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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 truefalse probiems worth one
point each one one shortenswer probiem worth one point, for an exam totai of 50 points. Part1. MuitigieChoice (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 52card 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 ngirX
(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. $0 they are currently inéistinguishable
from one another.) By mistake, the bulbs get (thoroughly) mixed together. If IoAnne buys 12
daffodil bulbs from this mixture, what is the probability that she gets at least 10 yellow bulbs? (A) .1166
(B) .2367 5:1: :22: E535” 5% :1 Heme} +—efx:s5}+vi>fx::2} 5;» ﬁwma Ema:6. Mag: 1.1m. (a) .390? .. 5/1.; af‘.25»\ {wag/25 {7?}525}
F .6393 .. a se a: f 92* 5.. ( 3 «a» W e W
(G) 5218 /}&€3\ {E 53:35} a (1/ iéé‘ >
3? 12:: 5: a} . 3‘ * (1) .8834 in a Earge manufactm'iug giant, there is an ﬁgury on the job, on average, once. every eight months“
Assume that this is a Poisson pracessk What is the probabii'ity that an entire3163: wiil g9 by with 119 injuries on the jab?
i W t. . k ; ¥ _ f
(A) .9830 Pg/WMWZ R :1 3,. ngMW #2? %{cﬂﬁ%b (B) .1175 5 1 {MM 5555551543.) ﬁwyM/L éﬁézjzﬁmS/A 5 (C) .2231 W i 7 35
(D) .3629 gas; ,54 : {ﬁg/355$; : §~
('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"$2 m > 0 m>§bw”) r/5 55“’ «x1 5 “:2?
(3558—1 riﬁﬁwjag 336' gig/)4 ” ”“5 ja
(®%a+*w : m 5;” m;
(1353“?1 6 ( >
(F) em“i m g W 2% was (F) 14:52} 5.. w (G) 2(1~e’1) 9. Consider a tandem variabke X whose éistribution is normal wiih ,u. r: 12, and suppese that
13:6 g X g 18:; z .58. Which of the feiiowing is clesesi to “£116 value of a? (3%} 3. ﬁg; “229%. {{wwwwiwté gtgéfzzimﬁgéf. fqm) aggfwwﬂiwgéﬁ
(B) 2 . ﬂ Q . 3 i ., ‘g . ‘ 4: §
§ 59’"? Q? m MQM. mam a. mmwwi swayer. M (C) 3 ..
w} 5‘ ’ ' a . ’ .. ’ _ {392% W>«z..*J/,§£QM :2”wa QQLQMQJMJ cwiﬁm 6:15 35%;}... WWﬁMK» . E 6 W m4... it?” if if», (G) 9 (H) 12 10. Consider the tw0~dimensi0nal discrete random variable (X, Y) Whose density table is as follows.
Find COX/(X, Y). x/y 4 5 2 ,6 1 '5 .1 2
. ”H” .i i E 3 I § W .
(A) M21. 55 XE 3 25st Qaéﬁw 3:2. S
(C) 4583 glaﬁj 2:: agﬂ?>+§/53§ 3: %3 (D) .5238 6% ‘ . »
(13)}.46 £33.53]: ganmﬂgfwzégégh 3.552({3}
(F) 1.8330 (G) 3.36 '1': {2%. 2 (H) 13.6 3 _, . _ r... w a. 1.2 may}: gg’ngéﬂa’j (j) 18‘? ~: 3%, '2. .. {3. 2:} 5m} 11. Consider the twodimensional continuous random variabie (X,Y) Whose density is given as fOEIOWS. Finii EDS] fX¥($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 2sigma “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. TrueFalse (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: éﬁmlgézitlC?
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. lfX 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;, fyé&;%£&% 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.) ...
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