STAT 400 EXAM 1 solutions - STAT 400/MATH 463 Ditlev Monrad...

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Unformatted text preview: STAT 400/MATH 463 Ditlev Monrad July6,2007 OPEN BOOK - SHOW ALL WORK Problem 1. (20 points) Here is a two—way table of all s of the victim and uicides committed in a recent year by sex method used: Suppose that a suicide victim is selected at random. Answer the following questions from the table. (a) What is the probability th that the suicide victim used a firearm? ide victim used a firearm, given that the at the suicide victim is male? (b) What is the probability (0) What is the conditional probability that the suic victim is male? ictim is male" and "a firearm was used in the suicide" independent? ((1) Are the events "v Why or why not? («\ ZHIZZé /30)‘10‘{ (53 [8953/ 3,0,qu (4 15,5;&/ 2mm, (0“) No ECCMSQ ith answers h) 0* MA (c> m’ not Hm Same, (Hon; that we at Hm LESS “MM M4 o-lr i’lng inmle 0.7836] z 78 % 0.587% x 9170 0,47% 5% 477% ll males used (1 gm. om.) Problem 2. (15 points) Recently the US. Senate Committee on Labor and Public Welfare investigated the feasibility of setting up a national screening program to detect child abuse. A team of consultants estimated the following probabilities: (l) 1 child in 90 is abused, (2) a physician can detect an abused child 90% of the time, (3) a screening program would incorrectly label 3% of all nonabused children as abused. (a) What is the probability that a randomly selected child would be diagnosed as abused? (b) If a randomly selected child is diagnosed as abused, what is the probability that the diagnosis is correct? Lg/(D 131)sz a/vwl A ate/Ml: ‘H’LL wail/A Holt/{lat is abuseal" “data is not abused “1 Mat “0124M is. Ottaywsjeai a/l aim/Jed H. (A PW = Pgsm/tl + Pagan/ti = P BQPWEQL + P(BZ)P(A(BZ) ~ I ' .. q + .5— O. = 0.0100 4‘ 0.0Z°l7 = 0.03‘?7 % We (17) Palm -_—. 1%?‘03—45— = 95% = 0.252 x 252, 5mg Ma pram/[mag (1% 90) is (we: aim #LL %a»Posi/iacemte (3%)) most [DosiOIiU-C gm,ng Mafia be, c a/a/i/ms Pros/W 3 M X mot: M Aux/W601 015 M25 sag/eds HIM Sizw 4 Amde MEIaW/sz‘ 44M éz‘ Y Mira 11M W501 075 If flag/€506 #um‘ do Moi’ Séw 4 MM&% I'm/mammal. (A) If M Ma 15 m/g €{fec/iofl 50% o’F 416 Mme] 241m 50% X mm! Yam bi/IWM MM I4 = 25 and p = 0,50, 100(217) = I—- P(Xé/4) = (—0.6244/ = 0.053? 58 5‘20 OK: (>(Xa175 ll ,0 /\ —.< (I\ 00 \./ H O O (A m .9 Problem 4. (15 points) In the gambling game “craps” a player called “the shooter” tosses a pair of dice and the shooter’s score is the sum of the points on the upsides of the two six-sided dice. The shooter wins on the first roll of the sum is 7 or 1 l. The shooter loses on the first roll if the sum is 2, 3,0r 12. If the sum is 4, 5, 6, 8, 9, or 10, that number is called the shooter’s “point.” Once the point is established, the rule is: The shooter continues to toss the pair of dice. If the shooter rolls a 7 before rolling the “point,” the shooter loses; but if the “point” is rolled before a 7, the shooter Wins. Assume that 10 has just been established as the shooter’s point on the first toss. (a) What is the chance that the shooter finishes the game (Wins or loses) on the next toss? (b) How many additional tosses (after establishing the point) can the shooter expect to make to finish the game? (Compute the expected value.) (c) What is the chance that the shooter needs more than 10 additional tosses to finish the game? (“B (> = Ported“ l’oss Amalia in 7 ar (0) (7+3 -._1__ 34“L( (Q Lg/l— X = Mamba oi MAAHOMJ {vase/s Wok in finish Jinx X is Weific, M p =- 1/4- 91 X \/ § “4 Problem 5. (15 points) On average, 35 out of every 10,000 births result in a pair of identical twins. Imagine that we randomly select the records of 3,000 births. (a) What is the expected number of pairs of identical twins resulting from these 3,000 births? (b) What is the chance that there are at most 5 pairs of identical twins? (0) What is the chance that there are between 5 and 16 pairs of identical twins, inclusive? (Use a calculator or statistical tables.) [.024 X = no. 01C Pairs 01C identical twins, X15 binomial (MA n = 3000 Mat {3 = 0.0035, (A; E(><§ = up = 30000.0035 =_(_;0_.__§_ (Q X is apymximaleia “Pagan mm A 40.5 WK é 53 x g (at N52 m (45 = Wen} —- Pas—ct) = 0.96;?) - 0.02( = 0.?3‘3 Problem 6. (25 points) Keno is the Las Vegas equivalent of bingo. There are 80 balls, numbered 1 through 80. On each play, 20 balls are chosen at random without replacement from the 80. If you bet a dollar on the single number 47, for example, you are betting that ball number 47 will be among the 20 that are chosen. If you win, you get your dollar back, plus 2 dollars more. If you lose, they keep your dollar. (a) If you play once, betting a dollar on 47, what is your chance of winning? (b) If you play once, betting a dollar on 47, what is your expected loss? (In dollars.) (c) If you play 25 times, betting a dollar on 47 each time, what is your expected loss? ((1) If you play 25 times, how many times do you have to win to come out ahead (to win more money than you lose)? (6 Compute our chance of comin out ahead if you plan to play 25 times. y g (Use calculator or statistical tables) (a) 0mm of = g; = :1— (b) 12chde - EXJmclcd C035 ’-' $ 0.25 (C) Expuch (95g = ZS, (at) Cl ,h/mw (€BMX=W6r/togf m%n=25mdp=3q POO/Ct) = 1—P()<é85 = [—0.85% = 0.14% a: 5% ‘ 0.25 Joya/ls u .L .clw 4. N 4P 9. 3; § V‘ II 625 £41 9 N (/1 ll Problem 77: (20 points) The world’s most active volcano above sea level is Kilauea nn Hawaii, which sincc thc mid—1980’s has cruptcd an average of3.5 times per year. (An eruptimi ilenntes vigorous activity eitliei l‘mm a new volcanic vcnt or from an established volcanic vent aftei a Slowdown.) (a) What us the expected nimiliei nf‘emptions during thc ncxt 24 months? (b) What is the chance that over the next 24 lllflnlhs, Kilauea will havc at most 3 eruptions‘! (e) VVlmt is the probability that over the next 24 months, Kilauea will have between 4 and l() eruptions. inclusive‘.7 (d) What is the chance that Kilauea will havc cxactly o eruptions over the next 24 months? (USE a calculaltn HI statistical tables.) (a> The, expected mcméer 01‘ erzAfJ/a‘oms over at :24 Moflffi pen/4‘04 is 2‘ 3-5 1 M M X: Hun/156:” 01" emphoms over 74w maiL 2% MOI?)th X is (appmximaI-e/g) R-iflom mam A z 7. 073 P(Xé3\ --= 0.082 (A PCLtéxé Io) -= Pthlo) —P(Xé:<> = O.°t0t ~ 0.082 = _Q._8_Lft__ (on P(X= é) = mag) — fixes) :: “ ‘1 w', ...
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This note was uploaded on 02/10/2009 for the course STAT 400 taught by Professor Tba during the Spring '05 term at University of Illinois at Urbana–Champaign.

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STAT 400 EXAM 1 solutions - STAT 400/MATH 463 Ditlev Monrad...

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