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Spring 2011 Final Exam - Probability and Statistics for...

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Unformatted text preview: Probability and Statistics for Engineering (ESE 326) Final Exam May 9, 2011 This exam contains 19 multiple—choice problems worth two points each and 12 true-false problems worth one point each, for an exam total of 50 points. Part I. Multiple-Choice (two points each) Clearly fill in the oval on your answer card which corresponds to the onlyr correct response. 1. What is the probability that a five—card hand selected randomly from a standard 52-card deck will be a “three-of—a~kind”? (Such a hand consists of three cards of the same kind together with two cards whose kinds do not match the first three or each other. An example is 9 0 310,91! ,Av,5 0 .) (A) .0070 (B) .0123 (C) .0211 (D) .0226 (E) .0423 (F) .0451 (G) .0595 (H) .0616 (I) .0824 (J) .0872 2. Suppose A and B are events such that P[AIB] = .7, P[A’ m B’] 2 .16, and P[B] 2 .6. Find PM]. (A) .24 (B) .40 (C) .42 (D) .504 (E) .588 (F) .6864 (G) .66 (H) .7143 (I) .8333 (J) .8571 3. Consider the discrete random variable X Whose density is given by the following table. Find ElX]. (A) g = 1.5 (B) 3 = 1.6 (C) :- = 1.75 (D) g- : 1.8 (E) 2 (F) 13: m 2.2 @ ll NJ to O1 @ W MlCfl 045 (Jr-Ice m 5| to ,4; (I) m 2.5 A 3 4. Gambler Gus has a weighted coin which comes up heads Only 40% of the time. He places a bet with Saloon Sally that if he flips his coin 10 times, more than half of the flips will come up tails. What is the probability that Gambler Gus will lose his bet? (A) .1662 (B) .3669 (o) .3823 (D) .4512 (E) .5 (F) .5488 (G) .6177 (H) .6331 (I) .8338 There are 101 students in a physics class, specifically, 22 sophomores, 50 juniors, and 29 seniors. If the instructor randomly selects three students to be on a speciai Advisory Panel, what is the probability that at least two are juniors? (A) .1176 (B) .2426 (C) .2723 (D) .3749 (E) .4926 (F) .5075 (G) .6251 (H) .7277 (I) .7574 (J) .8824 The arrival of patients at the Emergency Room of a small-town hospital is a Poisson process, where A 2 20 patients per day. What is the probability that the first patient of the day arrives during the second hour (1:00 am. to 2:00 am)? (Of course, the ER is open 24 hours a day, and each day begins at midnight.) (A) .1509 (B) .1574 (C) .1889 (D) .2457 (E) .3344 (F) .3622 (G) .4346 (H) .5654 (I) .6656 (J) .7543 7. Consider the continuous random variable X with the density function shown below. Find the median of X. f(93) I 82:”, a: > 2 (A) 2 (B) 2x/5 (C) 2 3 (D) 4 (E) 4V5 (I) 4\/§ (G) 8 (H) 8V5 (I) 8\/§ 8. Consider the continuous random variable X with the density function shown below. Find the variance of X. f(;c)=2—2x,0ga:51 (A) ~1— (B) 3— (C) ~1— (D) i (E) i (F) (G) (I) .1, 9 J; 8 (H) l; l 4 (J) g 9. 10. Consider the two»dimensional random variable (X, Y) with the joint density shown below. Set up the integral which would be needed to find P[Y > X]. You do not need to work out the integration. fxy(x,y)2§my,1s:cs3,osys2 on jfjfgxydydm (B) ffffgmydyda (C) jfjfgmydydx (D) jfjfgxydydx (E) jfj:%xydydx no jfjfgxydydx an jfjjéxydydx on jfjfgxydydm Consider the following theorem and proof. Which step in the proof uses the assumption that X1, . .. ,Xn is a random sample? (Note that a “step” is a transition from one line to the next. For example, (A) is the correct choice if the transition from the first line to the second line in the proof uses the random sample assumption.) Theorem. Let X be a random variable with mean n, let X1, . . . , X“ be a random sample of size n from the distribution of X, and let X = ZXg- - 111 be the sample mean. Then E[X] = [in 1'21 Proof. E[Y] n it (A) =E{Z& Q 121 fl it (B) 2% an] n l} (C) : lE [ 2X1] i=1 n ll (D) ixl n ii (E) =tzmn i=1 n ii (F) Zia“ it (G) ==%(nn) ll (H) ll. Suppose a random variable X has mean ,u : 80 and variance 0‘2 z 25. Let X1, ,X4 be a random sample of size n = 4 from the distribution of X. According to Chebyshev’s Inequality, at least 3 of the sample means in the distribution of X will lie in what interval? (A) 77.5 < 3?" < 82.5 (B) 75 < 1)? < 85 (C) 73.75 < 32'" < 85.25 (D) 72.5 < “X” < 87.5 (E) 70 < 3? < 90 (F) 57.5 < “X” < 92.5 (G) 65 < X < 95 (H) 51.25 < ”X“ < 98.75 (I) 55 < ”X" < 105 (J) 30 < f < 130 12. Suppose X is a normal random variable with unknown mean ,LL and unknown standard deviation 0. The statistic s : .8 is obtained from a random sample of size n m 10. Find the left endpoint L1 for a 92% confidence interval for the standard deviation 0. (A) .33 (B) .41 (C) .57 (D) .55 (E) .54 (F) 1.35 (G) 1.44 (H) 1.52 (I) 1.85 (I) 2.32 13. Suppose a two-tailed hypothesis test has null and alternative hypotheses as shown. H0: [1. z 7 H1: ii 75 7 A random sample yields the sample mean 233" 2 10. Which of the following is the correct set-up for computing the P value? (A) P[M%7|7:10l (B) 2P1M37IX7£101 (C) 2P[,u27l77£10] (D) 2P[ag7|f=1o] (E) 2Piu271fn10] (F) Plfflomzfl (G) 2Pl35510Imé7] (H) 2131?: 10mm (I) 2P[”X':10Iu:7} (I) mfziolurfl 14. Let X be a normal random variable, and consider a hypothesis test on the mean ,u, of X having the following characteristics. it 2 9 0: I .05 H92n=15 H1:,u<15 E = 13 s = 2.8 Find the probability of committing a Type II error given that ,u 2 12. (A) .1003 (B) .1048 (C) .1063 (D) .1260 (E) .1475 (F) .1624 (G) .1641 (H) .1871 (I) .1951 (I) .2267 15. 16. 17. Suppose the significance level for a hypothesis test is a = .05, and suppose the P value is .08. Which of the following statements is correct? (A) We are 95% confident that Hg is true. (B) We are 95% confident that H0 is false. (C) We are 95% confident that H0 is significant. (D) The probability that H; is true is .92. (E) The probability that H0 is false is .92. (F) The probability that H0 is significant is .92. (G) None of these statements is correct. Rose is interested in the proportion of American households which do not have a landline. She takes a random sample of 75 homes and finds that 21 of them do not have a landline. She uses this data to construct a confidence interval, which she then decides is unacceptably wide. If Rose starts her study over, what sample size should she choose in order to obtain a 97% confidence interval with radius at most .04? (A) 445 (B) 445 (C) 598 (D) 594 (E) 735 (F) 736 (G) 944 (H) 945 (I) 982 (J) 988 Consider the following random sample of points. Use the regression line for this data to estimate y(12). (5, 24) (10, 28) (15, 55) (20, 62) (25, 88) (A) 35.84 (8) 37.82 (C) 38.80 (D) 49.24 (E) 41.58 (F) 43.25 (G) 44.83 (H) 45.40 (I) 47.97 (1) 49.55 18. 19. A 2—sigma :3? control chart has been set up to monitor the sample means of a normal random variable X , using samples of size n = 9. When the process is in control, the mean and standard deviation of X are #0 2 40 and or = 6. The control limits can then be computed to be as follows. _ W i: 3 1L: LCL—4O 2V5 36 UCL 40+2\/§ 44 Find the average run length (to the nearest integer) if the mean shifts to a = 41.25. (A) 2 (B) 3 (C) 5 (D) 11 (E) 12 (F) 24 (G) 231 (H) 243 (I) 336 (J) 462 In the movie “Case Study” from the video series Against All Odds: Inside Statistics, an account is given of the clinical trials in which a new medication was tested. What was this medication? (A) Allegra (B) Amoxcil (C) AZT (D) Claritin (E) Lipitor (F) Prevacid (G) Ritalin (H) Vicodin (I) Zithromax (J) Zyrtec Part II. True-False (one point each) Mark “A” on your answer card if the statement is true; mark “B” if it is false. 20. For any events A and B, P[B|A] : 1 — P[A|B]. 21. If the two—dimensional random variable (X, Y) has joint density given by the table below, then X and Y are independent. 22. Suppose (X , Y) is a two-dimensional random variable such that the correlation between X and Y is PXY 2 —1. Then X and Y are exactly linearly related. 23. Let (X, Y) be a two-dimensional random variable, and consider iii/Mm), the curve of regression of Y on X. For each :13, am gives the mean of Y for that value of m. 24. If X and Y are independent normal random variables, then 0x+y : ox + 01/. 25. Let X be a random variable, and let X1, . .. ,Xn be a random sample of size n from the distribution of X. If n is large, then, by the Central Limit Theorem, the distribution of X is approximately normal. 26. Suppose a confidence interval for some parameter is being constructed. All other things being equal, increased reliablility leads to a longer confidence interval. 27. f9 2 if is an unbiased point estimator for p. Consider the usuai situation Where there appears to be a linear relationship between a variable X and a random variable Y, and where a random sample of points (331,y1), m , (33m yn) has been obtained for the purpose of finding a line of regression. Recall that for each i, Y; 2 fig + 6135 + Ei, where E; is an “error” random variable. Assume that the usual four assumptions on the 131- are satisfied. One of those assumptions is that the ET- all have the same variance, which we denote by 02, and it follows that the variance of each Y; is also 02. Use the above information for problems 28 through 30. 28. Si, is an unbiased point estimator for 02. 29. For any as, the confidence intervals for ny‘x and Y|zc have the same center. 30. For any 3:, the confidence interval for ”Yin: is shorter than the confidence interval for Y|ac. 31. For a control chart to give good results, the sample size should be large (n 2 20). ...
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