University of Iowa. Elliot A. Tanis. Hope College. Dale L. Zimmerman. University of Iowa

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1 INSTRUCTOR S SOLUTIONS MANUAL PROBABILITY AND STATISTICAL INFERENCE NINTH EDITION ROBERT V. HOGG University of Iowa Elliot A. Tanis Hope College Dale L. Zimmerman University of Iowa Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo

2 Copyright 2016 Pearson Education Limited All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps. ISBN-13: ISBN-10:

3 iii Contents Preface v 1 Probability Properties of Probability MethodsofEnumeration Conditional Probability IndependentEvents Bayes Theorem Discrete Distributions RandomVariablesoftheDiscreteType MathematicalExpectation SpecialMathematicalExpectations TheBinomialDistribution TheNegativeBinomialDistribution ThePoissonDistribution Continuous Distributions RandomVariablesoftheContinuousType TheExponential,Gamma,andChi-SquareDistributions TheNormalDistribution AdditionalModels Bivariate Distributions Bivariate Distributions of the Discrete Type TheCorrelationCoefficient ConditionalDistributions BivariateDistributionsoftheContinuousType TheBivariateNormalDistribution Distributions of Functions of Random Variables FunctionsofOneRandomVariable TransformationsofTwoRandomVariables SeveralRandomVariables TheMoment-GeneratingFunctionTechnique RandomFunctionsAssociatedwithNormalDistributions TheCentralLimitTheorem ApproximationsforDiscreteDistributions Chebyshev s Inequality and Convergence in Probability LimitingMoment-GeneratingFunctions... 59

4 iv Contents 6 Point Estimation Descriptive Statistics ExploratoryDataAnalysis OrderStatistics MaximumLikelihoodEstimation A Simple Regression Problem Asymptotic Distributions of Maximum LikelihoodEstimators SufficientStatistics BayesianEstimation MoreBayesianConcepts Interval Estimation Confidence Intervals for Means Confidence Intervals for the Difference of Two Means Confidence Intervals For Proportions SampleSize Distribution-Free Confidence Intervals for Percentiles More Regression ResamplingMethods Tests of Statistical Hypotheses TestsAboutOneMean TestsoftheEqualityofTwoMeans TestsaboutProportions TheWilcoxonTests PowerofaStatisticalTest BestCriticalRegions LikelihoodRatioTests More Tests Chi-Square Goodness-of-Fit Tests ContingencyTables One-FactorAnalysisofVariance Two-WayAnalysisofVariance General Factorial and 2 k FactorialDesigns Tests Concerning Regression and Correlation StatisticalQualityControl

5 Preface v Preface 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 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 ). You can download a copy of this manual at Our hope is that this solutions manual will be helpful to each of you in your teaching. If you find an error or wish to make a suggestion, send these to Elliot Tanis, tanis@hope.edu, and he will post corrections on his web page, R.V.H. E.A.T. D.L.Z.

6 vi

7 Chapter 1 Probability 1 Chapter 1 Probability 1.1 Properties of Probability (a) (b) (i) /, (ii) 0, (iii) /, (iv) /, (v) /, (vi) /, (vii) / (a) P (A B) ; (b) A (A B ) (A B) P (A) P (A B )+P(A B) 0.4 P (A B )+0.3 P (A B ) 0.1; (c) P (A B )P [(A B) ]1 P (A B) Let A {lab work done}, B {referral to a specialist}, P (A) 0.41, P(B) 0.53, P([A B] )0.21. P (A B) P (A)+P(B) P (A B) P (A B) P (A B) 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) (a) / ; (b) / ; (c) 0; (d) /.

8 2 Section 1.2 Methods of Enumeration P (A) 2[r r( 3/2)] 2r Note that the respective probabilities are p 0,p 1 p 0 /4, p 2 p 0 /4 2,. p 0 4 k 1 k0 p 0 1 1/4 1 p p 0 p Methods of Enumeration (a) (b) Number of experiments with each factor at 4 levels (4) (4) (4) 64. ( ) (a) 4 80; 3 (b) 4(2 6 ) 256; (c) (4 1+3)! (4 1)!3! 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 Total number of varieties of pizzas 4 3 (2) ( ) n 1 r + ( ) n 1 r (1 1) n (n 1)! r!(n 1 r)! + (n 1)! (r 1)!(n r)! (n r)(n 1)! + r(n 1)! n! r!(n r)! r!(n r)! n ( ) n n ( ) n ( 1) r (1) n r ( 1) r. r r r0 n ( ) n n ( ) n 2 n (1+1) n (1) r (1) n r. r r r0 r0 ( ) ! 36 36!9! 886,163,135. ( )( ) (a) ( ) 102, , ; 9 ( )( )( )( )( )( )( ) (b) ( ) 7, ,236, r0 ( ) n. r

9 Chapter 1 Probability Conditional Probability (a) ; (a) (b) ; (c) (d) The proportion of women who favor a gun law is greater than the proportion of men whofavoragunlaw. Probability of drawing 3 spades (b) Probability of drawing a spade on the first draw, a heart on the second draw, and a third draw (c) Probability of drawing a spade on the first draw, a heart on the second draw, and an ace on the third draw Let H {died from heart disease}; P {at least one parent had heart disease} (a) ; ( )( ) 3 17 P (H P ) N(H P ) N(P ) (b) (c) k ; 2 1 ( ) 20 3 ( )( 3 17 ) 9 2 2k 2 ( ) 20 2k k (d) Draw second. The probability of winning is (a) P (A) ,808,975 11,881, ; (b) P (A )1 P(A) (a) It doesn t matter because P (B 1 ) 4 2 (b) ( ) on each draw (a) P (A 1 )30/100; (b) P (A 3 B 2 )9/100; (c) P (A 2 B 3 )41/ /100 9/100 60/100; ( 5 ) ( 10 ) ( 20 ) ( 30 )

10 4 Section 1.4 Independent Events (d) P (A 1 B 2 )11/41; (e) P (B 1 A 3 )13/29. Probability that an employee has a college degree and works in sales Probability that an employee does not have a college degree and works in sales 0.8 (1 0.6) Probability that an employee chosen at random works in sales Independent Events (a) P (A B) P (A)P (B) (0.3)(0.6) 0.18; P (A B) P (A)+P(B) P (A B) (b) P (A B) P (A B) P (B) Proof of (b): P (A B) P (B)P (A B) P (B)[1 P (A B)] P (B)[1 P (A)] P (B)P (A ). Proof of (c): P (A B ) P [(A B) ] 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 (A )P (B ) P [A (B C)] P [A B C] P (A)P (B)P (C) P (A)P (B C). P [A (B C)] 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 [A (B C )] P (A C B) P (B)[P (A C ) B] P (B)[1 P (A C B)] P (B)[1 P (A C)] P (B)P [(A C) ] P (B)P (A C ) P (B)P (A )P (C ) P (A )P (B)P (C ) P (A )P (B C ) P [A B C ] P [(A B C) ] 1 P (A B C) 1 P (A) P (B) P (C)+P(A)P(B)+P(A)P(C)+ P (B)P (C) PA)P (B)P (C) [1 P (A)][1 P (B)][1 P (C)] P (A )P (B )P (C ).

11 Chapter 1 Probability Probability that exactly 2 of the 3 dice came up orange (a) ; (a) (b) ; (c) (b) (c) (d) (a) 1 (0.4) ; (b) 1 (0.4) (a) k0 ( ) 2k ; (b) (a) 7; (b) (1/2) 7 ; (c) 63; (d) No! (1/2) 63 1/9,223,372,036,854,775, No ! 1 4!4! Bayes Theorem (a) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (b) ( ) 0.37 ( ) ( ) Let event B denote an accident and let A 1 be the event that age of the driver is Then (0.1)(0.05) P (A 1 B) (0.1)(0.05) + (0.55)(0.02) + (0.20)(0.03) + (0.15)(0.04) Let B be the event that the policyholder dies. Let A 1,A 2,A 3 be the events that the deceased is standard, preferred and ultra-preferred, respectively. Then

12 6 Section 1.5 Bayes Theorem (0.60)(0.01) P (A 1 B) (0.60)(0.01) + (0.30)(0.008) + (0.10)(0.007) ; P (A 2 B) ; P (A 3 B) Let A be the event that the tablet is under warranty. (0.40)(0.10) P (B 1 A) (0.40)(0.10) + (0.30)(0.05) + (0.20)(0.03) + (0.10)(0.02) ; P (B 2 A) ; P (B 3 A) ; P (B 4 A) (a) P (D + )(0.02)(0.92) + (0.98)(0.05) ; (b) P (A D + ) ; P (A+ D + ) ; (c) P (A D (0.98)(0.95) ) (0.02)(0.08) + (0.98)(0.95) ; P (A + D ) (d) Yes, particularly those in part (b) Let D {has the disease}, DP {detects presence of disease}. Then P (D DP) P (D DP) P (DP) P (D) P (DP D) P (D) P (DP D)+P (D ) P (DP D ) (0.005)(0.90) (0.005)(0.90) + (0.995)(0.02) Let D {defective roll}. Then ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

13 Chapter 2 Discrete Distributions 7 Chapter 2 Discrete Distributions 2.1 Random Variables of the Discrete Type (a) (b), x 1, f(x), x, 1, x, Figure 2.1 2: Line graph (a) f(x) 1, x 0, 1, 2,, 9; 10 (b) N ({0})/150 11/ ; N ({5})/150 13/ ; N ({1})/150 14/ ; N ({6})/150 22/ ; N ({2})/150 13/ ; N ({7})/150 16/ ; N ({3})/150 12/ ; N ({8})/150 18/ ; N ({4})/150 16/ ; N ({9})/150 15/

14 8 Section 2.1 Random Variables of the Discrete Type (c) f(x), h(x) x Figure 2.1 4: Michigan daily lottery digits (a) f(x) 6 7 x,x2, 3, 4, 5, 6, 7, 8, 9, 10, 11, (b) f(x) x Figure 2.1 6: Probability histogram for the sum of a pair of dice

15 Chapter 2 Discrete Distributions (a) The space of W is S {1, 3, 5, 7 }. 1/12, 1 1/6, 3 1/6, 5 ) ( 1/6, 7 1/6, 9 1/6, 11 1 / 12, 13 (b) Figure 2.1 8: Probability histogram of sum of two special dice (a) Probability of the sample containing exactly one defective item 4 96 C1 C C20 (b) Probability that the sample contains at most one defective item C0 C20 C1 C C20 C

16 10 Section 2.2 Mathematical Expectation P (X 4 X 1) P (X 4) P (X 1) 1 P (X 3) 1 P (X 0) 1 [1 1/2+1/2 1/3+1/3 1/4+1/4 1/5] 1 [1 1/2] ( )( ) P (X 1) 1 P (X 0)1 0 5 ( ) (a) P (2, 1, 6, 10) means that 2 is in position 1 so 1 cannot be selected. Thus ( )( )( ) P (2, 1, 6, 10) ( ) ; 6 (b) P (i, r, k, n) ( )( )( ) i 1 1 n i r 1 1 k r ( ) n. k 2.2 Mathematical Expectation EX ( ) ( 02) + ( 01) + ( 0) + ( 1) + ( 2) EX ( ) ( 02) + ( 01 ) + (0) + (1) + (2) EX ( 03X+ 9) EX ( ) 03EX ( ) f(x) 9 ( c ) 6 x0 c 2 49 ; E(Payment) Note that E(X) x1 x1 6 π 2 x 2 6 π 2 x 6 π 2 x 2 6 π 2 ( ) units. x1 x1 1 x 2 6 π 2 π 2 6 1,sothisisapdf 1 x + and it is well known that the sum of this harmonic series is not finite E( X c ) 1 x c, wheres {1, 2, 3, 5, 15, 25, 50}. 7 When c 5, x S E( X 5 ) 1 [(5 1) + (5 2) + (5 3) + (5 5) + (15 5) + (25 5) + (50 5)]. 7

17 Chapter 2 Discrete Distributions 11 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) μ, the mean, minimizes E[(X b) 2 ]. You could also let h(c) E( X c ) and show that h (c) 0whenc (15) + ( 0 15) (15) + ( 015) (20) + ( 015) (a) Theaverageclasssizeis (16)(25) + (3)(100) + (1)(300) 20 50; (b) 0.4, x 25, f(x) 0.3, x 100, 0.3, x 300, (c) E(X) 25(0.4) + 100(0.3) + 300(0.3) Special Mathematical Expectations (a) μ E(X) 3 ( ) x ( ) 3 x 3! 1 3 x x!(3 x)! 4 4 x1 ( ) 2 ( ) k ( ) 2 k 1 2! k!(2 k)! 4 4 k0 ( )( ) ; 3 ( ) x ( ) 3 x 3! 1 3 E[X(X 1)] x(x 1) x!(3 x)! 4 4 x2 ( ) 2 ( ) (3) ( ) 2 ( )( ) ; σ 2 E[X(X 1)] + E(X) μ 2 ( )( ) ( ) ( ) (2) ( )( ) ( )( ) ( )( ) (2) + 3 ;

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