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Unformatted text preview: Probability and Statistics for Engineering (ESE 326) Exam 1 September 28, 2010 This exam contains 10 multipIechoice probiems worth two points each, six true—false problems worth one
point each, 11 shortanswer probiems worth one poth each, and one freeuresponse problems worth three points, for an exam total of 40 points.
Part1. MultipleChoice (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 rockand—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 roekvandmll 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 $55M am
(C) 3. 185.1100 w/iﬁbfga jw Wmﬂjmﬁaa i
W
t {D} 185,§9§j§
(E) 2003 +2503+3003 {W 35259;?) «a {mag/gm) w,» azaajfgaa}
(s) 58,0603 + 60 0003 + 75 0003
X f 95” @5335;
(G) 3[45002 + 5990 2 + 550 O2] 2
(H) 3[450P2 ‘ 500132 Ml“ 550132]
3 3g ' ~ {I  )9" l u»— I « '3 w“ '1' W»,
(I) [4500 3 590632 + .580 J3 W’ “gym 5 (/3333; 333333553 333 #3333553 3’ 31333333 3
(j) [45%192 + SLEEP? + 550P2] Suppose 15% of American adults play fantasy footbail, 40% of those who play fantasy football also
play fantasy basketball, and 81% play neither fantasy football net fantasy basketball. What percent
of American adults play one or the other but not both? (A) 4% g W Wgﬂ ghee? thaaéwj {WM (3) 6% (C) 8% 5):) Wk, 1Mémv; ﬂw MW 5&6};ng %ﬁ
(D) 9% Fiat: . , (E) 10% ‘
5 (F) 13%} P £5,334: at t
G 16% g . ) W f _ ' Nita
EH: 19% P 5a he 3 333 “W“ (I) 25% 3‘3 ta m : gialézilaaé . .0 p» f . I i
(I) 60/13 Efﬁfgﬂﬁéjfgﬁfgﬁfﬁawgff gwiagwlg3}:,§g Suppose 3% cf peepie weridwide have diabetes. A certain test that: is Often used to diagnose
diabetes is not always accurate. When a person with diabetes is tested, the test gives a positive result
90% of the time and when a persen without diabetes is tested, the test gives a positive result 5% of
the time. {A “positive result” means that the test says the person has diabetes.) If a eerson chosen
tandomiy is tested and there is a positive result, what is the probabiiity that the person actuality has diabetes? ’ 5 ’ 5 ”i .
E 5 ﬁgs, Ewe/MW gem, gigwggiemw (A) .001?
:2: :::Z i W?“ : gag” ging'sii {jg/em 5e me seem 5 VM . g 7 y . , E
(D).3927 gegbgees iii/“siege? PfTiﬁjﬁiﬁg"
(E) .4433 y {g g . m m e w WWW (G) .6073 f f ( {eyes} sg” 2
(H) .6424 § .53 "3"} I , Q ,7”? “:43“ m
(I) .9245 deﬁes} ew {gaeﬁffi‘tﬁ (J) .9983 Let f (e) be a density fer a discrete random variable X. What is the correct computationai formula
for EEK2 + 3]? (A) gas? (13} gngtw) (C) gem?)
m>§ﬁﬂf+$
(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&wgemwanmm 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 longterm 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 17ng «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 [4055 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 random 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 calculatorbutton—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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