Instructor’s Solutions Manual Probability and Statistical ...

1 Instructor s Solutions ManualProbability andStatistical InferenceEighth EditionRobert V. HoggUniversity of IowaElliot A. TanisHope CollegeThe author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expresses or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these by Pearson Prentice Hall from electronic files supplied by the 2010 Pearson Education, Inc.

2 Publishing as Pearson Prentice Hall, Upper Saddle River, NJ 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. Printed in the United States of : 978-0-321-58476-2 ISBN-10: 0-321-58476-7 ContentsPrefacev1 .. Probability .. of Enumeration.. Events.. 'sTheorem..72 theDiscreteType .. ,Variance,andStandardDeviation..243 Data.. theContinuousType ..544 Two RandomVariables.. cient ..665 Distributionsof Functionsof OneRandomVariable.

3 Two RandomVariables.. RandomVariables..86iiiiv6 Estimation.. denceIntervalsforMeans.. denceIntervalsfortheDi erenceof Two Means.. denceIntervalsforVariances.. denceIntervalsforProportions.. SimpleRegressionProblem.. 1077 Tests of .. theEquality of Two Means.. Variance.. Variance.. 1288 .. denceIntervalsforPercentiles.. Goodnessof FitTest.. 1499 Probability .. 15910 cient Statistics.. of a StatisticalTest.. RatioTests.. 'sInequality andConvergencein Probability .. MaximumLikelihood Estimators.. 17111 Quality Improvement .. Control.. 179 PrefaceThissolutionsmanualprovidesanswer sfortheeven-numberedexercisesinProbabili tyandStatisticalInference, 8thedition,by RobertV.

4 HoggandElliotA. formostof ,theinstructor,may decidehow many of theseanswersyouwant to makeavailableto the guresin thismanualweregeneratedusingMaple, a the guresweregeneratedandmany of thesolutions,especiallythoseinvolvingdat a,weresolvedusingproceduresthatwerewritt enby ZavenKarianfromDenisonUniversity. We theseproceduresareprovidedin the\MapleCard"thatis theseproceduresaregiveninProbabilityandS tatistics:ExplorationswithMAPLE, secondedition,1999,writtenby ZavenKarianandElliotTanis,publishedby PrenticeHall(ISBN0-13-021536-8).REMARKN otethatProbabilityandStatistics:Explorat ionswithMAPLE, secondedition,writtenby Zaven KarianandElliotTanis,is availablefordownloadfromPearsonEducation ' hasbeenslightlyrevisedandnow containsreferencesto severalof theexercisesin the8theditionofProbabilityandStatistical Inference.

5 Ourhope is thatthissolutionsmanualwillbe helpfulto each of youin you ndanerroror wishto make a suggestion,sendtheseto willpostcorrectionsonhiswebpage, (a)S=fbbb;gbb;bgb;bbg;bgg;gbg;ggb;gggg;( b)S=ffemale;maleg;(c)S=f000;001;002;003; : : : ; (a)Clutch size:4567891011121314 Frequency:35727263782011(b)xh(x) {4:Clutch sizesforthecommongallinule(c) (a) :456789101112131415161924 Frequency:1019138137952442211(b)xh(x) {6:Number of boxesof (a)f(1)=210; f(2)=310; f(3)=310; f(4)=210 (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)AlthoughJames'battingaverageis higherthatHrbek'sonbothgrassandarti cialturf,Hrbek'sis higherover erent numbersof at batsongrassandarti cialturfandhow thisa a gureand llin theprobabilitiesof each of thedisjoint ,P(A) = 0 sportscarg,P(B) = 0 ,P(C) = 0 is alsogiventhatP(A\B) = 0 (A\C) = 0, it followsthatP(A\B\C0) = 0 (A0\B\C0) = 0:06andP(A0\B0\C) = 0 (a)S=fHHHH;HHHT;HHTH;HTHH;THHH;HHTT;HTTH ;TTHH;HTHT;THTH;THHT;HTTT;THTT;TTHT;TTTH ;TTTTg;(b) (i)5/16,(ii)0,(iii)11/16,(iv)4/16,(v)4/1 6,(vi)9/16,(vii)4 (a)1/6;(b)P(B) = 1 P(B0) = 1 P(A) = 5=6;(c)P(A[B) =P(S) = of (a)P(A[B) = 0:4 + 0:5 0:3 = 0:6.]]}}

6 (b)A=(A\B0)[(A\B)P(A)=P(A\B0) +P(A\B)0:4=P(A\B0) + 0:3P(A\B)=0:1;(c)P(A0[B0) =P[(A\B)0] = 1 P(A\B) = 1 0:3 = 0 ,B=freferralto a specialistg,P(A) = 0:41; P(B) = 0:53; P([A[B]0) = 0 (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 [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)1/3;(b)2/3;(c)0;(d)1 (a)S=f(1;2);(1;3);(1;4);(1;5);(2;3);(2;4 );(2;5);(3;4);(3;5);(4;5)g;(b) (i)1/10;(ii)5 (A) =2[r r(p3=2)]2r= 1 p32 probabilitiesarep0; p1=p0=4; p2=p0=42; : : :.1Xk=0p04k=1p01 1=4=1p0=341 p0 p1= 1 1516= of (4)(3)(2)= (a)(4)(5)(2)= 40;(b)(2)(2)(2)= (a)4 63 = 80;(b)4(26) = 256;(c)(4 1 + 3)!]]]]]]]]]]]]]

7 (4 1)!3!= !5!= 3024 ,HHCH,HCHH,CHHH,HHCCH,HCHCH,CHHCH,HCCHH, CHCHH,CCHHH,CCC,CCHC,CHCC,HCCC,CCHHC,CHC HC,HCCHC,CHHCC,HCHCC,HHCCCgso 3 212= 36;864 n 1r + n 1r 1 =(n 1)!r!(n 1 r)!+(n 1)!(r 1)!(n r)!=(n r)(n 1)!+r(n 1)!r!(n r)!=n!r!(n r)!= nr (1 1)n=nXr=0 nr ( 1)r(1)n r=nXr=0( 1)r nr :2n=(1 + 1)n=nXr=0 nr (1)r(1)n r=nXr=0 nr nn1; n2; : : : ; ns = nn1 n n1n2 n n1 n2n3 n n1 ns 1ns =n!n1!(n n1)! (n n1)!n2!(n n1 n2)! (n n1 n2)!n3!(n n1 n2 n3)! (n n1 n2 ns 1)!ns!0!=n!n1!n2!: : : ns! (a) 193 52 196 529 =102;486351;325= 0:2917;(b) 193 102 71 30 51 20 62 529 =7;6951;236;664= 0:00622 4536 = 886;163; (a)10411456;(b)392633;(c)649823.

8 (d)Theproportionof womenwhofavor a gunlaw is greaterthantheproportionof menwhofavor a (a)P(HH)=1352 1251=117;(b)P(HC)=1352 1351=13204;(c)P(Non-AceHeart,Ace)+P(Aceo f Hearts,Non-HeartAce)=1252 451+152 351=5152 51= or 4 kingsg,B=f2, 3, or 4 (AjB)=P(A\B)P(B)=N(A)N(B)= 43 4810 + 44 489 42 4811 + 43 4810 + 44 489 = 0:170 ;P=fat leastoneparent (HjP0) =N(H\P0)N(P0)= (a)320 219 118=11140;(b) 32 171 203 117=1760;(c)9Xk=1 32 172k 2 202k 120 2k=3576= 0:4605:(d)Draw of winningin 1 0:4605= 0 20 85 105 25+ 21 84 105 15=15 (a)P(A) =5252 5152 5052 4952 4852 4752=8;808;97511;881;376= 0:74141;(b)P(A0) = 1 P(A) = 0 (a)It doesn'tmatterbecauseP(B1) =118; P(B5) =118; P(B18) =118;(b)P(B) =218=19oneach 58+25 48=2340 (a)P(A1) = 30=100;(b)P(A3\B2) = 9=100;(c)P(A2[B3) = 41=100+ 28=100 9=100= 60=100;(d)P(A1jB2) = 11=41;(e)P(B1jA3) = 13= (a)P(A\B)=P(A)P(B) = (0:3)(0:6) = 0:18.]

9 P(A[B)=P(A) +P(B) P(A\B)=0:3 + 0:6 0:18=0:72:(b)P(AjB) =P(A\B)P(B)=00:6= 0 of(b):P(A0\B)=P(B)P(A0jB)=P(B)[1 P(AjB)]=P(B)[1 P(A)]=P(B)P(A0):Proof of(c):P(A0\B0)=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(B0) [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[A0\(B\C0)]=P(A0\C0\ B)=P(B)[P(A0\C0)jB]=P(B)[1 P(A[CjB)]=P(B)[1 P(A[C)]=P(B)P[(A[C)0]=P(B)P(A0\C0)=P(B)P (A0)P(C0)=P(A0)P(B)P(C0)=P(A0)P(B\C0)P[A 0\B0\C0] =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(B0)P(C0) ' 26 36+16 46 36+56 26 36= (a)34 34=916;(b)14 34+34 24=916;(c)24 14+24 44= (a) 12 3 12 2;(b) 12 3 12 2;(c) 12 3 12 2;(d)5!]]]]]]]]]]]]]

10 3! 2! 12 3 12 (a)1 (0:4)3= 1 0:064= 0:936;(b)1 (0:4)8= 1 0:00065536= 0 (a)1Xk=015 45 2k=59;(b)15+45 34 13+45 34 23 12 11=35 (a)7;(b)(1=2)7;(c)63;(d)No!(1=2)63= 1=9;223;372;036;854;775; (a) (b) (c)Verylittlewhenn >15,samplingwithreplacementVerylittlewhe nn >10,samplingwithoutreplacement.(d)Conver genceis ' (a)P(G)=P(A\G) +P(B\G)=P(A)P(GjA) +P(B)P(GjB)=(0:40)(0:85)+ (0:60)(0:75)= 0:79;(b)P(AjG)=P(A\G)P(G)=(0:40)(0:85)0: 79= 0:43 ' andletA1be theevent thatageof thedriver is 16{ (A1jB)=(0:1)(0:05)(0:1)(0:05)+ (0:55)(0:02)+ (0:20)(0:03)+ (0:15)(0:04)=5050 + 110+ 60 + 60=50280= 0:179 theevent ; A2; A3be theevents thatthedeceasedis standard,preferredandultra-preferred,res pectively.}