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Fall 2010 Exam 1 Solutions - Prob-ability and Statistics...

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Unformatted text preview: Prob-ability and Statistics for Engineering (ESE 326) Exam 1 September 28, 2010 This exam contains 10 multipIe-choice probiems worth two points each, six true—false problems worth one point each, 11 short-answer probiems worth one poth each, and one freeuresponse problems worth three points, for an exam total of 40 points. Part1. Multiple-Choice (two points each) Cleariy fiil in the ovai on you: answer card which corresponds to the only correct response. I. At the horse races, 81 “Trifeota” is a bet which predicts the first place, second piece, and third piece horses in an upcoming race. If there are nine horses in the first race at Canterbury Park, in how many ways could Donald place a Trifecta bet? (A)6 x .,.,. ‘ (8)24 3§g§§ 3§3§8§*?5,§W% (C) 27 (D) 54 (H) 19.888 (1) 60,480 (J) 862.880 2. There are 98 students in a Probabiiity and Statistics class, of which 22 are sophomores, 46 are juniors, and 30 are seniors. If the instructor randomfy selects three students to be on a special Advisory Panel, what is the probability that exactiy one student from each class is chosen? (A) .0228 £33 a}; egg; 5 535:2 “ (B) .0328 { gjg ; £5: 3 ,} ,. (C) .0334 W” i » j%%é, (o) .1385 if, {731' /> (3&8004' (o) .8684 (H) .966? (I) .967“? (I) .9772 Mark, Ron, and Harry enter a small music stare which sells 208 different rock-and—rell CBS, 250 different jazz CD3, and 300 different hip hop CD8. (Te clarify, there are 300 different titles it: the hip hop category, for example, but there are multiple copies of each title.) Each of the three friends plans to buy two CDs) but none of them will buy two from the same category. (For example, Mark might buy one roekvand-mll CD and one jazz CD, but he will not buy two jazz CBS.) What is the total number of ways the three friends can make their purchases? (A) 3.3.350 a g f. 3?" t (B) 7503 “shag, 7*va .6le ma - ,a $55M am (C) 3. 185.1100 w/ifibfga jw Wmfljmfiaa i W t {D} 185,§9§j§ (E) 2003 +2503+3003 {W 35259;?) «a {mag/gm) w,» azaajfgaa} (s) 58,0603 + 60 0003 + 75 000-3 X f 95” @5335; (G) 3[45002 + 5990 2 + 550 O2] 2 (H) 3[450P2 ‘ 500132 Ml“ 550132] 3 3g ' ~ {I - )9" l u»— I « '3 w“ '1' W», (I) [4500 3 590632 + .580 J3 W’ “gym 5 (/3333; 333333553 333 #3333553 3’ 31333333 3 (j) [45%192 + SLEEP? + 550P2] Suppose 15% of American adults play fantasy footbail, 40% of those who play fantasy football also play fantasy basketball, and 81% play neither fantasy football net fantasy basketball. What percent of American adults play one or the other but not both? (A) 4% g W Wgfl ghee? thaaéwj {WM (3) 6% (C) 8% 5):) Wk, 1Mémv; flw MW 5&6};ng %fi (D) 9% Fiat: . , (E) 10% ‘ 5 (F) 13%} P £5,334: at t G 16% g . )- W f _ ' Nita EH: 19% P 5a he 3 333 “W“ (I) 25% 3‘3 ta m : gialézilaaé . .0 p» f . I i (I) 60/13 Effifgflfiéjfgfifgfiffiawgff gwiagwlg3}:,§g Suppose 3% cf peepie weridwide have diabetes. A certain test that: is Often used to diagnose diabetes is not always accurate. When a person with diabetes is tested, the test gives a positive result 90% of the time and when a persen without diabetes is tested, the test gives a positive result 5% of the time. {A “positive result” means that the test says the person has diabetes.) If a eerson chosen tandomiy is tested and there is a positive result, what is the probabiiity that the person actuality has diabetes? ’ 5 ’ 5 ”i . E 5 figs, Ewe/MW gem, gigwggiemw (A) .001? :2: :::Z i W?“ : gag” ging'sii {jg/em 5e me seem 5 VM . g 7 y . , E (D).3927 gegbgees iii/“siege? PfTifijfiifig" (E) .4433 y {g g .- m m e w WWW (G) .6073 f f ( {eyes} sg” 2 (H) .6424 § .53 "3"} I , Q ,7”? “:43“ m (I) .9245 defies} ew- {gaefiffi‘tfi (J) .9983 Let f (e) be a density fer a discrete random variable X. What is the correct computationai formula for EEK2 + 3]? (A) gas? (13} gngtw) (C) gem?) m>§fiflf+$ (E) ;x2<(f(m))2+3) (F) yew) WWW ”(6) me at we) i (14> 2e? + 3W?) m ZWW$flfi+a w 2e%%MKmF+s Which one of the following statements is true? (A) Cheby‘shev's Equality is a strong statement which applies to all random variables. (13) Chebyshev‘s Inequality is a strong statement which applies to all random variables. (C) Chebyshev‘ s Equality IS a rather weak statement which applies to all random variables WWW ”*2 w “HWMMKMW WKMMfiwMWM,WMH wwwfi W MW M (D) Chebyshev‘ s Inequality IS a rather weak statement which "agglie s to all random vanablesy, -m-»M&wgemwan-mm Wm! m. 4» nmwwmfllmw- (E) Chebyshev s Equality as a strong statement which applies only to some random variables (F) Chebyshev s Inequality IS a strong statement which applies only to some random variables. (G) Chebyshev’s Equality is a rather weak statement which applies only to some rantiom variables. (ll) Chebyshev’s Inequality is a rather weak statement which applies only to some random variables. For advertising purposes, a seafood restaurant runs a game at a local eamival. The game involves 500 hollow plastic fish floating in a pool, For $2.00: a player can purchase a fish from the pool. One hundred fish have $5.00 cash inside, and the remaining s90 contain a $2.00 restaurant coupon. Arnold is the first to arrive at this booth in the morning, and, to the annoyance of the testament employee, he buys 30 fish. What is the probability that Arnold will win exactly 6 $5.00 bills? l: :3: gfifw Mime; M: its ”234% ft, : gm AXE, r: 3g? , ‘ g) y} : 5W ”Stiwym ajw /§$z§}[&f&gfifi (D) .2009 2" is e e“ m , Z: I Egg”; (E) 3589 {f 5336 :2 . {g (F) .5000 3 (G) .6411 (H) .81le (I) .8205 (I) .8259 9. There are 38 stats in a roulette whee}, 18 of which are red, 18 of which are biaek, and. two of which are green. Spins of the Wheel are independent. What is the probability that it takes it) at more spins untii the ball lands in a green 3101:? 3 q 2 s (A) .0324 t W p, M W« 2:: W page“ we, 56> M 4 i t (13) .3342 5;?” a; 3g it: (C) .3853 M t} f < .. t . _ “it (D).4176 PiXfi‘i mg 5: 5’” Pi}? fat; (E) .4483 m ‘ r» a} (F) .5517 W EM 2“ 5“? it? at? (i) .9658 (J) .9676 :: y Mtg? 10. An antibiotic {Sad to treat msacea (a skin disease) is effective 70% of the time. Suppese this antibiotic is prescribed for a group of 50 people with rosacea. What is the probability that more than 35 of them Show impmvement? (A) 0 h“ t. Z? W p M» w (B) .32?‘ immfiw’”? “3% W 7% 3:3 W i j C) .4308 £3” , . W F g: I” 5:? tttttflwmmfij ii: :85? 1: E W £53535} (G) .5143 t: g M (5:32 (H) .5532 t t t} (LE) .5692 j , gig E? (J) .6721 Part Ii. Tme~Faise (one point each) Mark “A” on your answer card if the statement is true; mark “B” if it is faise. 11- Probebihty is long-term frequency. 55:55.55: 5:55 5: 55:5 ,, $555 We? 55555 5‘? {[55 5,5555% 2 555% &: wee/55.55,} ,5 12. If two events are mutuaily exclusive 551555551555 are independent ; 5: jyg #51355 5 55% 5:55 5556:555533: Wei/ff; V5555, 5:55:54: W3 5 I egbxfiegéé 55555355555555: M W. Wee/55255536 13. Suppose A and B are independent evente such that P[A[B] PM 51 B} m .12. 55:5 ”geeggi: 2:; M £3? m 35%»; ' 5:55.555: 55555 5355555555,} ”5";- 5?3 [533: P55555553: 555553? 555533 14. The foilowing function qualifies to be a density for a discrete random variable X. H {.6 E5. 3 E"; E H pm. W] 53‘ :3 f($):352+1 a: : 83.1323: u» x: jg fl 3 , 525m g s a; 4 I i 2: 5;: Mi.» 555%} 5 “E” 5:5 5;"? gm f” 3‘93 “’ r5 «5:, _ w, ~ 1 I 555555 555 5:5: 5:; 5~ :55 w 5 5’ {i 17-ng «39' A "fie/g: E I I I 3 15 Let X be a discrete random variabie. Then for any a: f (:12) < F(x). “m ,5 .5 5’55 -- . 51W % ,5 ”£55555: £35525? {Egg 33,553 § $5,}? 55' X2 ** ngjl 16. It would be appropriate to use €116 hinomiai distribution with p x .4 to approximate a hypergeomeirie distribution with 5V: 100 7' ~— 40 and n — 25 5.5 fl M05255 [4-055 55.: éme} aweiw 52% 55% eff/4W2; 3w; 5255;555:555, 5:55; 5535 «5.525454% 25%} 355% £551. W 455%: 9555: ., f Iflwéiéfi g! Part EH. Sheri: Answer (one point each) The answer to each of these is right or wrong: no work is required, and no partial credit win be given. Give only one answer 10 each. (If you give more than one answer, the peerer one wfi} count.) Numericai answers should be given in decimal form, rounded to four decimal places (if reunding is needed). 1?. Suppose A and B are events such that PM} 2 .8, PW} u .65, and PEA Q B} :2 .55. Find PEEL/4]. 5 m: A ,3 View @{eeegreiififiii : 935;:: g??? 9515;? E E3” 3’ ‘“ For problems 18 finengh 21, iet X be the discrete random variabie given by the following density tabie. a? 2 3 4 fie) .2 .1 .7 18. Find Fm). Ffe'} 2‘: $53} a3 19. Find 53X}. 35%] Mafia} «2:» 53%) E» em} 2 20. Find Var X. £532}? feEgi“ £EE5eE5 Eefeé/“EE fl 52,3? 1/51?“ E5 : gee 5.. 533’}; 2: ee” 21. Find ax. For probierns 2?. and 23, iet X be a discrete random valiable such that EiX] m 40 and Var X 2:: 5. 22. Find EEEGO u 2X]. {,1 iéémwexj :: fees w 25"ny :2 35} 23. Find Varmo w 23*). _ ”1 :J m M EérfleeWZXEfieflezxw 2m 24. Find the moment generating function mx (t) for the discrete rand-om variabie X which is given by the foilowing density table. a: 5 10 .EE— we 3 , 3 P. ”Elfiff’x é.él> :f a?» :ggéi- ”a?“ g 2 £2 25. Write out the density fimction fer a geometric random variable with p x: .75. g) ~ QXW T _ %Ze§:52§:§ Z?5”} exigefieww 26. Find the variance of a binomial tandem variabie with n 2 25 and p 2 .4. m . W {New} \/ “*2 w§§,méee}éejee§~ é 27. Correctly fiii in both of the fofiowing blanks. (Your choices for each blank are “geometric,” “binomial,” “negative binomial,” and “hypergeernetrie.”) Th6 Weierggew distribution is a specie} case of the {if we; a a. distribution“ Part W. Free Resyonse (three points) 28. A statistics teacher rolls a die every day at the beginning of eiass to determine if there W11} be a quiz 1 that titty, so the (faity probabitity of a quiz is 5-. There are 60 class days in the semester, and the maximum number of quizzes that wit} be given timing the semester is 10. What is the probebitity that the tenth quiz will be given on the 59th or 60th. day of class? You do not need to work out the final numerical answer, but please write out the cemgiete setuug which would he needed to compute this probability, so that the oniy step left undone is calculator-button—pushing. t Vfiggegfllfigfix Mmtwetwg} f; “i i {if} jg 3:” is: My: eel $3235 we; (/fJ Q J (”57? % ifs/é J eff; ...
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