Fall 2010 Final Exam Solutions - Probabiiity and Statisiics...

Info icon This preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 12
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Probabiiity and Statisiics for Engineering (ESE 326) Finai Exam December 20, 20H) This exam contains 20 multiple—choice probiems worth two points each, 9 true-false probiems worth one point each one one shortenswer probiem worth one point, for an exam totai of 50 points. Part1. Muitigie-Choice (two points each) Clearly fili in the owed on your angwer card which corresponds to the only correct response, I. Ex.) If five cards are seiected randomly from a Stoodard 52-card deck, what is the orobabiiity that there are three of one color and two of the other color? (A) .0325 ‘35 2;: {/3333 23: (B) .0650 { 23 2:) 3 ‘3 33/313 (c) .2235 :51) W #35233 (D) .3251 ( 5” (E) .3497 (F) .5503 (o) .6749 (H) .7?65 (I) .9350 (J) .9675 A new dessefi bar cefled Pie R Square recentiy opened in St. Louis. For $3.14, Pie R Square offers its customers the choice of one large square of pie, two medium squares of pie (which must be two different kinds), or three small squares of pie (which must be three different kinds). If there are twenty kfinds of pie from which to choose, how many different orders are possible? W3; &% {235)4, _ fizz/3‘3 512,33 M (B) 4050 ((3) me (D) 8320 (E) 21,320 (F) 253260 ((3) 57,680 (H) 33325000 (I) 51,9843080 (J) aeoocfioo £333 3 [3&3 3. In a certain pepuiaiien 0f celiege students, 15% Eike guacameie but disiike Neck been salsa, 38% Eike black bean saksa but disiike guacamole, and 13% dislike bath. What is the prebabiiity that a student who Iikes guacamole aise likes Mack bean saisa? w 25 M: M MMM MMM MMMMM E: :3 53 5 WM jMMMVMWM tam geek We ”W* (D) .45 (E) .58 (F) .55 (G) .60 (H) .6? 7m 4. Let X be a random. variabie whose moment generating function is mfit} m (1 —— 3t)“2r’13. Find the secenci moment of X. PM" . ) I“ j (A) V? ‘?‘fo {jg} 3: (3M 3&3 / fl ‘ “3;!!! z ,\”'§’_3 (B) ":39" x: Z. _ , 3M : 2 ngir-X (C) 33/; Mg:- «34,2; 5%; :3 W 7:?{5 ”3%); 6/ 3} C/ J (D) x/é {£11 m A'MMg/é/ [I “\g”§’f§ WV' mg i M “W ’2‘" ‘35 e m $2325., (E) fifi WM Mew; M {f ,M} (F) 3% a [595% we i . f { ~ 3 i W , {121% j (H) 2 (1) 6M Katy is screening grade school children for Vision problems. The children enter the screwing room in groups of four. If 89% of the children have good Vision, what is the probability that the fifth group to enter will be the firs: group of the day in whieh all foue children have gooé Vision? (A) .8115 r” (B) '0166 Ejza‘fié M we at f/fiifl‘éf AM; agreeimaeye] (C) .0372 .2: 2 Eye 2 , 413% $5; (E) .056: 6. . 5% f‘ :: seéafig; (F) .071? J _ ,3 (G) .0819 Fig: 5’}: {feee} {#aag): a}??? (H) .1021 (I) .1215 (J) .1555 A hardware Store receives a shipment of 75 yellow daffodil bulbs and 25 White daffodil bulbs. (Of course, these bulbs will not Show color until they bloom. $0 they are currently inéistinguishable from one another.) By mistake, the bulbs get (thoroughly) mixed together. If IoAnne buys 12 daffodil bulbs from this mixture, what is the probability that she gets at least 10 yellow bulbs? (A) .1166 (B) .2367 5:1: :22: E535” 5% :1 Heme} +—efx:s5}+vi>fx::2} 5;» fiwma Ema-:6. Mag:- 1.1m. (a) .390? .. 5/1.; af‘.25»\ {wag/25 {7?}525} F .6393 .. a se a: f 92* 5.. ( 3 «a» W e W (G) 5218 /}&€3\ {E 53:35} a (1/ iéé‘ > 3? 12:: 5: a} . 3‘ * (1) .8834 in a Earge manufactm'iug giant, the-re is an figury on the job, on average, once. every eight months“ Assume that this is a Poisson pracessk What is the probabii'ity that an entire-3163: wiil g9 by with 119 injuries on the jab? i W- t. . k ; ¥ _ f (A) .9830 Pg/WMWZ R :1 3,. ngMW #2? %{cflfi%b (B) .1175 5 1 {MM 5555551543.) fiwyM/L éfiézjzfimS/A 5 (C) .2231 W i 7 35 (D) .3629 gas; ,54 : {fig/355$; : §~ ('13) .4866 ,5 , . _. E u- ‘ a, ‘ 5 (l E . . jaw". ‘24:? , .3, 9 Kw} (G) .6321 5 , 55‘ a (H) .7?69 mmfimm HWMM Wfimai. a «x X I .8825 5,, M M U 5:555} ~ 1’ v 53 3‘3 (J) .9200 m i” w, ~ \ v“ If? , .P:Y>52 i5w4whwirfi : 2235 fix) m 2$e"$2 m > 0 m>§bw”) r/5 55“’ «x1 5 “:2? (3558—1 rififiwjag 336' gig/)4 ” ”“5 ja (®%a+*w : m 5;” m; (1353“?1 6 ( > (F) em“i m g W 2% was (F) 14:52} 5.. w (G) 2(1~e’1) 9. Consider a tandem variabke X whose éistribution is normal wiih ,u. r: 12, and suppese that 13:6 g X g 18:; z .58. Which of the feiiowing is clesesi to “£116 value of a? (3%} 3. fig; “229%. {{wwwwiwté gtgéfzzimfigéf. fqm) aggfwwfliwgéfi (B) 2 . fl Q . 3 i ., ‘g . ‘ 4: § § 59’"? Q? m MQM. mam a. mmwwi sway-er. M (C) 3 .. w} 5‘ ’ ' a . ’ .. ’ _ {392% W>«z..*J/,§£QM :2”wa QQLQMQJMJ cwifim 6:15 35%;}... WWfiMK» . E 6 W m4... it?” if if», (G) 9 (H) 12 10. Consider the tw0~dimensi0nal discrete random variable (X, Y) Whose density table is as follows. Find COX/(X, Y). x/y 4 5 2 ,6 1 '5 .1 2 . ”H” .i i E 3 I § W . (A) M21. 55 XE 3 25st Qaéfiw 3:2. S (C) 4583 glafij 2:: agfl?>+§/53§ 3: %3 (D) .5238 6% ‘ . » (13)}.46 £33.53]: ganmflg-fwzégégh 3.552({3} (F) 1.8330 (G) 3.36 '1': {2%. 2 (H) 13.6 3 _, . _ r... w a. 1.2 may}: gg’ngéfla’j (j) 18‘? ~: 3%, '2. .. {3. 2:} 5m} 11. Consider the two-dimensional continuous random variabie (X,Y) Whose density is given as fOEIOWS. Finii EDS] fX¥($yy§xS$§ 0§$<y<1 (B) 3%; <1} $2,: X :5; (C) “1% mjg 5" g ‘ z a 5 E f . a a 5 1 (m gmg ;& 4% Ex fig (E) f; A % 3. in”: Silt i a {w :1: Us} NR M W W Mm“— "(E226 M? g (H) EM if '3 P Jim "' i f; a; ® E W 5 f 5? (J) gfléf‘ % ”My“ MWJa JA”Mf§ Z“: i X} 17 (A) W (B) m (C) W; me E .4 : 1 f G» «83? $2»??ng + m w (E) V “r: 33; «g g»; :2 325’ W)n (©4Q §i$ Egg? 2% if m)w @ % (J) $21 9??? vii»?- § E F M. Suppose X and Y are independent random variables with 0X 2 9 and 03/ z: 2. Find ame. 13. 14. Suppose the distribution of a random variebie X is normai, and suppose org is unknown A random sample of size n x: 10 is ebtained from the distribution ef X. in erder to find a confidence intewai for the mean of X, What distribution Should be utilized? 5’ (A z , j ‘e E C‘ y ’I F < ) X :Le jI/‘Eer'wvggj 8”” £143 wwemffifihfi’? ) "Vi Lee femaég‘ (93mg <c> M) em- (D) Tm (E) XE (I?) x5 (G) Xe For many years, the percentage of 25~year~olds with measurable hearing E053 has been 7%. However, fit is feared that this percentage has increased in recent years due to the greater use of headphones. A research physician plans to perform a hypoehesis test (with n 2 275 and a: m .04) to determine if this is the case. If, in fact, the percentage of American 25~year~olds with measurabie hearing loss has increased to 12%, what is the probability that she will commit a Type IIerror‘? _ i: m i (”I {g 4;: frog»? fig 5 ff)» >éfi? ,ngbfmzigéé g (A) .9613 (B) .0669 fag“. PW? Whig/J fowéei: (C) .1159 (123) .1539 (F) .1595 fie (o) .1680 ’ (H) .1739 g”? :: gjfifiefl; (I) .2115 {gee (:51? flat,“ _ fl 9 (J) .3093 W ~:.:: Ljffiéggé f? W“ @g‘flée’Be \é {Eavgééfaéglfi-‘i g; 275” The reiatioaship between {ha weight of a car and its gas mileage was stufiied, yielding the foilowing éata. Use thiS informatitm far probiems i5 and 16. i Weighi in was (at) . Gas Mileage in mpg (31) 2 I 15. To the nearest one percent, what percenéage of the variabiiity in gas mileage can be attributed to its linear regression an weight? M 3 [(A) 63%‘2 (B) 70% J: x , g; 3?; (C) 715% (D) 719% a (E) 80% (F) 81% {Jam AVW m?§\1/ f2} (G) 88% (H) 96% (I) 98% (J) 100% 16. Find the right endpoint L2 for a 95% prediction intervai for Yim = 1.5. (Yen may assume that the four assumptions on the “error” random variables E are satisfied.) (A) 26.97 3 :22: 2g? mg w 514%. §?33 :36: I}? M (B) 29.02 , K (C) 30.66 g: {;V§§ : 2am D 31.28 ‘1’ _ ; U {"22 ’2fl??é~'1%1%5’ ELM??? W) 35.70- a}??? :2 35:52%AA% (G) 3136 5?“: 5L? : 15:22ifl? 523.?Sgg’fl (H) 38.17 A? a) 38.82 f1 3:; (1) 41.11 {J 111% :3 323;} Ma”; 5 s ”a x Raj m ‘4» W’_ ‘ $5”? :: 25$ 3f" f2 ’??A1%%§>"2?§ A?%j§} £34m £14? 1?. In order to determine if linear regression is significant, what hypothesis should be tested? (WA) 3 o 0- jg at J; x; m M We all {18) 51%0 (3 ‘ , N . If. g; (C) We #’ 0 go. grave} ewe W” M X M “M ,/ - _ 5 (D) #le 7% 0 Sew/23; MW gel/5&1? M g} 1%; m E C72 0 r) e: 3 a 2; . (G) Sm # 0 (H) SSE ¢ 0 18. Consider a 2-sigma “X“ control chart for a normal random variable X , where the target mean is #3 m 10 and the standard deviation is known to be a = 1. Note that if samples of size n m 4 are to be used, then the lower control limit and upper confi’ol limit are given respectively by LCL x 10 —— 23%; m 9 and UCL m 10 + 2517: 2 11. If the mean shifis to p m 10.6, What is the average run length (to the nearest integer)? (A) 1 {wayggmgmiiig {if a“, xdwhazy m 3 1 m w ,_ 3 ”SEEM” :PZXéémeE’ffil/aiffiéé (D)? f” rfifwffifie ii“'§§‘5é':t (3)9 '~ Pigé~ ( (F) ill ((3) 13 {v (H) 15 9" L (I) I?" (J) 19 :9. A quaiit’y centrei manager is Sewing up a 3—sigme 3;; central chart for the mean weight {in grams) ef the Widgeis manufaemeed in. his factory. He takes m r: 5 random samples, each of size n z: 3, ebt’aining the feiiowing data. {For the sake of Simpkicity, both m and n. have been chaser: to be smaiier than they reefly should be.) Use these ciafe to find the upper centrei iimii fer the 35 chart. You may agsme that Widget weight is nermaiiy distribufed. Hint: "i7 2 10.8. (Yen do not need :0 Verify this.) I * I: : :3: ” i _1:2.:.11.4 .' “5?; e 7“; = 11.8 11.2 2 :12: 0.35:: 3 i.693 0.833 _ 4 2.959 0.83:) (A) 11.00 We 5 2.326 0.86.: (B) 11.15 5:5,; :- :55? .: 3 V :5?” E 333;: 3:3: (C) 11.24 {mg } 3 8 2.84? 0.82% _ 9 29:0 0.808 (D) 11.3? .5 «a; 10 31:?8 9297 e m 5’ :: :2: :3: - {WW 3:: 3.335 @570 W :z: we? {am-:2 (G) 11.78 :5 3.4:”: {32:55: (H) 11.84 (E) 11.97 (J) 12.15 20. In the mevie “Significance Tests,” a hypothesis test was performed to test the hypothesis that a reeently~discovered poem attributed to Wiiliam Shakespeare was aetuaiiy not written by Shakespeare. What were the first Six words of this poem? (A) Shall 1 age? Shall I rage w Shaiiiggij g y & Zéeem; :iwjé‘e Magi aifiwififi (C) Sheilidreem‘? Shall {scheme “ngmue gfiwm 5* (D) Shani fight“? Shaiifsmite if (E) Shafi Ipeuse? Shalilcause (F) Shafilpray? Shafllstay... (G) Shalflqueff? Sheiiicieff... (H) Shafllquaii? Shaflifaii (E) Shah}. sing? Shafilbring (J) Shel} lweep? Shall I sleep “s 3’ ('3 2' 5 9 ’ ii , f s” a»; Part H. True-False (one point each) Mark “A” en your answer card if the Statement is true; mark “8” if it is faiee. 21. Let X be a discrete random variabie, and suppose EEXE— W 5 "I hen EE X E m 25. EW 35%"? I?” {gfng’jl 22. If (X, Y) is a twe-«dimensienai random variabie such that VarX + VarY m Var(X + Y}, then X and Y are independent. , é %.&geer msieéwaééw} @Aéfi ExféeX e» Eé *“ Vizr' [EX fig} 23. For a certain data set, 35 m 48, q] r 34, and Q3 2 63. For this data set, the observation 115 qualifies as an oatlier. if: éfimlgézitl-C? flaw 3?: 1*: 5:3 effiiifi/m E: Efiéfi; Egg“ ::> geese“ 24. The purpose of the Method of Maximum Likeiihood is to find the value of a parameter which £5 mes”: Eikely to have produced a particular sample. 25. Let: X be a tandem variabie. If X is 1101111211, ihen X is normal. '26. Let X be a random variable. lf-X- is normal. then X is normal. E ”file fem? ;j fine. fiefiiawf’? Ziwugw ” Meg/wt... FEM/ire... vafiz‘lgqu/ééwjrg méférfiw NEW) “’32sz M flee V; A M ”3,4an- 27. If the P value for a hypothesis test is .01 then the value .93 represents the probability that Hg; is true. “$21," if‘izé’wgam : €533 ngjfwgamxfifiie %w;, fy-é&;%£&% A I fiifieé- é} Maria? M ye axing... reggae... wepimfljw $315M Mae, *‘éJWyim; gram ”My: We“? 395:3 wfim r 42.. 28, Consider a one~sided hypothesis test on the mean ll of some random variable X. All other things being equal, if the value of a is increased, the critical point f moves closer to the null value #3. 3 \ .- . / r 3‘» m efim.wwewww&?\ gig, g: 5" “mix .1 r 29. If we think of each sample in a quality control process as a hypothesis test, then a “false alarm” is the same as a Type I error. 5. if Parr III. Short Answer (one point) 5 *‘ g i . 30. Fill in the blank with the correct one—word answer: A(n) f; £2; g: ,A gm. 1 is a random variable whose numerical value can he computed from a Sample. (Note that “estimator" is a closelymrclated word which is not the intended answer here.) ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern