d ln L(θ) dθ

Thus

n n 1 X (xi − µ)2 = 0 + 2 2θ 2θ i=1 n 1X (xi − µ)2 . n i=1

−

θb =

1X (Xi − µ)2 . n i=1

To see that θb is an unbiased estimator of θ, note that Ã ! n σ 2 X (Xi − µ)2 σ2 b E(θ) = E = · n = σ2, n i=1 σ2 n

since (Xi − µ)2 /σ 2 is χ2 (1) and hence the expected value of each of the n summands is equal to 1. For the unbiased property, it could also be noted that θb is the sample mean from a random sample of size n from the distribution of (X − µ)2 . The sample mean is always an unbiased estimator of the distribution mean, which in this case is σ 2 . 6.4–4 The joint probability of {3, 2, 3, 1} when θ = −1 is µ ¶µ ¶µ ¶µ ¶ 2 3 1 18 3 = 4. P ({3, 2, 3, 1}; θ = −1) = 6 6 6 6 6 In similar fashion we find that the joint probability of {3, 2, 3, 1} when θ = 0 is µ ¶µ ¶µ ¶µ ¶ 2 2 2 2 16 P ({3, 2, 3, 1}; θ = 0) = = 4, 6 6 6 6 6 and the joint probability of {3, 2, 3, 1} when θ = 1 is µ ¶µ ¶µ ¶µ ¶ 1 2 1 3 6 P ({3, 2, 3, 1}; θ = 1) = = 4. 6 6 6 6 6 Since the likelihood function is largest when θ = −1, the maximum likelihood estimate is θb = −1.

c2 = 5.0980. 6.4–6 θb1 = µ b = 33.4267; θb2 = σ

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Section 6.4 Maximum Likelihood Estimation

6.4–8 (a)

L(θ)

1 θn

¶Ã Y n

i=1

!1/θ−1

0<θ<∞

¶ Y n 1 xi − 1 ln ln L(θ) = −n ln θ + θ i=1 n Y d ln L(θ) −n 1 = − 2 ln xi = 0 dθ θ θ i=1 n 1 Y xi θb = − ln n i=1 n 1X = − ln xi ; n i=1 µ

(b) We first find E(ln X): E(ln X) =

Z 1

ln x(1/θ)x1/θ−1 dx.

Using integration by parts, with u = ln x and dv = (1/θ)x1/θ−1 dx, i1 h E(ln X) = lim x1/θ ln x − θx1/θ = −θ. a→0

Thus

6.4–10 (a) x = 1/p so pe = 1/X = n/

E( θb ) = −

1X (−θ) = θ. n i=1

i=1 Xi ;

(b) pe equals the number of successes, n, divided by the number of Bernoulli trials, Pn i=1 Xi ; (c) 20/252 = 0.0794.

6.4–12 (a) E( X ) = E(Y )/n = np/n = p; (b) Var( X ) = Var(Y )/n2 = np(1 − p)/n2 = p(1 − p)/n; (c)

E[ X(1 − X )]

= = =

[E( X ) − E( X )]

{p − [p2 + p(1 − p)/n]} = [p(1 − 1/n) − p2 (1 − 1/n)] (1 − 1/n)p(1 − p)/n = (n − 1)p(1 − p)/n;

(d) From part (c), the constant c = n/(n − 1). ( · ¸1/2 ) cσ (n − 1)S 2 6.4–14 (a) E(cS) = E √ σ2 n−1 Z ∞ 1/2 (n−1)/2−1 −v/2 v v e cσ µ ¶ dv = √ n − 1 n−1 0 Γ 2(n−1)/2 2 √ cσ 2 Γ(n/2) , = √ n − 1 Γ[(n − 1)/2)] √ n − 1 Γ[(n − 1)/2] √ so c = ; 2 Γ(n/2) √ √ √ (b) When n = 5, c = 8/(3 2π ) and when n = 6, c = 3 5π/(8 2 );

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Chapter 6 Point Estimation (c)

c 1.18 1.16 1.14 1.12 1.10 1.08 1.06 1.04 1.02 1.00 0.98 0.96 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Figure 6.4–14: c as a function of n We see that lim c = 1.

n→∞

6.4–16 The experiment has a hypergeometric distribution with n = 8 and N = 64. From the sample, x = 1.4667. Using this as an estimate for µ we have µ ¶ N1 f1 = 11.73. 1.4667 = 8 implies that N 64 A guess for the value of N1 is therefore 12.

6.4–18 L=

1 2πσ 2

¶kn/2

  µ ¶¸2 k · n X  X  k+1 xij − c − d j − exp − /(2σ 2 ) .   2 i=1 j=1

To find b c and db try to minimize S=

n X k · X i=1 j=1

We have

¶¸2 µ k+1 . xij − c − d j − 2

· µ ¶¸ k n X X k+1 ∂S 2 xij − c − d j − (−1) = 0. = ∂c 2 i=1 j=1

Because

¶ µ k X k+1 = 0, d j− 2 j=1

b c=

i=1

j=1 xij

= x, say.

· µ ¶¸· µ ¶¸ k n X X k+1 k+1 ∂S 2 xij − c − d j − − j− = 0. = ∂d 2 2 i=1 j=1 Thus db =

j=1 (xij − x)(j − {k + 1}/2) = Pn P k 2 j=1 (j − {k + 1}/2) i=1

i=1

j=1 xij (j − {k + 1}/2)

j=1 n(j − {k + 1}/2)

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Section 6.5 A Simple Regression Problem

6.4–20 (a) F (y) =

Z y 0

(1/θ) dx = [(x/θ)]0 = y/θ, so the desired pdf is gn (y) = n(y/θ)n−1 (1/θ) = ny n−1 /θn ,

(b) E(Yn ) =

Z θ

(ny n /θn ) dy =

0 < y < θ;

¸θ nθ ny n+1 ; = n (n + 1)θ 0 n+1

E(Yn2 ) =

Z θ 0

· ¸θ ny n+2 ny n+1 nθ2 dy = , so = θn (n + 2)θ n 0 n+2 "

¶2 # µ 2 nθ nθ = c2 − n+2 n+1 · ¸ n n2 θ2 (n + 1)2 2 θ − ; = = n2 n + 2 (n + 1)2 n(n + 2)

Var(cYn )

(e) Var(2X ) = (4/n)Var(X1 ) = (4/n)(θ 2 /12) = θ 2 /(3n). Since 3n < n(n + 2) for n > 1, Var(2X ) is larger.

6.5

A Simple Regression Problem

¸ (yi − βxi )2 √ 6.5–2 (a) L(β, σ ) = exp − 2σ 2 2πσ 2 i=1 µ ¶n/2 ¸ · Pn 2 1 i=1 (yi − βxi ) = . exp − 2πσ 2 2σ 2 Maximizing L or equivalently maximizing Pn (yi − βxi )2 n ln L(β, σ 2 ) = ln(2πσ 2 ) + i=1 2 2σ 2 Pn requires selecting β to minimize H(β) = i=1 (yi − βxi )2 . n X 0 (yi − βxi )(−xi ) = 0 H (β) = 2 n Y

i=1

βb =

Pn x i yi Pi=1 n 2 i=1 xi

∂ ln L(β, σ 2 ) ∂(σ 2 )

b = 2 Pn x2 > 0. which is a point of minimum since H 00 (β) i=1 i

(b) βb =

Ã n X i=1

c2 σ x2i

n − 2σ 2

i=1 (yi − βxi ) 2(σ 2 )2

= 0

!−1 n X i=1

1X b i )2 ; (yi − βx n i=1 xi Yi . Since βb is a linear combination of independent normal

random variables, it is normal with mean

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Chapter 6 Point Estimation

b E(β)

77 Ã n X

x2i

i=1

Ã n X

x2i x2i

i=1

xi E(Yi )

i=1

Ã n X

!−1 n X

xi (βxi )

x2i

!−1

i=1

Ã n X i=1

and variance b Var(β)

Ã n X

x2i

i=1

Ã n X

x2i Var(Yi )

i=1

x2i

i=1

= σ2

!−2 n X

n ±X

!−2 n X

x2i σ 2

i=1

x2i .

i=1

Also n X i=1

(Yi − βxi )2

n X i=1

b i + (βb − β)xi ]2 [Yi − βx

b i )2 + (βb − β)2 (Yi − βx

n X

x2i .

i=1

Pn Because Yi ∼ independent N (βxi , σ 2 ) and βb ∼ N (β, σ 2 i=1 x2i ), Pn

i=1 (Yi − βxi ) σ2

∼ χ2 (n)

and

Pn (βb − β)2 i=1 x2i ∼ χ2 (1). σ2

By Theorem 9.3-1, n X i=1

b i )2 ∼ χ2 (n−1); (Yi − βx

3 c2 [1 − (3/5)(1)]2 + [1 − (3/5)(2)]2 1 (1)(1) + (2)(1) = , σ = = ; 12 + 2 2 5 2 10

(d) y1 − yb1 = 1−(3/5) = 2/5, y2 − yb2 = 1−(6/5) = −1/5, and their sum is (2/5)−(1/5) = 1/5 6= 0. c 2020 Pearson Education, Inc. Copyright °

Section 6.5 A Simple Regression Problem 6.5–4 (a) x

2.0 3.3 3.7 2.0 2.3 2.7 4.0 3.7 3.0 2.3

1.3 3.3 3.3 2.0 1.7 3.0 4.0 3.0 2.7 3.0

4.00 10.89 13.69 4.00 5.29 7.29 16.00 13.69 9.00 5.29

2.60 10.89 12.21 4.00 3.91 8.10 16.00 11.10 8.10 6.90

1.69 10.89 10.89 4.00 2.89 9.00 16.00 9.00 7.29 9.00

0.361716 0.040701 0.027725 0.009716 0.228120 0.206231 0.006204 0.217630 0.014900 0.676310

29.0

27.3

89.14

83.81

80.65

1.849254

α b

y = 27.3/10 = 2.73;

βb =

(b)

yb =

(y − yb)2

83.81 − (29.0)(27.3)/10 4.64 = = 0.9206; 89.14 − (29.0)(29.0)/10 5.04 2.73 + (4.64/5.04)(x − 2.90); y 4.0 3.5 3.0 2.5 2.0 1.5 1.5

2.0

2.5

3.0

3.5

4.0

Figure 6.5–4: Earned grade (y) versus predicted grade (x) c2 = (c) σ

6.5–6 (a) α b

1.849254 = 0.184925. 10 =

βb = yb =

395 = 26.333, 15 9292 − (346)(395)/15 180.667 = = 0.506, 8338 − (346)2 /15 356.933 µ ¶ 180.667 346 26.333 + x− 356.933 15 0.506x + 14.657;

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Chapter 6 Point Estimation

(b)

y 32 30 28 26 24 22 20 18 16 16 18 20 22 24 26 28 30 32

Figure 6.5–6: ACT natural science (y) versus ACT social science (x) scores (c)

c2 nσ

26.33, βb = 0.506,

211.8861,

c2 σ

α b

10, 705 −

3952 − 0.5061636(9292) + 0.5061636(346)(395)/15 15

211.8861 = 14.126. 15

6.5–8 (a) y = 3.575 + 1.225x; (b) 35 34 33 32 31 30 29 28 27 26 18 19

Figure 6.5–8: (b) Highway mpg (y) versus City mpg (x)

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Section 6.6 Asymptotic Distributions of Maximum Likelihood Estimators (c) y = 55.6745 − 0.0075x.

35 34 33 32 31 30 29 28 27 26 3100 3200 3300 3400 3500 3600 3700 3800 3900

Figure 6.5–8: (c) Highway mpg (y) versus Curb Weight (x)

6.5–10 (b) The least squares regression line for y = a + b versus b is yb = 1.360 + 1.626b; (c) y = φx = 1.618x is added on the right figure below;

280 260 240 220 200 180 160 140 120 100 80 60 40

280 260 240 220 200 180 160 140 120 100 80 60 40 20

100 120 140 160 180

Figure 6.5–10: Scatter plot of a + b versus b with least squares regression line and with y = φx

(d) The sample mean of the points (a+b)/b is 1.647 which is close to the value of φ = 1.618.

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Chapter 6 Point Estimation

6.6

Asymptotic Distributions of Maximum Likelihood Estimators f (x; p)

px (1 − p)1−x ,

ln f (x; p)

x ln p + (1 − x) ln(1 − p)

6.6–2 (a)

∂ ln f (x; p) ∂p ∂ 2 ln f (x; p) ∂p2 · ¸ X −1 X E 2− p (1 − p)2

= =

x = 0, 1

x x−1 + p 1−p x x−1 − 2+ p (1 − p)2

p p−1 1 − = . 2 2 p (1 − p) p(1 − p) p(1 − p) ; Rao-Cramér lower bound = n (b)

p(1 − p)/n = 1. p(1 − p)/n

6.6–4 (a)

ln f (x; θ) ∂ ln f (x; θ) ∂θ ∂ 2 ln f (x; θ) ∂θ2 ¸ · 2X 2 E − 2+ 3 θ θ

−2 ln θ + ln x − x/θ

2 x − + 2 θ θ 2 2x = − 3 2 θ θ 2(2θ) 2 2 = 2 = − 2+ θ θ3 θ θ2 ; Rao-Cramér lower bound = 2n =

(b) Very similar to (a); answer = (c)

ln f (x; θ) ∂ ln f (x; θ) ∂θ ∂ 2 ln f (x; θ) ∂θ2 E[ ln X ]

= = = = =

− ln θ +

¶ 1−θ ln x θ

1 1 − − 2 ln x θ θ 1 2 + 3 ln x θ2 θ Z 1 ln x (1−θ)/θ 1 x dx. Let y = ln x, dy = dx. x 0Z θ ∞ y −y(1−θ)/θ −y − e e dy = −θ Γ(2) = −θ. θ 0

Rao-Cramér lower bound =

6.7

θ2 ; 3n

θ2 1 ¶= . 1 2 n n − 2+ 2 θ θ µ

Sufficient Statistics

6.7–2 The distribution of Y is Poisson with mean nλ. Thus, since y = Σ x i ,

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Section 6.7 Sufficient Statistics

P (X1 = x1 , X2 = x2 , · · · , Xn = xn | Y = y)

(λΣ xi e−nλ )/(x1 ! x2 ! · · · xn !) (nλ)y e−nλ /y!

y! , x1 ! x 2 ! · · · x n ! n y

which does not depend on λ. 6.7–4 (a) f (x; θ) = e(θ−1) ln x+ln θ ,

0 < θ < ∞;

0 < x < 1,

so K(x) = ln x and thus Y =

n X i=1

is a sufficient statistic for θ;

ln Xi = ln(X1 X2 · · · Xn )

L(θ)

θ n (x1 x2 · · · xn )θ−1

ln L(θ)

n ln θ + (θ − 1) ln(x1 x2 · · · xn )

n + ln(x1 x2 · · · xn ) = 0. θ

(b)

d ln L(θ) dθ Hence

θb = −n/ ln(X1 X2 · · · Xn ),

which is a function of Y ; (c) Since θb is a single valued function of Y with a single valued inverse, knowing the value of θb is equivalent to knowing the value of Y , and hence it is sufficient.

6.7–6 (a)

f (x1 , x2 , . . . , xn )

(x1 x2 · · · xn )α−1 e−Σ xi /θ [Γ(α)]n θαn

µ −Σ xi /θ ¶µ ¶ e (x1 x2 · · · xn )α−1 . θαn [Γ(α)]n

The second Pnfactor is free of θ. The first factor is a function of the xi s through only, so i=1 xi is a sufficient statistic for θ; Pn (b) ln L(θ) = ln(x1 x2 · · · xn )α−1 − i=1 xi /θ − ln[Γ(α)]n − αn ln θ d ln L(θ) dθ

αnθ

Y =

n X

θb =

i=1 xi /θ

i=1 xi

− αn/θ = 0

i=1 xi n

1 X Xi . αn i=1

Xi has a gamma distribution with parameters αn and θ. Hence

i=1

b = E(θ)

6.7–8 tZ

E(e ) =

Z ∞

−∞

···

Z ∞µ −∞

1 √ 2πθ

¶n

exp −

1 (αnθ) = θ. αn

n X x2 i

i=1

2θ

exp

¸ · Pn t i=1 ai xi Pn dx1 dx2 . . . dxn . i=1 xi

√ √ Let xi / θ = yi , i = 1, 2, . . . , n. The Jacobian is ( θ )n . Hence

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Chapter 6 Point Estimation

E(e )

" n # ¶n ¸ · Pn Z ∞ √ µ X y2 t ai yi 1 i n exp − dy1 dy2 . . . dyn . ··· ( θ) √ exp Pi=1 n 2 2πθ −∞ −∞ i=1 yi i=1 " n # ¶n · Pn ¸ Z ∞ Z ∞µ X y2 t i=1 ai yi 1 i √ exp − ··· exp Pn dy1 dy2 . . . dyn 2 2π −∞ −∞ i=1 yi i=1 Z ∞

which is free of θ. Since the distribution of Z is free of θ, Z and Y = statistics, are independent.

2 i=1 Xi , the sufficient

6.7–10 The joint pdf can be written " n Y

i=1

xi (1 − xi )

so by definition

#θ ·

Γ(2θ) {Γ(θ)}2

n Y

i=1

is a sufficient statistics for θ.

¸n

1 ; [ i=1 xi (1 − xi )] Qn

Xi (1 − Xi )

6.7–12 The joint pdf of Y1 , Y2 , . . . , Yn , letting c=

√

1 2πσ 2

¶n

is c · exp = c · exp = c · exp = c · exp = φ

µ Pn

i=1 [yi − α − β(xi − x)] 2σ 2

µ Pn

2 2 2 2 i=1 [yi + α + β (xi − x) − 2αyi − 2β(xi − x)yi + 2αβ(xi − x)] 2σ 2

µ Pn

2 i=1 yi − 2α

µ Pn

Ã n X i=1

i=1 yi − 2β

2 i=1 yi − 2(α + βx)

yi ,

n X i=1

yi2 ,

n X i=1

i=1 yi (xi − x) + quantities that don’t involve the yi ’s 2σ 2

i=1 yi − 2β

xi yi ; α, β, σ

i=1 yi xi + quantities that don’t involve the yi ’s 2σ 2

· 1.

The result follows immediately by the Factorization Theorem.

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Section 6.8 Bayesian Extimation

6.8

Bayesian Estimation

6.8–2 (a)

g(τ | x1 , x2 , . . . , xn )

∝

(x1 x2 · · · xn )α − 1 τ nα τ α0 − 1 e−τ /θ0 e−Σxi /(1/τ ) [Γ(α)]n Γ(α0 )θ0α0

τ αn + α0 − 1 e−(1/θ0 + Σxi )τ ¶ θ0 which is Γ nα + α0 , ; 1 + θ0 Σxi ∝

(b)

E(τ | x1 , x2 , . . . , xn )

θ0 1 + θ0 Xn

(nα + α0 )

α0 θ0 αnθ0 + 1 + θ0 nX 1 + nθ0 X

nα + α0 ; 1/θ0 + nX

(c) The posterior distribution is Γ(30 + 10, 1/[1/2 + 10 x ]). Select a and b so that P (a < τ < b) = 0.95 with equal tail probabilities. Then Z b

(1/2 + 10 x )40 40−1 −w(1/2 + 10 x) dw = w e Γ(40) a

Z b(1/2+10x) a(1/2+10x)

1 z 39 e−z dz, Γ(40)

making the change of variables w(1/2 + 10 x ) = z. Let v0.025 and v0.975 be the quantiles for the Γ(40, 1) distribution. Then a=

v0.025 ; 1/2 + 10 x

v0.975 . 1/2 + 10 x

It follows that P (a < τ < b) = 0.95.

6.8–4 3 3 g(θ | x1 , x2 , . . . , xn ) ∝ (3θ)n (x1 x2 · · · xn )2 e−θΣxi · θ4 − 1 e−4θ ∝ θn + 3 e−(4 + Σxi )θ

µ which is Γ n + 4,

¶ 1 . Thus 4 + Σ x3i E(θ | x1 , x2 , . . . , xn ) =

n+4 . 4 + Σ x3i

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Chapter 6 Point Estimation

g(y | θ)

ny n−1 , θn

g(y | θ)h(θ)

∝

θn+β+1

k(θ | y)

6.8–6 (a)

1 c θn+β+1

0<y<θ ,

max(α, y) < θ < ∞

(n + β)[max(α, y)]n+β by

Z ∞

max(α,y)

E(θ | y)

Z ∞

= =

w(Y )

[max(α, y)]n+β (n + β) dθ θn+β+1 ¶ max(α, y) = w(y)

(θ)

max(α,y)

k(θ | y) dθ = 1

n+β n+β−1 n+β n+β−1

max(α, Y );

(b) With n = 4, α = 1, β = 2, k(θ | y) =

[max(1, y)]6 (6) , θ7

max(1, y) < θ.

Since Loss = |θ − w(Y )|, we want w(y) 1 2

median

Z w(y)

1−

[max(1, y)]6 (6) dθ θ7 max(1,y) [max(1, y)]6 [w(y)]6

so w(Y ) = 6.8–8 L=

√

1 2πσ 2

¶n

(

exp −

√ 6

n X i=1

2 max(1, Y ).

[yi − α − β(xi − x)] /2σ

)

The summation in the exponent can be written as follows: n X i=1

α − α)2 + (βb − β)2 [yi − α − β(xi − x)] = n(b

n X i=1

(xi − x)2 +

n X i=1

b i − x)]2 . [yi − α b − β(x

Note: Let’s say σ 2 is known in the exercise. Then α has prior N (α1 , σ12 ); so posterior of α (eliminating factors not involving α) is

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Section 6.8 Bayesian Extimation

posterior of α

¸ · ¸ −n(α − α b )2 −(α − α1 )2 exp 2σ12 2σ 2 2πσ12 · ¸ (α2 − 2αα1 ) (nα2 − 2nb αα) ∝ exp − − + terms not involving α 2σ12 2σ 2 · ½ µ ¶ µ ¶¾¸ n nb α α1 1 + + = exp − α2 − 2α + 2σ12 2σ 2 2σ12 2σ 2

∝

exp

terms not involving α ¶½ ¾¸ · µ α1 /(2σ12 ) + nb α/(2σ 2 ) n 1 2 α − 2α + = exp − + 2σ12 2σ 2 1/(2σ12 ) + n/(2σ 2 ) terms not involving α " µ ¶½ ¾2# α1 /(2σ12 ) + nb α/(2σ 2 ) n 1 = exp − + + 2 α− 2σ12 2σ 1/(2σ12 ) + n/(2σ 2 ) terms not involving α. This is normal with posterior mean equal to α1 nb α + 2 σ12 σ . n 1 + σ12 σ2 Now β has prior N (β0 , σ02 ); so posterior of β is " # b 2 Pn (xi − x)2 (β − β) (β − β0 )2 i=1 − posterior of β ∝ exp − 2σ 2 2σ02 Ã P ! # " µP ¶ n n 2 βb i=1 (xi − x)2 1 β0 i=1 (xi − x) 2 ∝ exp − + 2 β +2 + 2 β 2σ 2 2σ0 2σ 2 2σ0  ( )2  ¶ µ Pn 2 b Pn (xi − x)2 /σ 2 + β0 /σ 2 β x) (x − 1 i 0 i=1 . β − Pni=1 + 2 ∝ exp− 2 2 2 2σ 2 2σ0 i=1 (xi − x) /σ + 1/σ0 This is normal with posterior mean

Pn βb i=1 (xi − x)2 /σ 2 + β0 /σ02 Pn 2 . 2 2 i=1 (xi − x) /σ + 1/σ0

Hence the posterior mean of α + β(x − x) is

Pn βb i=1 (xi − x)2 nb α β0 α1 + + 2 σ12 σ2 σ2 σ0 + Pn (x − x). 2 1 n 1 i=1 (xi − x) + + σ12 σ2 σ2 σ02

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Chapter 7 Interval Estimation

Chapter 7

Interval Estimation 7.1

Confidence Intervals for Means

7.1–2 (a) [77.272, 92.728];

(b) [79.12, 90.88];

(d) [81.154, 88.846].

7.1–4 (a) x = 56.8; √ √ ¤ £ (b) 56.8 − 1.96(2/ 10 ), 56.8 + 1.96(2/ 10 = [55.56, 58.04] ; ¶ µ 52 − 56.8 = P (Z < −2.4) = 0.0082. (c) P (X < 52) = P Z < 2 · µ ¶ µ ¶¸ 11.80 11.80 7.1–6 11.95 − 1.96 √ , 11.95 + 1.96 √ = [8.15, 15.75]. 37 37 If more extensive t-tables are available or if a computer program is used, we have ¶ µ ¶¸ · µ 11.80 11.80 , 11.95 + 2.028 √ = [8.016, 15.884]. 11.95 − 2.028 √ 37 37 7.1–8 (a) x = 46.42; √ (b) 46.42 ± 2.132s/ 5

[40.26, 52.58].

7.1–10 (a) x = 3.580; (b) s = 0.512;

√ (c) [0, 3.580 + 1.833(0.512/ 10 ] = [0, 3.877].

7.1–12 (a) x = 245.80, s = 23.64, so a 95% confidence interval for µ is √ √ [245.80 − 2.145(23.64)/ 15 , 245.80 + 2.145(23.64)/ 15 ] = [232.707, 258.893];

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Section 7.2 Confidence Intervals for the Difference of Two Means (b)

220

240

260

280

Figure 7.1–12: Box-and-whisker diagram of signals from detectors (c) The standard deviation is quite large. √ √ √ 7.1–14 (a) ( x + 1.96σ/ 5 ) − ( x − 1.96σ/ 5 ) = 3.92σ/ 5 = 1.753σ; √ √ √ (b) ( x + 2.776s/ 5 ) − ( x − 2.776s/ 5 ) = 5.552s/ 5. √ √ 2 Γ(5/2)σ 3 πσ From Exercise 6.4–14 with n = 5, E(S) = √ = 5/2 = 0.94σ, so that 2 4 Γ(4/2) √ E[5.552S/ 5 ] = 2.334σ. 7.1–16

P (X − 1 < µ < X + 1)

0.90

P (−1 < µ − X < 1) = 0.90 ¶ X −µ 1 −1 √ < √ < √ P = 0.90 3/ n 3/ n 3/ n √ n = 1.645 3 √ n = 3(1.645) µ

7.2

24.35 % 25.

Confidence Intervals for the Difference of Two Means

7.2–2 x = 539.2, s2x = 4,948.7, y = 544.625, s2y = 4,327.982, sp = 67.481, t0.05 (11) = 1.796, so the confidence interval is [−74.517, 63.667]. 7.2–4 (a) x − y = 1511.714 − 1118.400 = 393.314;

(b) s2x = 49, 669.905, s2y = 15, 297.600, r = [8.599] = 8, t0.025 (8) = 2.306, so the confidence interval is [179.148, 607.480].

7.2–6 (a) x = 2.584, y = 1.564, s2x = 0.1042, s2y = 0.0428, sp = 0.2711, t0.025 (18) = 2.101. Thus a 95% confidence interval for µX − µY is [0.7653, 1.2747];

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Chapter 7 Interval Estimation

(b)

1.5

2.0

2.5

3.0

Figure 7.2–6: Box-and-whisker diagrams, wedge on (X) and wedge off (Y ) (c) Yes. 7.2–8 From (a), (b), and (c), we know s· ¸Á (nX − 1)SX2 (nY − 1)SY2 X − Y − (µX − µY ) s (nX + nY − 2) ÷ (d) + dσY2 σY2 dσY2 σY2 + nX nY has a t(nX +nY −2) distribution. Clearly, this ratio does not depend upon σY2 ; so s ¶ µ 1 d (nX − 1)s2X /d + (nY − 1)s2Y + x − y ± tα/2 (nX +nY −2) nX + n Y − 2 nX nY provides a 100(1 − α)% confidence interval for µX − µY . 7.2–10 (a) d = 0.07875;

√ (b) [ d − (1.714)(0.25492)/ 24, ∞) = [−0.0104, ∞); (c) not necessarily.

7.2–12 (a) x = 136.61, y = 134.87, s2x = 3.2972, s2y = 1.0957; (b) Using Welch with r = 18 degrees of freedom, the 95% confidence interval is [0.436, 3.041]. Assuming equal variances with r = 20 degrees of freedom, the 95% confidence interval is [0.382, 3.095];

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Section 7.3 Confidence Intervals for Proportions (c) The five-number summary for the X observations is 133.30, 135.625, 136.95, 137.80, 139.40. The five-number summary for the Y observations is 132.70, 134.15, 134.95, 135.825, 136.00;

133 134 135 136 137 138 139 Figure 7.2–12: Hardness of hot (X) and cold (Y ) water (d) The mean for hot seems to be larger than the mean for cold.

7.3

Confidence Intervals For Proportions

7.3–2 (a) pb =

142 = 0.71; 200 r

# r (0.71)(0.29) (0.71)(0.29) , 0.71 + 1.645 = [0.657, 0.763]; (b) 0.71 − 1.645 200 200 p 0.71 + 1.6452 /400 ± 1.645 0.71(0.29)/200 + 1.6452 /(4 · 2002 ) = [0.655, 0.760]; (c) 1 + 1.6452 /200 "

142 + 1.6452 /2 142 + 1.3530 = = 0.7072; 200 + 1.6452 200 + 2.7060 " # r (0.7072)(0.2928) 0.7072 ± 1.645 = [0.6546, 0.7598]; 202.7060 # " r (0.71)(0.29) , 1 = [0.669, 1]. (e) 0.71 − 1.282 200 " # r r (0.70)(0.30) (0.70)(0.30) 7.3–4 0.70 − 1.96 , 0.70 + 1.96 = [0.674, 0.726]. 1234 1234 " # r r (0.26)(0.74) (0.26)(0.74) , 0.26 + 2.326 7.3–6 0.26 − 2.326 = [0.247, 0.273]. 5757 5757 (d) pe =

388 = 0.3796; 1022 r (0.3796)(0.6204) (b) 0.3796 ± 1.645 1022

7.3–8 (a) pb =

[0.3546, 0.4046].

7.3–10 (a) pb1 = 28/194 = 0.144; c 2020 Pearson Education, Inc. Copyright °

Chapter 7 Interval Estimation (b) 0.144 ± 1.96

91 (0.144)(0.856)/194 or [0.095, 0.193];

7.3–12 (a) pbA = 170/460 = 0.37, pbB = 141/440 = 0.32, r (0.37)(0.63) (0.32)(0.68) 0.37 − 0.32 ± 1.96 + 460 440 (b) yes, the interval includes zero.

7.4

[0.044, 1].

[−0.012, 0.112];

Sample Size

7.4–2 n =

(1.96)2 (169) = 288.5 so the sample size needed is 289. (1.5)2

7.4–4 Approximately n =

(1.96)2 (34.9) = 537, rounded up to the nearest integer. (0.5)2

7.4–6 Approximately n =

(1.96)2 (33.7)2 = 175, rounded up to the nearest integer. 52

7.4–8 n =

(1.645)2 (0.394)(0.606) = 404, rounded up to the nearest integer. (0.04)2 2 zα/2 p∗ (1 − p∗ )

(1.6452 )(0.1)(0.9) = 2435; 0.012 2 zα/2 pb (1 − pb ) (1.6452 )(0.16)(0.84) (b) n = = = 3637; 2 ε 0.012 2 zα/2 1.6452 = = 6765. (c) n = 2 4ε (4)(0.012 )

7.4–10 (a) n =

7.4–12 m =

ε2

(1.96)2 (0.5)(0.5) = 601, rounded up to the nearest integer. (0.04)2

(a) n =

601 = 430; 1 + 600/1500

(b) n =

601 = 578; 1 + 600/15,000

601 = 587. 1 + 600/25,000

7.4–14 For the difference of two proportions with equal sample sizes r p∗1 (1 − p∗1 ) p∗2 (1 − p∗2 ) ε = zα/2 + n n or 2 zα/2 [p∗1 (1 − p∗1 ) + p∗2 (1 − p∗2 )] . n= ε2 For unknown p∗ , 2 2 zα/2 zα/2 [0.25 + 0.25] = . n= ε2 2ε2

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Section 7.5 Distribution-Free Confidence Intervals for Percentiles

So n =

7.5

1.2822 = 329, rounded up. 2(0.05)2

Distribution-Free Confidence Intervals for Percentiles

7.5–2 (a) (y3 = 5.4, y10 = 6.0) is a 96.14% confidence interval for the median, m; (b) (y1 = 4.8, y7 = 5.8); P (Y1 < π0.3 < Y7 )

6 µ ¶ X 12 (0.3)k (0.7)12−k k

k=1

0.9614 − 0.0138 = 0.9476,

using Table II with n = 12 and p = 0.30. 7.5–4 (a) (y4 = 80.28, y11 = 80.51) is a 94.26% confidence interval for m; (b) (y6 = 80.32, y12 = 80.53); 11 µ ¶ X 14 (0.6)k (0.4)14−k = k k=6

8 µ ¶ X 14 (0.4)k (0.6)14−k k

k=3

= 0.9417 − 0.0398 = 0.9019. The interval is (y6 = 80.32, y12 = 80.53).

7.5–6 (a)

Leaves Stems 20f 5 20s 6677 88999 20• 21∗ 0000011 21t 2222233333333 21f 444444455555555 21s 66666666666667777777777 8888889999 21• 22∗ 000 (Multiply numbers by 10−1 .)

(b) (i) q1

Frequency 1 4 5 7 13 15 23 10 3

y20 + y21 21.2 + 21.2 = = 21.2 2 2 m = y41 = 21.5

21.7 + 21.7 y61 + 762 = = 21.7; 2 2 (ii) π0.60 = 0.8y49 + 0.2y50 = 21.6; (iii) π0.15 = 0.7y12 + 0.3y13 = 21.0; q3

(c) (i) We first find i and j so that P (Yi < π0.25 < Yj ) ≈ 0.95. Let the distribution of W be b(81, 0.25). Then P (Yi < π0.25 < Yj ) = P (i ≤ W ≤ j − 1) µ ¶ j − 1 + 0.5 − 20.25 i − 0.5 − 20.25 √ √ ≈ P . ≤Z≤ 15.1875 15.1875 If we let i − 20.75 j − 20.75 √ = −1.96 and √ = 1.96 15.1875 15.1875 we find that i ≈ 13 and j ≈ 28. Furthermore P (13 ≤ W ≤ 28 − 1) ≈ 0.9453. Also note that the point estimate of π0.25 , π e0.25 = (y20 + y21 )/2 c 2020 Pearson Education, Inc. Copyright °

Chapter 7 Interval Estimation

falls near the center of this interval. So a 94.53% confidence interval for π0.25 is (y13 = 21.0, y28 = 21.3); (ii) Let the distribution of W be b(81, 0.5). Then P (i ≤ W ≤ 81 − i) µ ¶ i − 0.5 − 40.5 81 − i + 0.5 − 40.5 √ √ ≈ P . ≤Z≤ 20.25 20.25

P (Yi < π0.5 < Y82−i )

i − 41 = −1.96, 4.5

then i = 32.18 so let i = 32. Also 81 − i − 40 = 1.96 4.5 implies that i = 32. Furthermore P (Y32 < π0.5 < Y50 ) = P (32 ≤ W ≤ 49) ≈ 0.9544. So an approximate 95.44% confidence interval for π0.5 is (y32 = 21.4, y50 = 21.6); (iii) Similar to part (a), P (Y54 < π0.75 < Y69 ) ≈ 0.9453. Thus a 94.53% confidence interval for π0.75 is (y54 = 21.6, y69 = 21.8). 7.5–8 A 95.86% confidence interval for m is (y6 = 14.60, y15 = 16.20). = P (b(25, 0.5) ≤ 15) − P (b(25, 0.5) ≤ 8) = 0.8852 − 0.0539 = 0.8313; (b) Want P (Yr < π0.90 ) ≥ 0.95, that is, P (b(25, 0.9) ≥ r) ≥ 0.95. Now,

7.5–10 (a)

P (Y9 < π0.50 < Y16 )

P (b(25, 0.9) ≥ 21) = 0.9020,

and

P (b(25, 0.9) ≥ 20) = 0.9666.

So the desired r is 20, and y20 = 190, and the desired interval is [190, ∞). Because this interval straddles the water quality standard, it does not provide strong evidence of a violation. 8 µ ¶ X 8 (0.7)k (0.3)8−k = 0.2553; 7.5–12 (a) P (Y7 < π0.70 ) = k k=7 7 µ ¶ X 8 (b) P (Y5 < π0.70 < Y8 ) = (0.7)k (0.3)8−k = 0.7483. k k=5

7.6

More Regression

7.6–2 The mean of the observations is ∗

Y =

1 X ∗ Y , m i=1 i

∗

where Y1∗ , Y2∗ , . . . Ym∗ are the future observations at X = x∗ . Since Y is a linear combination of independent and normally distributed random variables, it is normally distributed with mean Ã

1 X ∗ Y E m i=1 i

1 X E(Yi∗ ) m i=1 m

= =

1 X [α + β(x∗ − x)] m i=1

α + β(x∗ − x)

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Section 7.6 More Regression and variance Ã ! m 1 X ∗ Var Y m i=1 i

= =

Furthermore,

m 1 X Var(Yi∗ ) m2 i=1

m σ2 1 X 2 σ = . 2 m i=1 m ∗

b ∗ − x) Y −α b − β(x

is a linear combination of independent and normally distributed random variables, so it is normally distributed with mean

and variance

∗ b ∗ − x)] = α + β(x∗ − x) − α − β(x∗ − x) = 0 E[Y − α b − β(x

σ2 (x∗ − x)2 σ 2 σ2 + + P . m n (xi − x)2 ∗ c2 , we can form the t-statistic b, and βb are independent of σ Since Y , α ∗ b ∗ − x) Y −α b − β(x s 1 (x∗ − x)2 1 σ + +P (xi − x)2 m n s T = ∼ t(n−2). c2 nσ n−2

We can then obtain the following expression for the endpoints of the desired prediction interval: b ∗ − x) ± htα/2 (n−2) α b + β(x where

h=σ b

n n−2

7.6–4 (a) In Exercise 6.5–4 we found that βb = 4.64/5.04,

1 1 (x∗ − x)2 + +P . m n (xi − x)2

c2 = 1.84924, nσ

10 X i=1

(xi − x)2 = 5.04.

So the endpoints for the confidence interval are given by r r 1.8493 1 (x − 2.90)2 4.64 (x − 2.90) ± 2.306 + , 2.73 + 5.04 8 10 5.04 x=2:

[1.335, 2.468],

x=3:

[2.468, 3.176],

x = 4 : [3.096, 4.389]; (b) The endpoints for the prediction interval are given by r r 4.64 (x − 2.90)2 1.8493 1 2.73 + (x − 2.90) ± 2.306 + , 1+ 5.04 8 10 5.04 x=2:

[0.657, 3.146],

x=3:

[1.658, 3.986],

x=4:

[2.459, 5.026].

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Chapter 7 Interval Estimation

y 4.5

4.0

3.5

3.0

3.0 2.5

2.5

2.0

1.5

1.5 2.0

2.5

3.0

3.5

4.0

1.0 2.0

2.5

3.0

3.5

Figure 7.6–4: A 95% confidence interval for µ(x) and a 95% prediction band for Y

7.6–6 (a) For these data, 10 X

xi = 55,

10 X

yi = 9811,

i=1

10 X

x2i = 385,

xi yi = 65,550,

i=1

10 X

yi2 = 11,280, 031.

i=1

Thus α b = 9811/10 = 981.1 and

11589.5 65,550 − (55)(9811)/10 = = 140.4788. βb = 2 385 − (55) /10 82.5

The least squares regression line is

(b)

yb = 981.1 + 140.4788(x − 5.5) = 208.467 + 140.479x;

1600 1400 1200 1000 800 600 400

Figure 7.6–6: Number of programs (y) vs. year (x) (c) 1753.733 ± 160.368 or [1593.365, 1914.101].

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4.0

Section 7.6 More Regression

7.6–8 Let K(β1 , β2 , β3 ) =

n X i=1

∂K ∂β1

∂K ∂β2

∂K ∂β3

n X i=1 n X i=1 n X i=1

(yi − β1 − β2 x1i − β3 x2i )2 . Then

2(yi − β1 − β2 x1i − β3 x2i )(−1) = 0; 2(yi − β1 − β2 x1i − β3 x2i )(−x1i ) = 0; 2(yi − β1 − β2 x1i − β3 x2i )(−x2i ) = 0.

Thus, we must solve simultaneously the three equations ! ! Ã n Ã n n X X X nβ1 + x2i β3 = x1i β2 + yi i=1

i=1

Ã n X

x1i β1 +

i=1

Ã n X

We have

i=1

x2i β1 +

i=1

Ã n X

x21i β2 + !

x1i x2i β3 =

i=1

x1i x2i β2 +

i=1

Ã n X

x22i

i=1

n X

x1i yi

i=1

β3 =

n X

x2i yi .

i=1

12β1 + 4β2 + 4β3 = 23 4β1 + 26β2 + 5β3 = 75 4β1 + 5β2 + 22β3 = 37 so that 3852 βb2 = = 2.587, 1489

4373 = 0.734, βb1 = 5956

1430 βb3 = = 0.960. 1489

10 8

4 0

–4 –5

–8 –2

–1

x0 1

–2

–1

–10

–2

–1

0 x

0 y

Figure 7.6–8: Two views of the points and the regression plane 7.6–10 (c) and (d)

(e) Linear regression is not appropriate. Finding the least-squares quadratic regression line using the raw data yields yb = −1.88 + 9.86x − 0.995x2 ; c 2020 Pearson Education, Inc. Copyright °

Chapter 7 Interval Estimation

y 8

24 22 20 18 16 14 12 10 8 6 4 2

6 4 2 0 –2

–4 –6 –8 1

Figure 7.6–10: (y) versus (x) with linear regression line and residual plot (f ) and (g)

y 1.8

24 22 20 18 16 14 12 10 8 6 4 2

1.4 1.0 0.6 0.2 –0.2

–0.6 –1.0 1

x –1.4

Figure 7.6–10: (y) versus (x) with quadratic regression curve and residual plot

7.6–12 The x is part of the exponent of e. However ln µ(x) = ln β1 + β2 x is linear in x. 7.6–14 Recall that

√

n( α b − α) α b−α =s T = T2 = s σ c2 c2 nσ σ σ 2 (n − 2)

n−2

has a t distribution with n − 2 degrees of freedom. Thus

c 2020 Pearson Education, Inc. Copyright °

Section 7.6 More Regression 



P −tγ/2 (n−2) ≤ q

P −b α − tγ/2 

α b−α

c2 /(n − 2) σ

≤ tγ/2 (n−2)

c2 σ ≤ −α ≤ −b α + tγ/2 n−2

b − tγ/2 P α



c2 σ ≤α≤α b + tγ/2 n−2

 c2 σ  n−2  c 2 σ  n−2

1−γ

1 − γ.

It follows that the endpoints for a 100(1 − γ)% confidence interval for α are s c2 σ α b ± tγ/2 (n−2) . n−2

c2 = 0.184925, n = 10. The endpoints for the 95% 7.6–16 Recall that α b = 2.73, βb = 4.64/5.04, σ confidence interval are r 0.184925 2.73 ± 2.306 or [2.379, 3.081] for α; 8 s 1.84925 or [0.4268, 1.4145] for β; 4.64/5.04 ± 2.306 8(5.04) · ¸ 1.84925 1.84925 , = [0.105, 0.848] for σ 2 . 17.54 2.180 Pn 7.6–18 (a) The given facts, plus the result from Exercise 6.5-2 that βb has a N (β, σ 2 / i=1 x2i ) distribution, imply that βb − β Pn 2 βb − β i=1 xi s =q c2 /[(n − 1) Pn x2 ] c2 /σ 2 nσ nσ i=1 i n−1

σ2 /

has a t(n−1) distribution. Thus, for γ ∈ (0, 1),   b− β β ≤ tγ/2 (n−1) = 1 − γ, P −tγ/2 (n−1) ≤ q Pn 2 c 2 nσ /[(n−1) i=1 xi ]

and algebraic manipulation of the inequality to get β alone in the middle yields the 100(1 − γ)% confidence interval for βb as s c2 nσ Pn ; βb ± tγ/2 (n−1) (n − 1) i=1 x2i Pn (b) Yn+1 is N (βxn+1 , σ 2 ) and βb is [as noted in the solution to part (a)] N (β, σ 2 / i=1 x2i ), P n so Ybn+1 Yn+1 is N (βxn+1 − βxn+1 , σ 2 + x2n+1 σ 2 / i=1 x2i ), that is, N [0, σ 2 (1 + P− n 2 2 xn+1 / i=1 xi )]. This, plus the facts given in part (a), imply that q

Ybn+1 − Yn+1 Pn σ 2 (1 + x2n+1 / i=1 x2i ) Ybn+1 − Yn+1 s =q c2 /(n − 1)](1 + x2 / Pn x2 ) c2 /σ 2 [nσ nσ n+1 i=1 i n−1 c 2020 Pearson Education, Inc. Copyright °

Chapter 7 Interval Estimation

has a t(n−1) distribution. Thus, for γ ∈ (0, 1),   bn+1 − Yn+1 Y P −tγ/2 (n−1) ≤ q ≤ tγ/2 (n−1) = 1 − γ, Pn 2 2 c 2 [nσ /(n − 1)](1 + xn+1 / i=1 xi )

and algebraic manipulation of the inequality to get Yn+1 alone in the middle yields the 100(1 − γ)% prediction interval for Yn+1 as v s n c2 u X u n σ t1 + x 2 / x2i . Ybn+1 ± tγ/2 (n−1) n+1 n−1 i=1 7.6–20 (a) W

βb + tγ/2 (n−2) 2tγ/2 (n−2)

(n − 2)

E(W )

i=1 (xi − x)

(n − 2)

c2 nσ Pn

4t2γ/2 (n−2)

(b) 2

c2 nσ Pn

i=1 (xi − x̄)

4t2γ/2 (n−2)



− βb − tγ/2 (n−2)

(n − 2)

c2 nσ Pn

i=1 (xi − x)

;

(n − 2)

c2 nσ Pn

nσ 2 Pn

i=1 (xi − x)

;

Pn (c) For the first data set, i=1 (xi − x)2 ) = 24 so E(W 2 ) = [4t2γ/2 (4)] · [6σ 2 ]/[(4)(24)] = Pn (1/4)t2γ/2 (4), while for the second data set, i=1 (xi −x)2 ) = 8 so E(W 2 ) = [4t2γ/2 (4)]· [6σ 2 ]/[(4)(8)] = (3/4)t2γ/2 (4). The expected squared width is smallest for the first data Pn set because its value of i=1 (xi − x)2 is larger.

7.7

Resampling Methods

7.7–2 (a) > read ‘C:\\HTZ-CD\\Maple Examples\\stat.m‘: with(plots): read ‘C:\\HTZ-CD\\Maple Examples\\HistogramFill.txt‘: read ‘C:\\HTZ-CD\\Maple Examples\\ScatPlotCirc.txt‘: read ‘C:\\HTZ-CD\\Maple Examples\\Chapter_07.txt‘: XX := Exercise_7_7_2; XX := [12.0, 9.4, 10.0, 13.5, 9.3, 10.1, 9.6, 9.3, 9.1, 9.2, 11.0, 9.1, 10.4, 9.1, 13.3, 10.6] > Probs := [seq(1/16, k = 1 .. 16)]: XXPDF := zip((XX,Probs)-> (XX,Probs), XX, Probs): > for k from 1 to 200 do X := DiscreteS(XXPDF, 16): Svar[k] := Variance(X): od: Svars := [seq(Svar[k], k = 1 .. 200)]: > Mean(Svars); 1.972629584

 

100

Section 7.7 Resampling Methods > xtics := [seq(0.4*k, k = 1 .. 12)]: ytics := [seq(0.05*k, k = 1 .. 11)]: P1 := plot([[0,0],[0,0]], x = 0 .. 4.45, y = 0 .. 0.57, xtickmarks=xtics, ytickmarks=ytics, labels=[‘‘,‘‘]): P2 := HistogramFill(Svars,0 .. 4.4, 11): display({P1, P2}); The histogram is shown in Figure 7.7-2(ab). (b) > theta := Mean(XX) - 9; for k from 1 to 200 do Y := ExponentialS(theta,21): Svary[k] := Variance(Y): od: Svarys := [seq(Svary[k], k = 1 .. 200)]: θ := 1.31250000 > Mean(Svarys); 1.747515570 > xtics := [seq(0.4*k, k = 1 .. 14)]: ytics := [seq(0.05*k, k = 1 .. 15)]: P3 := plot([[0,0],[0,0]], x = 0 .. 5.65, y = 0 .. 0.62, xtickmarks=xtics, ytickmarks=ytics, labels=[‘‘,‘‘]): P4 := HistogramFill(Svarys,0 .. 5.6, 14): display({P3, P4}); 0.6

0.5

0.4

0.3

0.2

0.1

0.1 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4

0.4

1.2

2.0

2.8

3.6

4.4

5.2

Figure 7.7–2: (ab) Histogram of S s: Resampling on Left, From Exponential on Right

> Svars := sort(Svars): Svarys := sort(Svarys): > xtics := [seq(k*0.5, k = 1 .. 18)]: ytics := [seq(k*0.5, k = 1 .. 18)]: P5 := plot([[0,0],[5.5,5.5]], x = 0 .. 5.4, y = 0 .. 7.4, color=black, thickness=2, xtickmarks=xtics, ytickmarks=ytics, labels=[‘‘,‘‘]): P6 := ScatPlotCirc(Svars,Svarys): display({P5, P6});

Chapter 7 Interval Estimation

101

7 6 5 4 3 2 1 1

Figure 7.7–2: (c) q-q Plot of Exponential S 2 s Versus Resampling S 2 s Note that the variance of the sample variances from the exponential distribution is greater than the variance of the sample variances from the resampling distribution. 7.7–4 (a) > with(plots): read ‘C:\\HTZ-CD\\Maple Examples\\stat.m‘: read ‘C:\\HTZ-CD\\Maple Examples\\HistogramFill.txt‘: read ‘C:\\HTZ-CD\\Maple Examples\\Chapter_07.txt‘: XX := Example_7_7_4; XX := [261, 269, 271, 274, 279, 280, 283, 284, 286, 287, 292, 293, 296, 300, 304, 305, 313, 321, 322, 329, 341, 343, 356, 364, 391, 417, 476]; Probs := [seq(1/27, k = 1 .. 27)]: XXPDF := zip((XX,Probs)-> (XX,Probs), XX, Probs); XXPDF := [261, 1/27, 269, 1/27, 271, 1/27, 274, 1/27, 279, 1/27, 280, 1/27, 283, 1/27, 284, 1/27, 286, 1/27, 287, 1/27, 292, 1/27, 293, 1/27, 296, 1/27, 300, 1/27, 304, 1/27, 305, 1/27, 313, 1/27, 321, 1/27, 322, 1/27, 329, 1/27, 341, 1/27, 343, 1/27, 356, 1/27, 364, 1/27, 391, 1/27, 417, 1/27, 476, 1/27]; for j from 1 to 500 do X := DiscreteS(XXPDF, 27): Y := sort(X): YY[j] := Y[7]: od: Y7 := [seq(YY[k], k = 1 .. 500)]: Y7sorted := sort(Y7): Y7sorted[1]; 271 Y7sorted[500]; 304 >xtics := [seq(273 + 5*k, k = 0 .. 7)]: ytics := [seq(0.01*k, k = 1 .. 7)]: P1 := plot([[265,0],[265,0]], x = 268 .. 306, y = 0 .. 0.0771, xtickmarks=xtics, ytickmarks=ytics, labels=[‘‘,‘‘]): P2 := HistogramFill(Y7,270.5 .. 304.5, 7): display({P1, P2});

102

Section 7.7 Resampling Methods

0.07 0.06 0.05 0.04 0.03 0.02 0.01 273 278 283 288 293 298 303 Figure 7.7–4: (a) A histogram of N = 500 seventh order statistics (b)

>Here is an estimate of the first quartile. Mean(Y7); 141443 = 282.886; 500

(c)

>Here is an 82% confidence interval for the first quartile using, in an ordered Y7, the 44th number and the 206th number. [279, 283];

(d) In Exercise 7.5-2, the 82% confidence interval for π0.25 was [274, 287]. 7.7–6 (a) > with(plots): read ‘C:\\HTZ-CD\\Maple Examples\\stat.m‘: read ‘C:\\HTZ-CD\\Maple Examples\\ScatPlotPoint.txt‘: read ‘C:\\HTZ-CD\\Maple Examples\\EmpCDF.txt‘: read ‘C:\\HTZ-CD\\Maple Examples\\HistogramFill.txt‘: read ‘C:\\HTZ-CD\\Maple Examples\\ScatPlotCirc.txt‘: read ‘C:\\HTZ-CD\\Maple Examples\\Chapter_07.txt‘: Pairs := Exercise_7_7_6; Pairs := [[2.500, 72], [4.467, 88], [2.333, 62], [5.000, 87], [1.683, 57], [4.500, 94], [4.500, 91], [2.083, 51], [4.367, 98], [1.583, 59], [4.500, 93], [4.550, 86], [1.733, 70], [2.150, 63], [4.400, 91], [3.983, 82], [1.767, 58], [4.317, 97], [1.917, 59], [4.583, 90], [1.833, 58], [4.767, 98], [1.917, 55], [4.433, 107], [1.750, 61], [4.583, 82], [3.767, 91], [1.833, 65], [4.817, 97], [1.900, 52], [4.517, 94], [2.000, 60], [4.650, 84], [1.817, 63], [4.917, 91], [4.000, 83], [4.317, 84], [2.133, 71], [4.783, 83], [4.217, 70], [4.733, 81], [2.000, 60], [4.717, 91], [1.917, 51], [4.233, 85], [1.567, 55], [4.567, 98], [2.133, 49], [4.500, 85], [1.717, 65], [4.783, 102], [1.850, 56], [4.583, 86],[1.733, 62]]: > r := Correlation(Pairs); r := .9087434803 > xtics := [seq(1.4 + 0.1*k, k = 0 .. 37)]: ytics := [seq(48 + 2*k, k = 0 .. 31)]:

Chapter 7 Interval Estimation

103

P1 := plot([[1.35,47],[1.35,47]], x = 1.35 .. 5.15, y = 47 .. 109, xtickmarks = xtics, ytickmarks=ytics, labels=[‘‘,‘‘]): P2 := ScatPlotCirc(Pairs): display({P1, P2});

100 90 80 70 60 50 1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Figure 7.7–6: (a) Scatterplot of the 50 Pairs of Old Faithful Data (b) > Probs := [seq(1/54, k = 1 .. 54)]: EmpDist := zip((Pairs,Probs)-> (Pairs,Probs), Pairs, Probs): > for k from 1 to 500 do Samp := DiscreteS(EmpDist, 54); RR[k] := Correlation(Samp): od: R := [seq(RR[k], k = 1 .. 500)]: rbar := Mean(R); rbar := .9079354926 (c) > xtics := [seq(0.8 + 0.01*k, k = 0 .. 20)]: ytics := [seq(k, k = 1 .. 25)]: > P3 := plot([[0.79, 0],[0.79,0]], x = 0.79 .. 1.005, y = 0 .. 23.5, xtickmarks=xtics, ytickmarks=ytics, labels=[‘‘,‘‘]): P4 := HistogramFill(R, 0.8 .. 1, 20): display({P3, P4}); The histogram is plotted in Figure 7.7-6 ce;

104

Section 7.7 Resampling Methods (d) Now simulate a random sample of 500 correlation coefficients, each calculated from a sample of size 54 from a bivariate normal distribution with correlation coefficient r = 0.9087434803. > for k from 1 to 500 do Samp := BivariateNormalS(0,1,0,1,r,54): RR[k] := Correlation(Samp): od: RBivNorm := [seq(RR[k], k = 1 .. 500)]: AverageR := Mean(RBivNorm); AverageR := .9073168034 > P5 := plot([[0.79, 0],[0.79,0]], x = 0.79 .. 1.005, y = 0 .. 18.5, xtickmarks=xtics, ytickmarks=ytics, labels=[‘‘,‘‘]): P6 := HistogramFill(RBivNorm, 0.8 .. 1, 20): display({P5, P6}); (e)

15 10 10 5

0.80

0.85

0.90

0.95

0.80

0.85

0.90

0.95

Figure 7.7–6: (ce) Histograms of Rs: From Resampling on Left, From Bivariate Normal on Right

Chapter 7 Interval Estimation

105

(f ) > R := sort(R): RBivNorm := sort(RBivNorm): xtics := [seq(0.8 + 0.01*k, k = 0 .. 20)]: ytics := [seq(0.8 + 0.01*k, k = 0 .. 20)]: P7 := plot([[0.8, 0.8],[1,1]], x = 0.8 .. 0.97, y = 0.8 .. 0.97, color=black, thickness=2, labels=[‘‘,‘‘], xtickmarks=xtics, ytickmarks=ytics): P8 := ScatPlotCirc(R, RBivNorm): display({P7, P8});

0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.82 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96

Figure 7.7–6: (f ) q-q Plot of the Values of R from Bivariate Normal Versus from Resampling

> StDev(R); StDev(RBivNorm); .01852854051 .02461716901 The means are about equal but the standard deviation of the values of R from the bivariate normal distribution is larger than that of the resampling distribution.

Chapter 8 Tests of Statistical Hypotheses

107

Chapter 8

Tests of Statistical Hypotheses 8.1

Tests About One Mean

8.1–2 (a) The critical region is z=

x − 13.0 x − 13.0 √ ≤ −1.96; √ = 0.2/ n 0.2/ 25

(b) The observed value of z, z=

12.9 − 13.0 = −2.5, 0.04

is less that -1.96 so we reject H0 . (c) The p-value of this test is P (Z ≤ −2.50) = 0.0062. 8.1–4 (a) |t| =

| x − 7.5 | √ ≥ t0.025 (9) = 2.262. s/ 10 0.4

0.3

T, r = 9 d.f.

0.2

0.1 α/2 = 0.025 –3

–2

α/2 = 0.025 –1

Figure 8.1–4: The critical region is | t | ≥ 2.262 | 7.55 − 7.5 | √ = 1.54 < 2.262, do not reject H0 . 0.1027/ 10 (c) A 95% confidence interval for µ is · µ ¶ µ ¶¸ 0.1027 0.1027 √ √ 7.55 − 2.262 , 7.55 + 2.262 = [7.48, 7.62]. 10 10

(b) |t| =

108

Section 8.1 Tests About One Mean Hence, µ = 7.50 is contained in this interval. We could have obtained the same conclusion from our answer to part (b). 8.1–6 (a) H0 : µ = 3.4; (b) H1 : µ > 3.4; (c) t = (x − 3.4)/(s/3);

(d) t ≥ 1.860;

0.4

0.3

T, r = 8 d.f.

0.2

0.1 α = 0.05 –3

–2

–1

Figure 8.1–6: The critical region is t ≥ 1.860 (e) t =

3.556 − 3.4 = 2.802 ; 0.167/3

(f ) 2.802 > 1.860, reject H0 ; (g) 0.01 < p-value < 0.025, p-value = 0.0116. 8.1–8 (a) | t | =

| x − 125 | √ ≥ t0.025 (14) = 2.145; s/ 15 0.4

0.3

T, r = 14 d.f.

0.2

0.1 α/2 = 0.025 –3

–2

α/2 = 0.025 –1

Figure 8.1–8: The critical region is | t | ≥ 2.145 (b) | t | =

| 127.667 − 125 | √ = 1.076 < 2.145, do not reject H0 . 9.597/ 15

Chapter 8 Tests of Statistical Hypotheses

109

8.1–10 (a) The test statistic and critical region are given by t=

x − 5.70 √ ≥ 1.895; s/ 8

(b) The observed value of the test statistic is t=

5.869 − 5.70 √ = 2.42; 0.19737/ 8

0.4

0.3

0.2

0.1

0.1 α = 0.05

–3

–2

–1

–3

–2

–1

Figure 8.1–10: A T (7) pdf showing the critical region on the left, p-value on the right 8.1–12 The critical region is t=

d−0 √ ≥ 1.746 = t0.05 (16). sd / 17

Because d = 4.765 and sd = 9.087, t = 2.162 > 1.746 and we reject H0 .

8.2

Tests of the Equality of Two Means

8.2–2 (a) t = s

x−y

¶ ≥ t0.01 (27) = 2.473; 1 1 + 27 16 13 415.16 − 347.40 (b) t = s µ ¶ = 5.570 > 2.473, reject H0 . 1 15(1356.75) + 12(692.21) 1 + 27 16 13 1356.75 (c) c = = 0.662, 1356.75 + 692.21 1 r r

15s2X + 12sy Y 2

= =

0.6622 0.3382 + = 0.0387, 15 12 25.

The critical region is therefore t ≥ t0.01 (25) = 2.485. Because t = 5.570 > 2.485, we again reject H0 .

110

Section 8.2 Tests of the Equality of Two Means 8.2–4 (a) t = s (b) t = s

x−y µ ¶ ≤ −t0.05 (27) = −1.703; 12s2X + 15s2Y 1 1 + 27 13 16 72.9 − 81.7

(12)(25.6)2 + (15)(28.3)2 27

1 1 + 13 16

¶ = −0.869 > −1.703, do not reject H0 ;

|t| = s

|x − y| µ ¶ ≥ t0.025 (18) = 2.101; 1 9sX + 9s2Y 1 + 18 10 10 2

(b) | − 2.151 | > 2.101, reject H0 ;

(d) Yes. X

110

120

130

140

150

Figure 8.2–6: Box-and-whisker diagram for stud 3 (X) and stud 4 (Y ) forces

8.2–8 t = s

x−y µ ¶ = 3.380 > 2.400 = t0.01 (52), 1 24s2X + 28s2Y 1 + 52 25 29

reject µX = µY . 8.2–10 t = r

4.1633 − 5.1050

11(0.91426) + 7(2.59149) 18

1 1 + 12 8

= −1.648. Because −1.330 < −1.648 < −1.734,

0.05 < p-value < 0.10. In fact, p-value = 0.058. We would reject H0 at an α = 0.05 significance level and fail to reject H0 at an α = 0.10 significance level. 8.2–12 (a) s

y−x

> 1.96;

s2 s2Y + X nY nX

(b) 9.01 > 1.96, reject µX = µY ;

Chapter 8 Tests of Statistical Hypotheses

111

Figure 8.2–12: Lengths of male (X) and female (Y ) green lynx spiders 8.2–14 (a) For these data, x = 5.9947, y = 4.3921, s2X = 6.0191, s2Y = 1.9776. Using the number of degrees of freedom given by Equation 7.2-1 (Welch) we have that r = b28.68c = 28. We have t= p

5.9947 − 4.3921

6.0191/19 + 1.9776/19

= 2.47 > 2.467 = t0.01 (28)

so we reject H0 . (b)

Beech

Maple

2 4 6 8 10 Figure 8.2–14: Tree dispersion distances in meters

8.3

Tests for Variances 10s2Y ≤ χ20.95 (10) = 3.940. 5252 The observed value of the test statistic, χ2 = 4.223 > 3.940, so we fail to reject H0 . s2 113108.4909 (b) F = X = = 0.9718 so we clearly accept the equality of the variances. 2 sY 116388.8545 (c) The critical region is | t | ≥ t0.025 (20) = 2.086.

8.3–2 (a) Reject H0 if χ2 =

x−y−0 3385.909 − 3729.364 t= p = = −2.378. 2 2 144.442 sX /11 + sY /11 Because | − 2.378 | > 2.086, we reject the null hypothesis. The p-value for this test is 0.0275.

112

Section 8.2 Tests of the Equality of Two Means 8.3–4 (a) The critical region is χ2 =

19s2 ≤ 10.12. (0.095)2

The observed value of the test statistic, χ2 =

19(0.065)2 = 8.895, (0.095)2

is less than 10.12, so the company was successful. (b) Because χ20.975 (19) = 8.907, p-value ≈ 0.025. 8.3–6 (a) Because χ20.975 (22) = 10.98

and

χ20.025 (22) = 36.78,

22s2 ≤ 10.98 100

χ2 =

100(10.98) = 49.91 22

s 2 ≥ c2 =

H0 will be rejected if χ2 =

22s2 ≥ 36.78 100

or, equivalently, if s2 ≤ c 1 =

100(36.78) = 167.18. 22

(b)

0.06

χ 2 (22)

0.04

0.02 α /2 = 0.025 10

χ2 = 32.52 20

α /2 = 0.025 40

Figure 8.3–6: Test about the variance of IQ scores (c) Because 49.91 < s2 = 147.82 < 167.18, H0 is not rejected. ¶ µ ¶2 µ 100 100 22S 2 (d) Var(S 2 ) = Var · = (2)(22) = 10,000/11. 22 100 22

(e) The 95% confidence interval for σ 2 , · ¸ (22)(147.82) (22)(147.82) , = [88.42, 296.18], 36.78 10.98 contains σ 2 = 100.

Chapter 8 Tests of Statistical Hypotheses

113

s2X 9.88 = 2.42 < 3.28 = F0.05 (12, 8), so fail to reject H0 . = s2Y 4.08 ¸ · ¸ · 9.88 9.88 9.88/4.08 9.88/4.08 , F0.05 (8, 12) , (2.85) = = [0.738, 6.901]. (b) F0.05 (12, 8) 4.08 3.28 4.08

8.3–8 (a)

s2X 12.92 = = 3.30 > 3.07 = F0.05 (9−1, 11−1), so we reject H0 . s2Y 7.12 93 − 132 = −8.12 < −1.645 so H0 is clearly rejected. A modification (b) z = p 12.92 /9 + 7.12 /11 of this test would be to use Welch’s t with 11 degrees of freedom. (See Equation 7.2-1.) This also leads to rejection of H0 .

8.3–10 (a) F =

8.3–12 F =

8.4

9201 = 1.895 < 3.37 = F0.05 (6, 9), so we fail to reject H0 . 4856

Tests about Proportions

8.4–2 (a) C = {x : x = 0, 1, 2}; (b) α = P (X = 0, 1, 2; p = 0.6) = (0.4)4 + 4(0.6)(0.4)3 + 6(0.6)2 (0.4)2 = 0.5248; β

= P (X = 3, 4; p = 0.4) = 4(0.4)3 (0.6) + (0.4)4 = 0.1792.

OR (a0 ) C = {x : x = 0, 1}; (b0 ) α = P (X = 0, 1; p = 0.6) = (0.4)4 + 4(0.6)(0.4)3 = 0.1792; β

= P (X = 2, 3, 4; p = 0.4) = 6(0.4)2 (0.6)2 + 4(0.4)3 (0.6) + (0.4)4 = 0.5248.

8.4–4 Using Table II in the Appendix, (a) (b)

α = P (Y ≥ 13; p = 0.40) = 1 − 0.8462 = 0.1538;

= P (Y ≤ 12; p = 0.60) = P (25 − Y ≥ 25 − 12) = 1 − 0.8462 = 0.1538.

where 25 − Y is b(25, 0.40)

8.4–6 The value of the test statistic is z=p

0.70 − 0.75

(0.75)(0.25)/390

= −2.280.

(a) Because z = −2.280 < −1.645, reject H0 .

(b) Because z = −2.280 > −2.326, do not reject H0 .

8.4–8 Test H0 : p = 0.334 against H1 : p > 0.334. The computed test statistic is Z=p

(179/480) − 0.334

0.334(1 − 0.334)/480

= 1.81,

which exceeds the critical value of z0.05 = 1.645. So H0 is rejected at the 0.05 level of significance, indicating that the educational requirement is met. 8.4–10 (a) z = p

y/n − 0.65

(0.65)(0.35)/n

≥ 1.96;

114

Section 8.5 The Wilcoxon Tests 414/600 − 0.65 = 2.054 > 1.96, reject H0 at α = 0.025; (b) z = p (0.65)(0.35)/600 (c) Because the p-value ≈ P (Z ≥ 2.054) = 0.0200 < 0.0250, reject H0 at an α = 0.025 significance level; (d) A 95% one-sided confidence interval for p is p [0.69 − 1.645 (0.69)(0.31)/600 , 1] = [0.659, 1].

8.4–12 (a) | z | = p

| pb − 0.20 |

≥ 1.96; (0.20)(0.80)/n (b) Only 5/54 for which z = −1.973 leads to rejection of H0 , so 5% reject H0 ; (c) 5%; (d) 95%; 219/1124 − 0.20 (e) z = p = −0.43, so fail to reject H0 . (0.20)(0.80)/1124

pb1 − pb2 8.4–14 (a) z = p ≥ 1.645; pb(1 − pb)(1/n1 + 1/n2 ) 0.15 − 0.11 = 2.341 > 1.645, reject H0 ; (b) z = p (0.1325)(0.8675)(1/900 + 1/700) (c) z = 2.341 > 2.326, reject H0 ; (d) The p-value ≈ P (Z ≥ 2.341) = 0.0096.

−0.05 204/300 − 0.73 = = −1.95; 8.4–16 z = p 0.02563 (0.73)(0.27)/300 p-value ≈ P (Z < −1.95) = 0.0256 < α = 0.05 so we reject H0 . That is, the test indicates that there is progress.

8.5

Some Distribution-Free Tests

8.5–2 In the following display, those observations that were negative are underlined. |x| :

Ranks :

|x| :

Ranks :

3.5

14.5

20.5

The value of the Wilcoxon signed rank statistic is w

−1 − 3.5 − 3.5 − 3.5 + 3.5 − 6 − 8 − 8 − 8 − 10 − 12 + 12 + 12 + 14.5 + 14.5 + 16 + 17 + 18 + 19 − 20.5 + 20.5 + 22 + 23 + 24

132.

For a one-sided alternative, the approximate p-value is, using the one-unit correction, ! Ã W −0 132 − 1 − 0 ≥ P (W ≥ 132) = P p 70 24(25)(49)/6 ≈

P (Z ≥ 1.871) = 0.03064.

For a two-sided alternative, p-value = 2(0.03064) = 0.0613 > 0.05. Thus, we do not reject H0 .

Chapter 8 Tests of Statistical Hypotheses

115

8.5–4 In the following display, those observations that were negative are underlined. |x| :

0.0790

0.5901

0.7757

1.0962

1.9415

|x| :

3.0678

3.8545

5.9848

9.3820

74.0216

Ranks : Ranks :

The value of the Wilcoxon signed rank statistic is w = −1 + 2 − 3 − 4 − 5 − 6 + 7 − 8 + 9 − 10 = −19. Because

¯ ¯ ¯ −19 + 1 − 0 ¯ ¯ ¯ |z| = ¯ p ¯ = 0.917 < 1.96, ¯ 10(11)(21)/6 ¯

we do not reject H0 .

8.5–6 (a) In the following display, those differences that were negative are underlined. |xi − 50| :

2.5

4.5

Ranks :

1.5

5.5

|xi − 50| :

7.5

14.5

15.5

11.5

Ranks :

The value of the Wilcoxon signed rank statistic is w

= 1.5 − 1.5 + 3 + 4 + 5.5 − 5.5 − 7 − 8 + 9 + 10 + 11.5 + 11.5 − 13 + 14 + 15 = 50.

Because z=p

we do not reject H0 ;

50 − 1

15(16)(31)/6

= 1.392 < 1.645,

(b) The approximate p-value is, using the one-unit correction, p-value

= ≈

P (W Ã ≥ 50) P Z≥p

50 − 1

15(16)(31)/6

= P (Z ≥ 1.3915) = 0.0820.

8.5–8 The 24 ordered observations, with the x-values underlined and the ranks given under each observation are:

Ranks : Ranks : Ranks : Ranks :

0.7494

0.7546

0.7565

0.7613

0.7615

0.7701

0.7712

0.7719

0.7720

0.7731

8.5

10.5

0.7741

0.7750

0.7776

0.7795

0.7811

14.5

0.7815

0.7816

0.7851

0.7870

0.7876

0.7972

116

Section 8.5 The Wilcoxon Tests (a) The value of the Wilcoxon rank sum statistic is w

1 + 2 + 3 + 5 + 6 + 7 + 8.5 + 8.5 + 10.5 + 13 + 14.5 + 16

95.

Thus Ã

p-value = P (W ≤ 95) ≈ P Z ≤ p

95.5 − 150

12(12)(25)/12

= P (Z ≤ −3.15) < 0.001,

so that we clearly reject H0 ; (b)

y 0.80 0.79 0.78 0.77 0.76 0.75 0.75

0.76

0.77

0.78

0.79

0.80

Figure 8.5–8: q-q plot of pill weights, (good, defective) = (x, y)

8.5–10 The ordered combined sample with the x observations underlined are: Ranks: Ranks: Ranks:

67.4

69.3

72.7

73.1

75.9

77.2

77.6

78.9

82.5

83.2

83.3

84.0

84.7

86.5

87.5

87.6

88.3

88.6

90.2

90.4

92.7

94.4

95.0

20.5

The value of the Wilcoxon rank sum statistic is w = 1 + 2 + 3 + 5 + 6 + 7 + 10 + 11 + 12 + 16 + 19 + 20.5 = 112.5. Because Ã

113 − (12)(25)/2 p-value = P (W ≤ 112.5) ≈ P Z ≤ p 12(12)(25)/12

= P (Z ≤ −2.136) < 0.05,

we reject H0 .

Chapter 8 Tests of Statistical Hypotheses

117

8.5–12 The ordered combined sample with the 48-passenger bus values underlined are: Ranks: Ranks: Ranks:

104

184

196

197

248

253

260

279

300

308

323

331

355

386

393

396

414

432

450

452

The value of the Wilcoxon statistic is w = 1 + 6 + 9 + 10 + 11 + 12 + 16 + 17 + 20 = 102. Because Ã

101.5 − (9)(21)/2 p-value = P (W ≥ 102) ≈ P Z > p (9)(11)(21)/12

= P (Z > 0.532) > 0.05,

we do not reject H0 .

8.5–14 (a) Here is the two-sided stem-and-leaf display. Young Subjects 97 33 98 88 85 68 50 04 86 81 79 77 76 69 65 61 96

Stems 3• 4∗ 4• 5∗ 5• 6∗ 6• 7∗ 7• 8∗ 8• 9∗

Older Subjects

63 38 49 73 88 90 94 26 27 53 74 28 18 35 63 88 36

(b) The value of the Wilcoxon rank sum statistic, the sum of the ranks for the younger subjects, is w = 1 + 2 + 3 + 5 + 6 + 7 + 8 + 9 + 12 + 13 + 14 + 16 + 17 + 18 + 19 + 20 + 28 = 198. Because Ã

198.5 − (17)(35)/2 p-value = P (W ≤ 198) ≈ P Z ≤ p 17(17)(35)/12

= P (Z ≤ −3.410) < 0.05,

we clearly reject H0 .

(c) The t test leads to the same conclusion. 8.5–16 (a) Using the Wilcoxon rank sum statistic, the sum of the ranks for the normal air is w = 2 + 3 + 4 + 5 + 6 + 7 + 8 + 10 + 12 + 13 + 15 + 17 = 102.

118

Section 8.6 Power of a Statistical Test Because Ã

102.5 − (12)(21)/2 p-value = P (W ≤ 102) ≈ P Z ≤ p 12(8)(21)/12

= P (Z ≤ −1.813) = 0.035,

we reject the null hypothesis.

(b) Using a t statistic, we failed to reject the null hypothesis at an α = 0.05 significance level. (c) For these data, the results are a little different with the Wilcoxon rank sum statistic leading to rejection of the null hypothesis while the t test did not reject H0 .

8.6

Power of a Statistical Test

8.6–2 (a)

K(µ)

P ( X ≤ 354.05; µ) ¶ µ 354.05 − µ √ ; µ = P Z≤ 2/ 12 µ ¶ 354.05 − µ √ = Φ ; 2/ 12

354.05 − 355 √ (b) α = K(355) = Φ 2/ 12 (c)

= Φ(−1.645) = 0.05;

K(354.05)

Φ(0) = 0.5;

K(353.1)

Φ(1.645) = 0.95;

K(µ) 1.00

0.75

0.50

0.25

353

353.5

354

354.5

355 √ Figure 8.6–2: K(µ) = Φ([354.05 − µ]/[2/ 12 ])

(d) x = 353.83 < 354.05, reject H0 ; (e)

p-value

= =

P ( X ≤ 353.83; µ = 355) P (Z ≤ −2.03) = 0.0212.

Chapter 8 Tests of Statistical Hypotheses

119

P10 8.6–4 (a) α = P ( i=1 Xi ≤ 14; p = 0.4) = P (10 ≤ Y ≤ 14) where Y is distributed as negative binomial with parameters r = 10 and p = 0.4. So µ ¶ µ ¶ µ ¶ 9 10 11 10 0 10 1 α = 0.4 0.6 + 0.4 0.6 + 0.410 0.62 9 9 9 µ ¶ µ ¶ 12 13 10 3 + 0.4 0.6 + 0.410 0.64 9 9 =

0.0175;

(b) β = 1 − P (Y ≤ 14) where Y is distributed as negative binomial with parameters r = 10 and p = 0.7. So µ ¶ µ ¶ µ ¶ 11 10 9 10 1 10 0 0.710 0.32 0.7 0.3 − 0.7 0.3 − β = 1− 9 9 9 µ ¶ µ ¶ 13 12 10 3 0.710 0.34 0.7 0.3 − − 9 9 = 0.4158.

P (S 2 < c | σ 2 = 10) ¶ µ (n − 1)c 2 (n − 1)S 2 < | σ = 10 = P σ2 σ2 µ ¶ 23c = P Y < , 10 23c where Y is distributed as χ2 (23). So = χ20.95 (23), i.e., c = (10/23)13.09 = 5.69; 10 (b) 0.9 = P (S 2 < c | σ 2 = 2) µ ¶ µ ¶ (n − 1)S 2 (n − 1)c 2 23c = P < |σ = 2 = P Y < σ2 σ2 2

8.6–6 (a)

0.05

23c = χ20.10 (23), i.e., c = (2/23)32.01 = 2.78. 2 µ ¶ 668.94 − µ 8.6–8 (a) K(µ) = P ( X ≤ 668.94; µ) = P Z ≤ ; µ 140/5 ¶ µ 668.94 − µ ; = Φ 140/5 µ ¶ 668.94 − 715 (b) α = K(715) = Φ 140/5 where Y is distributed as χ2 (23). So

(c)

Φ(−1.645) = 0.05;

K(668.94)

Φ(0) = 0.5;

K(622.88)

Φ(1.645) = 0.95;

120

Section 8.6 Power of a Statistical Test K(µ) 1.00

(d)

0.75

0.50

0.25

600

620

640

660

680

700

720

µ 740

Figure 8.6–8: K(µ) = Φ([668.94 − µ]/[140/5]) (e) (f )

x = 667.92 < 668.94, reject H0 ; p-value = P ( X ≤ 667.92; µ = 715) P (Z ≤ −1.68) = 0.0465.

8.6–10 (a) and (b) K(p) 1.00

K(p) 1.00

0.75

0.50

0.25

0.3

0.4

0.5

p 0.7

0.6

0.2

0.3

0.4

0.5

0.6

0.7

Figure 8.6–10: Power functions corresponding to different critical regions and different sample sizes

8.6–12 Let Y =

8 X

Xi . Then Y has a Poisson distribution with mean µ = 8λ.

P (Y ≥ 8; λ = 0.5) = 1 − P (Y ≤ 7; λ = 0.5)

i=1

(a)

1 − 0.949 = 0.051.

(b) K(λ) = 1 − (c)

7 X (8λ)y e−8λ y=0

K(0.75)

K(1.00)

K(1.25)

1 − 0.744 = 0.256,

1 − 0.453 = 0.547,

1 − 0.220 = 0.780.

Chapter 8 Tests of Statistical Hypotheses

121

K(λ) 1.00

0.75

0.50

0.25

0.5

1.0

1.5

2.0

Figure 8.6–12: K(λ) = 1 − P (Y ≤ 7; λ) 8.6–14

K(θ)

= =

K(0)

K(10)

c−0 = c − 10 = 10 = n = c

8.7

P (X − (Y ) ≥ c; θ) ! Ã c−θ X −Y −θ √ ;θ ≥ P p 25/ n 400/n + 225/n c−0 √ = 1.645 0.05 ⇒ 25/ n c − 10 √ = −1.282 0.90 ⇒ 25/ n √ 1.645(25/ n) √ −1.282(25/ n) √ 2.927(25/ n) 53.55 % 54 √ 1.645(25/ 54) = 5.60.

Best Critical Regions L(4) L(16)

8.7–2 (a)

−

n X

3 x2 32 i=1 i n X i=1

(b)

0.05

√ (1/2 2π )n exp[−Σ x2i /8 ] √ (1/4 2π )n exp[−Σ x2i /32 ]

2n exp[−3Σ x2i /32 ] ≤ k

≤

ln k − ln 2n µ

¶ 32 ≥ − (ln k − ln 2n ) = c; 3 ! Ã 15 X 2 2 P Xi ≥ c; σ = 4

x2i

i=1

ÃP

15 2 i=1 Xi

c ≥ ; σ2 = 4 4

λ 2.5

122

Section 8.7 Best Critical Regions c = χ20.05 (15) = 25 and c = 100. 4 ! Ã 15 X Xi2 < 100; σ 2 = 16 = P

Thus (c)

i=1

8.7–4 (a)

ÃP

15 2 i=1 Xi

100 < = 6.25 16

L(0.9) (0.9)Σxi (0.1)n−Σxi = L(0.8) (0.8)Σxi (0.2)n−Σxi ·µ ¶µ ¶¸Pn xi · ¸n 1 2 1 9 8 1 2 ! Ã n X xi ln(9/4) i=1

n X

i=1

(b)

≈ 0.025.

≤

ln k + n ln 2

≤

ln k + n ln 2 = c. ln(9/4)

Recall that the distribution of the sum of Bernoulli trials, Y , is b(n, p). 0.10 = P [ Y ≤ n(0.85); p = 0.9 ] " # Y − n(0.9) n(0.85) − n(0.9) = P p ≤ p ; p = 0.9 . n(0.9)(0.1) n(0.9)(0.1) It is true, approximately, that

(c)

n(−0.05) √ n(0.3) n

β = P [ Y > n(0.85) = 51; p = 0.8 ]

= ≈

−1.282

= "

59.17 or n = 60.

Y − 60(0.8) 51 − 48 ; p = 0.8 P p > √ 9.6 60(0.8)(0.2) P (Z ≥ 0.97) = 0.166.

(d) Yes. 8.7–6 (a)

0.05

c1 − 80 X − 80 = P ≥ 3/4 3/4 µ ¶ c1 − 80 = 1−Φ . 3/4

Thus c1 − 80 3/4 c1

1.645

81.234.

−1.645

78.766;

c3 3/4

1.96

1.47.

Similarly, c2 − 80 3/4 c2

Chapter 8 Tests of Statistical Hypotheses (b)

123

K1 (µ)

1 − Φ([81.234 − µ]/[3/4]);

K2 (µ)

Φ([78.766 − µ]/[3/4];

K3 (µ)

1 − Φ([81.47 − µ]/[3/4]) + Φ([78.53 − µ]/[3/4]).

K(µ) 1.00 K1

0.75

0.50

0.25

K3 77

Figure 8.7–6: Three power functions 8.7–8 Let L(θ) = θ

n Y

i=1

thus L(θ0 ) = θ0n

n Y

i=1

(1 − xi )θ−1 ;

(1 − xi )θ0 −1 = 1,

because θ0 = 1. So when θ > 1, a best critical region is 1 L(θ0 ) = n Qn ≤ k. L(θ) θ (1 − xi )θ−1 i=1

That is,

" n Y

i=1

n Y

i=1

(1 − xi )

(1 − xi ) ≥

#θ−1

1 kθn

≥

1 k

¶1/(θ−1)

= c.

This is true for all θ > 1; so it is a uniformly most powerful test. 8.7–10 By the Neyman-Pearson Theorem, the desired critical region is of the form L(1) ≤ k L(2) Pn (1/2)n exp (− i=1 |xi |) Pn ≤ k (1/4)n exp (− i=1 |xi |/2) Pn exp(− i=1 |xi |/2) ≤ (1/2)n k Pn

i=1 |xi |

≥

−2 log[(1/2)n k] = c.

124

Section 8.8 Likelihood Ratio Tests

8.8

Likelihood Ratio Tests

8.8–2 (a) If µ ∈ ω (that is, µ ≥ 10.35), then µ b = x if x ≥ 10.35, but µ b = 10.35 if x < 10.35. Thus λ = 1 if x ≥ 10.35; but, if x < 10.35, then Pn [1/(0.3)(2π)]n/2 exp[− 1 (xi − 10.35)2 /(0.06)] Pn λ= [1/(0.3)(2π)]n/2 exp[− 1 (xi − x)2 /(0.06)] h n i exp − (x − 10.35)2 0.06 n − (x − 10.35)2 0.06 x − 10.35 p 0.03/n

≤

ln k

≤

√ − −2 ln k

−z0.05

−1.645.

10.31 − 10.35 p = −1.633 > −1.645; do not reject H0 . 0.03/50

(b)

(c) p-value = P (Z ≤ −1.633) = 0.0513. Pn [1/(2πσ02 )]n/2 exp[− i=1 (xi − µ0 )2 /(2σ02 )] Pn 8.8–4 (a) λ = [1/(2πσ02 )]n/2 exp[− i=1 (xi − x)2 /(2σ02 )] Pn ¸ · Pn 2 − i=1 (xi − x + x − µ0 )2 i=1 (xi − x) + = exp 2σ02 2σ02 · ¸ −n(x − µ0 )2 = exp ≤ k 2σ02 −n(x − µ0 )2 2σ02

≤

ln k

|x − µ0 | √ σ0 / n

≥

|z|

| x − 59 | √ ≥ 1.96. 15/ n

| 56.13 − 59 | = | − 1.913| < 1.96, do not reject H0 ; 15/10 (c) p-value = P (|Z| ≥ 1.913) = 0.0558.

(b) |z| =

8.8–6 t =

324.8 − 335 √ = −1.051 > −1.337, do not reject H0 . 40/ 17

8.8–8 In Ω, µ b = x. Thus,

Pn (1/θ0 )n exp[− 1 xi /θ0 ] Pn (1/x)n exp[− 1 xi /x] µ ¶n x exp[−n(x/θ0 − 1)] θ0

λ=

≤

Plotting λ as a function of w = x/θ0 , we see that λ = 0 when x/θ0 = 0, it has a maximum when x/θ0 = 1, and it approaches 0 as x/θ0 becomes large. Thus λ ≤ k when x ≤ c1 or x ≥ c2 .

Chapter 8 Tests of Statistical Hypotheses

125 n

Because the distribution of

2 X Xi is χ2 (2n) when H0 is true, we could let the critical θ0 i=1

region be such that we reject H0 if n

2 X Xi ≤ χ21−α/2 (2n) θ0 i=1

2 X Xi ≥ χ2α/2 (2n). θ0 i=1

1.00

0.75

0.50

0.25

0.5

1.0

1.5

2.0

2.5

3.0

Figure 8.8–8: Likehood functions: solid, n = 5; dotted, n = 10

8.8–10 Referring to Exercise 6.4-19, we test H0 : γ 2 = 1 against H1 : γ 2 6= 1. # Ã n ! " n Y X (yi − µ 1 b)2 p exp − 2x2i 2πx2i i=1 i=1  λ =  # ≤ k " n n 2 Y X (yi − µ 1 b )   exp − q c2 2 c2 x2 i=1 i=1 2 γ xi 2π γ i =

# " n n 2 2 X X (yi − µ (y − µ b ) b ) i n/2 c ≤ k + ( γ2 ) exp − c2 2 2x2i i=1 2 γ x i=1 i

= =

c2 )n/2 exp[−n γ c2 /2 + n/2] (γ

c2 )n/2 exp[n(1 − γ c2 )/2] ≤ k. (γ

This is a function of

where

≤ k

n X (yi − µ b)2 c2 = 1 γ n i=1 x2i

Pn yi /x2i µ b = Pi=1 n 2 . i=1 1/xi

Chapter 9 More Tests

127

Chapter 9

More Tests 9.1

Chi-Square Goodness-of-Fit Tests

9.1–2

= =

(119 − 116)2 (130 − 116)2 (48 − 58)2 (59 − 58)2 (224 − 232)2 + + + + 232 116 116 58 58 3.784.

The null hypothesis will not be rejected at any reasonable significance level. Note that E(Q4 ) = 4 when H0 is true. 9.1–4

(30 − 39)2 (43 − 39)2 (11 − 13)2 (124 − 117)2 + + + 117 39 39 13 = 0.419 + 2.077 + 0.410 + 0.308 = 3.214 < 7.815 = χ20.05 (3). Thus we do not reject the Mendelian theory with these data. q3

9.1–6 Using Table II in Appendix B with p = 0.30, the hypothesized probabilities are p 0 = P (X = 0) = 0.343, p1 = P (X = 1) = 0.441, p2 = P (X = 2) = 0.189, p3 = P (X = 3) = 0.027. Thus the respective expected values are 68.6, 88.2, 37.8, and 5.4. The value of the chi-square goodness of fit statistic is: q

(95 − 88.2)2 (38 − 37.8)2 (10 − 5.4)2 (57 − 68.6)2 + + + 68.6 88.2 37.8 5.4

= 6.405 < 7.815 = χ20.05 (3). Do not reject the hypothesis that X is b(3, 0.30) at a 5% significance level. Limits for the p-value are 0.05 < p-value < 0.10 because χ20.10 (3) = 6.251 < 6.405 < χ20.05 (3) = 7.815. We find that pb = 201/600 = 0.335. Thus a 95% confidence interval for p is p 0.335 ± 1.96 (0.335)(0.665)/600 or [0.297, 0.373].

The pennies that were used were minted 1998 or earlier. See Figure 9.1-6. Repeat this experiment with similar pennies or with newer pennies and compare your results with those obtained by these students.

128

Section 9.1 Chi-Square Goodness-of-Fit Tests

0.4 0.3 0.2 0.1 0

Figure 9.1–6: The b(3, 3/10) probability histogram and the relative frequency histogram (shaded)

9.1–8 (a) Estimate λ by x = 50/40 = 1.25. Then [12 − (0.3581)(40)]2 [15 − (0.2865)(40)]2 + + q3 = (0.2865)(40) (0.3581)(40) [7 − (0.2238)(40)]2 [6 − (0.1316)(40)]2 + (0.2238)(40) (0.1316)(40) = 1.999 < χ20.05 (2) = 5.991; do not reject H0 ; (b) Estimate p by 1/y = 1/2.25 = 4/9. Then [15 − (4/9)(40)]2 [12 − (20/81)(40)]2 q3 = + + (4/9)(40) (20/81)(40) [6 − (125/729)(40)]2 [7 − (100/729)(40)]2 + (100/729)(40) (125/729)(40) = 1.415 < χ20.05 (2) = 5.991; do not reject H0 . 9.1–10 The respective probabilities and expected frequencies are 0.050, 0.149, 0.224, 0.224, 0.168, 0.101, 0.050, 0.022, 0.012 and 15.0, 44.7, 67.2, 67.2, 50.4, 30.3, 15.0, 6.6, 3.6. The last two cells could be combined to give an expected frequency of 10.2. From Exercise 3.5–12, the respective frequencies are 17, 47, 63, 63, 49, 28, 21, and 12 giving q7 =

(47 − 44.7)2 (12 − 10.2)2 (17 − 15.0)2 + + ··· + = 3.841. 15.0 44.7 10.2

Since 3.841 < 14.07 = χ20.05 (7), do not reject. The sample mean is x = 3.03 and the sample variance is s2 = 3.19 which also supports the hypothesis. The following figure compares the probability histogram with the relative frequency histogram of the data.

Chapter 9 More Tests

129

0.20 0.15 0.10 0.05

Figure 9.1–10: The Poisson probability histogram, λ = 3, and relative frequency histogram (shaded) 9.1–12 (a) For the infected snails, x = 84.74, sx = 64.79; For the control snails, y = 113.16, sy = 87.02; (b)

100

150

200

250

300

Figure 9.1–12: Box-and-whisker diagrams for infected (X) and control (Y )

Observed

Expected

[0, 25.25) [25.25, 57.81) [57.81, 103.69) [103.69, 182.13) [182.13, ∞)

4 9 5 6 7

6.2 6.2 6.2 6.2 6.2

0.781 1.264 0.232 0.006 0.103

31.0

2.386

The p-value for 5 − 1 − 1 = 3 degrees of freedom is 0.496 so we fail to reject the null hypothesis;

130

Section 9.2 Contingency Tables (d) We shall use 10 classes with equal probability for the infected snalis. Ai

Observed

Expected

[0, 22.34) [22.34, 34.62) [34.62, 46.09) [46.09, 57.81) [57.81, 70.49) [70.49, 84.94) [84.94, 102.45) [102.45, 125.76) [125.76, 163.37) [163.37, ∞)

3 3 8 4 2 4 4 4 2 5

3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9

0.208 0.208 4.310 0.003 0.926 0.003 0.003 0.003 0.926 0.310

39.0

6.900

The p-value for 10 − 1 = 9 degrees of freedom is 0.648 so we fail to reject the null hypothesis.

9.2

Contingency Tables

9.2–2 10.18 < 20.48 = χ20.025 (10), accept H0 . 9.2–4 In the combined sample of 45 observations, the lower third includes those with scores of 61 or lower, the middle third have scores from 62 through 78, and the higher third are those with scores of 79 and above. low

middle

high

Totals

Class U

9 (5)

4 (5)

2 (5)

Class V

5 (5)

Class W

1 (5)

6 (5)

8 (5)

Totals

Thus q = 3.2 + 0.2 + 1.8 + 0 + 0 + 0 + 3.2 + 0.2 + 1.8 = 10.4. Since q = 10.4 > 9.488 = χ20.05 (4), we reject the equality of these three distributions. (p-value = 0.034.) 9.2–6 q = 8.410 < 9.488 = χ20.05 (4), fail to reject H0 . (p-value = 0.078.) 9.2–8 q = 4.268 > 3.841 = χ20.05 (1), reject H0 . (p-value = 0.039.) 9.2–10 q = 7.683 < 9.210 = χ20.01 (2), fail to reject H0 . (p-value = 0.021.) 9.2–12 q = 8.792 > 7.378 = χ20.025 (2), reject H0 . (p-value = 0.012.)

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131

9.2–14 The contingency table is: Eyes

Totals

Blue

22 (20)

17 (20)

21 (20)

20 (20)

Brown

14 (17.50)

20 (17.50)

16 (17.0)

Other

14 (12.50)

13 (12.50)

9 (12.50)

14 (12.50)

Totals

200

Because q = 3.603 < 12.59 = χ20.05 (6), do not reject the hypothesis of independent attributes. (p-value = 0.730.)

9.3

One-Factor Analysis of Variance

9.3–2 Source

p-value

Treatment Error

388.2805 316.4597

3 12

129.4268 26.3716

4.9078

0.0188

Total

704.7402

F = 4.9078 > 3.49 = F0.05 (3, 12), reject H0 . 9.3–4 (a) Source

p-value

Treatment Error

184.8 102.0

2 17

92.4 6.0

15.4

0.00015

Total

286.8

F = 15.4 > 3.59 = F0.05 (2, 17), reject H0 ; (b)

C 74

Figure 9.3–4: Box-and-whisker diagrams of strengths of different beams

132

Section 9.3 One-Factor Analysis of Variance

9.3–6 (a) F ≥ F0.05 (3, 24) = 3.01; (b)

Source

p-value

Treatment Error

12,280.86 28,434.57

3 24

4,093.62 1,184.77

3.455

0.0323

Total

40,715.43

F = 3.455 > 3.01, reject H0 ; (c) 0.025 < p-value < 0.05. (d)

X1 X2 X3 X4 140

160

180

200

220

240

260

280

Figure 9.3–6: Box-and-whisker diagrams for cholesterol levels 9.3–8 (a) F ≥ F0.05 (4, 30) = 2.69; (b)

Source

p-value

Treatment Error

0.00442 0.01157

4 30

0.00111 0.00039

2.85

0.0403

Total

0.01599

F = 2.85 > 2.69, reject H0 ;

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133

(c)

X1 X2 X3 X4 X5 1.02

1.04

1.06

1.08

1.10

Figure 9.3–8: Box-and-whisker diagrams for nail weights

9.3–10 (a) t = s F =

92.143 − 103.000 µ ¶ = −2.55 < −2.179, reject H0 . 6(69.139) + 6(57.669) 1 1 + 12 7 7

412.517 = 6.507 > 4.75, reject H0 . 63.4048

The F and the t tests give the same results since t2 = F ; (b) F =

86.3336 = 0.7515 < 3.55, do not reject H0 . 114.8889

9.3–12 (a) Source

p-value

Treatment Error

122.1956 860.4799

2 30

61.0978 28.6827

2.130

0.136

Total

982.6755

F = 2.130 < 3.32 = F0.05 (2, 30), fail to reject H0 ; (b) D6

D 22 165

170

175

180

185

Figure 9.3–12: Box-and-whisker diagrams for resistances on three days

134

Section 9.4 Two-Way Analysis of Variance

9.3–14 (a) Source

p-value

Worker Error

1.5474 17.2022

2 24

0.7737 0.7168

1.0794

0.3557

Total

18.7496

F = 1.0794 < 3.40 = F0.05 (2, 24), fail to reject H0 ; (b)

C 2 2.5 3 3.5 1 1.5 4 4.5 Figure 9.3–14: Box-and-whisker diagrams for workers A, B, and C The box plot confirms the answer from part (a).

9.4

Two-Way Analysis of Variance

9.4–2

µ + βj

6 10 8 8

3 7 5 5

7 11 9 9

8 12 10 10

µ + αi 6 10 8 µ=8

So α1 = −2, α2 = 2, α3 = 0 and β1 = 0, β2 = −3, β3 = 1, β4 = 2. 9.4–4

b a X X i=1 j=1

( X i· − X ·· )(Xij − X i· − X ·j + X ·· ) =

a X i=1

= =

a X

i=1 a X i=1

b a X X i=1 j=1

( X i· − X ·· ) ( X i· − X ·· )

b X

[(Xij − X i· ) − ( X ·j − X ·· )]

j=1  b X



j=1

(Xij − X i· ) −

b X j=1

( X ·j − X ·· )

( X i· − X ·· )(0 − 0) = 0;

( X ·j − X ·· )(Xij − X i· − X ·j + X ·· ) = 0, similarly; ( X i· − X ·· )( X ·j − X ·· ) =

( a X i=1

( X i· − X ·· )

) b X 

j=1

( X ·j − X ·· )

  

= (0)(0) = 0 .

Chapter 9 More Tests

135

9.4–6

µ + βj

6 10 8 8

7 3 5 5

7 11 9 9

12 8 10 10

µ + αi 8 8 8 µ=8

So α1 = α2 = α3 = 0 and β1 = 0, β2 = −3, β3 = 1, β4 = 2 as in Exercise 9.4–2. However, γ11 = −2 because 8 + 0 + 0 + (−2) = 6. Similarly we obtain the other γij ’s : −2 2 0

2 −2 0

−2 2 0

2 −2 0

9.4–8 Source

p-value

Row (A) Col (B) Int(AB) Error

5,103.0000 6,121.2857 1,056.5714 28,434.5714

1 1 1 24

5,103.0000 6,121.2857 1,056.5714 1,184.7738

4.307 5.167 0.892

0.049 0.032 0.354

Total

40,715.4286

(a) Since F = 0.892 < F0.05 (1, 24) = 4.26, do not reject HAB ; (b) Since F = 4.307 > F0.05 (1, 24) = 4.26, reject HA ; (c) Since F = 5.167 > F0.05 (1, 24) = 4.26, reject HB .

9.5

General Factorial and 2k Factorial Designs

9.5–4 (a)

(b)

[A] = −28.4/8 = −3.55, [B] = −1.45, [C] = 3.2, [AB] = −1.525, [AC] = −0.525, [BC] = 0.375, [ABC] = −1.2; Identity of Effect [A] [AB] [B] [ABC] [AC] [BC] [C]

Ordered Effect −3.550 −1.525 −1.450 −1.200 −0.525 0.375 3.20

Percentile 12.5 25.0 37.5 50.0 62.5 75.0 87.5

Percentile from N(0,1) −1.15 −0.67 −0.32 0.00 0.32 0.67 1.15

The main effects of temperature (A) and concentration (C) are significant.

136

Section 9.6 Tests Concerning Regression and Correlation

3 2

–3

[ABC] –2[B] –1 [AB]

[A]

[AC]

[C]

[BC]

–1 –2 –3

Figure 9.5–4: q-q plot

9.6

Tests Concerning Regression and Correlation

9.6–2 The critical region is t1 ≥ t0.025 (8) = 2.306. From Exercise 6.5–4, 10 X c2 = 1.84924; also βb = 4.64/5.04 and nσ (xi − x)2 = 5.04, so t1

0.9206 4.64/5.04 s = 4.299. = 0.2142 1.84924 8(5.04)

i=1

Since t1 = 4.299 > 2.306, we reject H0 . 9.6–4 For these data, r = −0.413. Since |r| = 0.413 < 0.7292, do not reject H 0 . 9.6–6 Following the suggestion given in the hint, the expression equals (n − 1)SY2 −

2RsX SY R2 s2X SY2 (n − 1)s2X (n − 1)Rs S + X Y s2X s4X u(R)

≈

u(ρ) + (R − ρ)u0 (ρ),

Var[u(ρ) + (R − ρ)u0 (ρ)]

[u0 (ρ)]2 Var(R)

9.6–8

u(ρ) Thus, taking k = 1,

(n − 1)SY2 (1 − 2R2 + R2 )

(n − 1)SY2 (1 − R2 ).

(1 − ρ2 )2 = c, which is free of ρ, n k/2 k/2 + , = 1−ρ 1+ρ µ ¶ k k k 1+ρ = − ln(1 − ρ) + ln(1 + ρ) = ln . 2 2 2 1−ρ =

u0 (ρ)

[u0 (ρ)]2

u(R) =

¸ µ ¶ · 1+R 1 ln 2 1−R

has a variance almost free of ρ.

Chapter 9 More Tests

137

9.6–10 (a) r = −0.4906, | r | = 0.4906 > 0.4258, reject H0 at α = 0.10; (b) | r | = 0.4906 < 0.4973, fail to reject H0 at α = 0.05.

9.6–12 (a) r = 0.339, | r | = 0.339 < 0.5325 = r0.025 (12), fail to reject H0 at α = 0.05; (b) r = −0.821 < −0.6613 = r0.005 (12), reject H0 at α = 0.005;

9.7

Statistical Quality Control

9.7–2 (a) x = 67.44, s = 2.392, R = 5.88; (b) UCL = x + 1.43(s) = 67.44 + 1.43(2.392) = 70.86; LCL = x − 1.43(s) = 67.44 − 1.43(2.392) = 64.02;

UCL

3 _ x

68 66

_ s

2 1

LCL 2

10 12 14 16 18 20

LCL 10 12 14 16 18 20

Figure 9.7–2: (b) x-chart using s and (c) s-chart

138

Section 9.7 Statistical Quality Control (d) UCL = x + 0.58(R) = 67.44 + 0.58(5.88) = 70.85; LCL = x − 0.58(R) = 67.44 − 0.58(5.88) = 64.03;

(e) UCL = 2.11(R) = 2.11(5.88) = 12.41; LCL = 0;

UCL

10 8

70 _ x

6 4

_ R

LCL 2

10 12 14 16 18 20

LCL 10 12 14 16 18 20

Figure 9.7–2: (d) x-chart using R and (e) R-chart (f ) Quite well until near the end. p 9.7–4 With p = 0.0254, UCL = p + 3 p(1 − p)/100 = 0.073; p with p = 0.02, UCL = p + 3 p(1 − p)/100 = 0.062;

In both cases we see that problems are arising near the end.

0.08

UCL

0.06

0.08 UCL

0.06

0.04

_ p

0.02

0.04 _ p

0.02

LCL 8 10 12 14 16 18 20 22 24

Figure 9.7–4: p-charts using p = 0.0254 and p = 0.02

Chapter 9 More Tests

139

√ √ 9.7–6 (a) UCL = c + 3 c = 1.80 + 3 1.80 = 5.825;

LCL = 0; UCL

6 5 4 3

_ c

2 1 2

LCL 10 12 14 16 18 20

Figure 9.7–6: c-chart (b) The process is in statistical control. 5101 = 34.0067, s = 2.8244, R = 7; 150 (b) UCL = x + 1.43(s) = 38.046 for the x chart; LCL = x − 1.43(s) = 29.968 for the x chart;

9.7–8 (a) x =

UCL

6 5

36 4

__ x

_ s

3 2

32 LCL

30 0

1 0

Figure 9.7–8: (b) x-chart using s and (c) s-chart

LCL 30

140

Section 9.7 Statistical Quality Control (d) UCL = x + 0.58(R) = 38.0667; LCL = x − 0.58(R) = 29.9467;

(e) UCL = 2.11(R) = 2.11(7) = 14.77; LCL = 0(R) = 0; UCL

UCL 14 12

_ _ x

_ r

8 6

4 LCL

30 0

2 5

Figure 9.7–8: (d) x-chart using R and (e) R-chart (f ) Yes. √ 9.7–10 LCL = 1.4 − 3 1.4 < 0 so LCL = 0; √ U CL = 1.4 + 3 1.4 = 4.9496; (a) If λ = 3, P (X ≥ 5) = 1 − P (X ≤ 4) = 1 − 0.815 = 0.185;

(b) Y is b(10, 0.185). Thus P (at least 1) = 1 − P (Y = 0) =

1 − p0 q 10

1 − 0.81510

0.871.

LCL 30