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Fall 2010 Final Exam - Probabiiity and Statistics for...

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Unformatted text preview: Probabiiity and Statistics for Engineering (ESE 326) Final Exam December 20, 201% This exam eomains 20 muifipiechoice problems worth two (Joints each, 9 {me-false problems worth one point each, and (me Shortmanswer problem worth one point. for an exam total of 50 points. Part1. MultigieuChoice (two points each) Cieariy fiii in the ova} on your answer card which corresponds to the only correct response. 1. If five cards are seiected randomly from a standard 52—card deck; what is the probability that there are three of one color and two of the other color? (A) 10325 (B) .0650 (C) .2235 (D) .3251 (E) .3497 (F) .6503 (6} -5749 (H) .7765 (I) .9350 (J) .9575 2. A new dessert bar called 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 smaii Squares of pie (which must be three different kinds). If there are twenty kinds of pie from which to choose, how many different orders are possibie? (A) 1350 (B) 4050 (C) 7240 (D) 8420 (E) 21:720 (F) 25,260 (6) 57,680 (H) 423329000 ('1) 5149843000 (3') eaooofooo DJ In a certain popuiation of coilege students, 15% iike guacamoie bot disiike black bean saisa, 30% ii-ke black bean salsa but disfiko guacamole, and 18% disiike both What is the probability that a student who Iikes guacamole aiso likes biack boon saisa? (A) .25 (B) .33 (C) .240 (o) .45 (E) .50 (F) .55 (o) .60 (H) .67 (I) .125 Let X be a, random variabie whose moment generating function is mxfit) m (1 —~ SEQ/“’3. Find the second moment of X. (A) V? (B) V? (C) V35 (D) V6 <5) Vi (F) ? 5. Katy is screening gracie school children for Vision problems. The children enter the screening room in groups of foot“. If 80% of the children have good Vision, what is the probability that the fifth group to enter will be the first group of the do}? io which all four children have good Vision? (A) .3115 (B) .0166 (C) .032? (o) .0498 (E) 0501 (F) .071? (o) 0:319 (H) .1021 (I) .1215 (J) .1555 6. A hardware store receives a shipment of 7’5 yellow daffodil bulbs and 25 white {iaffodil bulbs. (Of course, these bulbs will not Show color until they bloom, so they are currently indistinguishable from one another.) By mistake, the bulbs get (thoroughly) mixed together. If JoAnne buys 12 daffodil bulbs from this mixture. what is the probability that she gets at least 10 yellow bulbs? (A) .1166 (B) .236? (C) .2511 (D) .3732 (E) .3907 (F) .5093 (o) .6218 (H) .7489 (1) X633 (J) .8834 7. In a large manufacturing plant, there is an injury on the 30%), an average, mace every" eight months, Assume that this is a Peisson process. What is the prebabflity that an entire year wiil go by with no injuries on {he job? 0%) .8800 @)3n5 m)2%1 63) .3679 (E) .4866 (a 5B4 a3) .6321 (PS .7769 g) .8825 (n emu 8. Consider the continuous random variable X Whose density is given below. Find F(1). 12 f(:c) m 2336—3: $ > 0 (A) (B) (C) (D) (E) 6”} (F) 3 + 8—}. @)2Uwe”) @3261 (D m1+afl A E find-4 i (B E \.::‘ Mir—4 Mir—4 G) F“ NIH ,v-m. 7—4 (“G i V l m: ,‘_. 9. Comsider 3 random variabie X Whose distribution is {karma} with (2 m 12, and suppose that PEG 3 X g ES} 2 .68. Which Ofihe following is ciesest to the vaiue Ufa? (A) 1 (B) 2 (C) (33) (B) (F) 8 (G) 9 (H) 12 Chi-Rh“ 10- Consider the tw0~dimensionai discrete random vafiable ( X , Y) whose density table is as foliows. Find COVCX} Y). (A) .21 (B) .44 (C) ,4583 (D) ,5238 (E) 1.46 (F) 1.8330 (G) 3.36 (H) 13.6 (I) 14.2 (3) 18.7 11. Consider the Ewe—dimensionai cominuous random variabie (X,Y) whcse density is given as foilows. Find E§X3 fXY<$1§3mgxy fiéxflyfil (A) 1% 2% (B) 1%; (C) {3 =13- (111 1% 1% (E) {g (F) «1% (G) 1% 11% (H) 3% $.23? a) 1%- 111 1% m% 12. Suppose X and Y are inéependen’z random variabies with a); = 9 and fly m 2. Find Effigy. .13. 14. Suppose- the distribution of a random variabie X is nermal, and suppcese ex is unknown. A randem sampie ef size n z 10 is Obtaineé from the distribution ef X. In order to find a confidence intervai fer the mean ofX, What” disfi‘ibutien sheuid be utiiized? (A) Z (B) Ts (C) T. (D) Tm (E) xi: (1“) X3 (G) x333 For many years, the percentage of 25wyear~0Eds with measurable hearing loss has been 7%. However, it is feared that this percentage has increased in recent years due to the greater use of headphones. A research physician plans to perform a hypothesis test (with n z 275 and a m .04) to determine if this is the case. If, in fact, the percentage of American 25-yearuelds with measurable hearing 1053 has increased to 12%, What is the probability that she wili commit a Type I} error? (A) .0613 (B) .0669 (C) .1159 (D) .1196 (E) .1539 (F) .1596 (G) .1680 (H) .1739 (I) 42116 (J) .3093 The reiatienship between the weight (if a car and its gas mileage was studied, yieiding {he foliewing data. Use this infen’riation for problems i5 and 36. ' Weight iii tens (x) Gas Mileage in mpg (y) 21 15. To the nearest one percent, What percentage of the variability in gas mileage can be attributed to its iinear regression en weight? (A) 63% (B) 70% (C) ?e% (D) 79% (E) 80% (F) 81% (e) 88% (H) 96% (I) 98% (I) 100% 16. Find the right endpoint L2 for a 95% prediction interval for Yja: x 1.5. (You may assume that the four assumptions on the “error” random variables 132- are satisfied.) (A) 26.97 (B) 29.02 (C) 30.66 (D) 31.28 (E) 34.42 (F) 35.0? (G) 37.36 (H) 38.17 (E) 38.82 (I) 41.11 37’. i8. In erder to determine if fines: regression is significant, What hypeshesis should be tested? (A) :30 $5" 0 {B} £91 % 9 (C) Yix is 0 (D) we e 0 (E) a2 s s {13‘} 3'3 5%.. s (G) S} #3 0 (H) SSE 71 3 Consider a 2-«sigma TX: central Chart for a normal random variable X , Where the target mean is [L3 m 19 and the standard deviation is known to be a m 1. Note that if samples of size n m 4 are to be used, then the lower controi iimit and upper contrel limit are given respectively by LCL m 10 — 2% x 9 and UCL : 10 mi“ 2;}: m 11. If the mean shifts to 11 m 16.6, what is the average run length (to the nearest integer)? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 (1:) 11 (G) 13 (H) 15 (I) 1? (1) 1s 19. A quality oontroi manager is setting up a 3—sigma X control cheat for the mean weight (in grams) of the widgets manufactured in his factory. He takes m z 5 random samples, each of Size n z: 3, obtaining the fofiowing {iata‘ {For the sake of simplicity, both we and n have been chosen to be smafler than they realty Shouid be.) Use these data to fine the upper control iimit for the if Chart. You may assume that wtd'get weight i5 normality distrihuted Hint: “e“ m 10.8. (You do not need to verify this.) , 2 1.228 0.353 3 1.693 mg 4 2.059 ease (A) 11.05 3 2.326 08:34 6 2.534 0.848 (B) 11'15 7 2.?(24 0.333 a 2.847 9.828 (C) 1124 9 2.9?0 0.808 (D) 11.37 10 am (we? ‘ 11 3.173 {178? (E) 1144 2 3.258 0378 13 3.336 0.770 (F) 11-58 M. 3.40? 0%2 (G) $1.78 25 3.472 8.?55 (E) 11.84 (I) 11.97 (J) 12.15 20. In the movie “Significance Tests,” a hypothesis test was performed to test the hypothesis that a recentlyudiscovered poem attributed to William Shakespeare was actuajly not written by Shakespeare. What were the first six words of this poem? (A) Shafllage'? Shalilrage (B) Shaliidie‘? Shaflifly (C) Shaiilctream? ShaHI scheme... (D) Shail I fight? Shel} I smite ... (E) Shafi I pause? Shafl 1 cause ... (F) Shall I pray? Shah I stay . t. (G) Shailiquaff? Shel} Idoff... (H) Shafl I quaii‘? Shaii I fail (I) Shel} I sing? Shaikibring (I) Shajilweep? Shalileieep... Part1}. True-False {one point each.) Mark “A” or: your answer card if the siarement is true; mark “B” if it is false. 21. Let X be a discrete random variable, and suppose EEXE 2* 5. Then EEX’gE 2:: 25. 22. If (X: Y) is a two-dimensional random variable such that VarX + VarY m Var(X + Y), then X and Y are independent. 23. For a certain data set, E = 48, q; 2: 34, and Q3 2 63. For this data set, the observation 115 qualifies as an outlier. 24. The purpose of the Method of Maximum Likelihood is to find the value of a parameter which is most likely to have produced a particular sample. 25. Let X be a random variable. If X is normal, then 3:: is normal. 26. Let X be a random variabie. If 32: is normed, then X is Home}. 2?. If the P vaiue for a hypothesis test is .0?) then the value .93 represents the probability that Hg is {1118. 28. Consider a one-sided hypothesis test on the mean ,u of some random variabie X. Ail other things being equal, if the value of a is increased, the eriticai point f moves closer to the {31111 value pg. 29. If we think of each sample in. a quality control process as a hypothesis test, then a “false alarm” Es the same as a Type I error“ Part III. Short Answer (one point) 30. Fifi in the biaok with the correct: enemword answer: A01) is a random variable whose numerieai vafue can be computed from a sample. (Note that “estimator" is a cioseiy~reiated word which is not the iotended answer here.) ...
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