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Unformatted text preview: Frobabiiity and Statistics for Engineering (ESE 326) Exam 3 April 15, 2010 This exam contains nine muitipie—choice probiems worth two points each, nine twenfalse probiems worth
one point each, seven Shomanswer probiems worth one point each and one freeresponse problem
worth six points, for an exam iota} of 40 points. 93:: I. WinnieChoice (two points each) Clearly iii: in the ovai on your answer card which corresponds to the onIy correct response. 1. Let X be a random variabie, and iet X1, . .. , X” be a random sample from the distribution of X. Which ofthese m n, X, or n ~—— is an exampEe of a statistic?
(A) it 031131 (B) X only (C) 71 only (D) n and 3? only (E) $1 and in only (F) Y and n oniy (G) 1.5, "Xand n (H) none of these 2. A random vanablo X has the following density function with 6 unknown: f (x) m m Let X;,... ,Xﬁ, be a random sampie of size n ﬁ'om the distribution of X. Find the maximum
likeiihood estimator fox 6. (A) “‘5"?
(B) J?
((2) 2f")?
(D) £2?—
(E) “X”
(F) 2“”X“
(G) 993
(H) (3??
(I) 2(3‘5)‘Z 3. Suppose X and Y are independent normai random variabies with the foiiowing characteristics.
ELK] = 50 0X m 10 E[Y] 2 (£0 cry m 5
Find PiX > Y}.
M. First ﬁnd the distribution (shape, mean, standard deviation) of the random variable X — Y. {A} .9222
(3) .0777
(C) .1241
o)) .2855
(E) .2525
(F) .2475
(G) .8245
(H) .8759
(2) .9223
(J) .9772 4. Soppose a sampie of size n = 6 is taken from a norm} population with 11. unknown and a = 3.5.
The sample mean is computed to be E m 29.2. What is the correct computation for the upper
endpoint L2 of a 96% conﬁdence intervai for ,2? (A) 29.2 +(1.751)(1.065)
(B) 29.2e(1.751)(1.229)
(C) 29.2 +(1.751)(1.676)
(D) 292+ (2.054)(1.065)
(E) 29.2 2» (2.0590429)
(2‘) 29.2+(2.054)(1.676)
(G) 29.2 + (2.752)(1.065)
(H) 29.2 + (2.257)(1.429)
(1) 29.2 + (2.757)(1.6’?6) Suppose X is a normai random variable with o" m 15. How Iarge a sample size is needed to
estimate is to within 2 units with 98% conﬁdence? (A) 17
(B) 18
((3) m
(D) 77
(E) 237
(F) 238
(o) 304
(H) 305
(I) 1217
(J) 1218 Suppose we age using a. random sampie of size n : 9 to ﬁnd an 88% conﬁdence intervai for the
variance 0'2 of a normai random variable X. Find the vaiue of (X2 )* which wiil be used in the
computation of the tight endpoint L2. (A) 2.908
(B) 3.52}.
(C) 3.73?
(o) 4443
(E) 5.189
(F) 11.2763
(G) 12.770
(H) 14.066
(I) 14.956
(J) 16.346 '3. Let X be a random variabie, and suppese we ate perfemting a hypothesis test on the mean pt (33? X ,
where a m .65. What probability is represented by the number 1 ~— 0: m .95? (A) the probabiiity that Hg is tree (B) the probability that H} is true (C) the probabitity that we Wiii {eject Hg} (D) the probability that we wt}! fail to reject Hg) (E) the prebability that we Witt reject H9 given that Hg is tme (F) the probabiiity that "we will faii to reject HG given that H0 is true
(G) the preeabiiity that we wiii reject Hg given that If; is true (H) the probabiiity that we will faii to reject Hg given that H; is true 8. Supgaese X is a normal tandem variabie. Consider a hypethesis test with the feiiowing
characteristics. Hg:nm80 ﬂ;:n<80 0:.19 31:21 §m76 3:13
Find ,8 given that the true popuiation mean is pt 2 ‘74:. (A) .0013
(B) .0026
(C) .0064.
(n) .0123
(E) .0270
(F) .1600
(G) .2195
(H) .2710
(t) .3487
(J) .3502 9. In the movie Exgerimental Design, a contrast is drawn between an observational study and an
experiment. What matine animal is featured as the object of an observational study? (A) barracuda
(B) clownﬁeh
(C) dolphin
(D) lobster (E) manatee (F) octopus
(G) seahorse (H) shark
(I) shrimp
(J) Stingray Part Ii. E‘meaFalse (one point each)
Mark “A” on your answer card if the statement is true; mark “B” if it is false. 30. Let X be a random variable. A random sample from the éistribution of X is a collection of
independent random variables, each having the same distribution as X. 11. Let X be a random variable, and let X1, , Xn be a random sample from the distribution of X.
Then the sample variance is computed as follows: 32 mmXémX)? i 31.
V TL Fun2 12. get X be a random variable? aed let X1, . . i ,Xn be a random sample from the éistlibution of X. If
X is approximately normal, then X is approximately normal. 13. 144 15. 16. 17. 18. Suppose you are computing the bounds for a conﬁdence intervai. A11 other things being eqoaig a
higher conﬁdence levei will resuit in a Ionger conﬁcienee intervai. Suppose you are computing the bounds for a conﬁdence interval. Ali other things being equal, an
unknown popuiation variance (as compared to a known population variance) will tesuit in a longer
conﬁdence intervai. (You may assume that the {mother for s in the “onimown variance” case
happens to be the same as the number for a“ in the “known variance” case.) Vﬁi‘(T3) < Vadng) 32 is an unbiased estimator for 02, If { 25, 36] is a 95% conﬁdence intervai for 02, then [5, 6] is a 95% conﬁdence interval for a. A researcher performs a hypothesis test. Unknown to her, the aitemative hypothesis H1 is true. If
she regects Hg, this is a Type I error. Fart Hi. Short Answer (one point each)
The answer to each of these is right or wrong: no work is required, and no partiai credit wit} be given. For several years, EnCounoil has been keeping records of the number X of pizzas purchased each week
for Cheap Lunch. The population mean is {1. m 15‘? and the popuiation standard deviation is 0‘ = 21.
Someone who does not know these oararueters obtains a random sampie X1, , X}; from the
distribution of X, resulting in the foliowing data Use the information above or below for problems 19
through 23. If you wish to round any numbers, round to one éeeimai piece. 172 172 180 168 174: 150 150 90 £69 155 127 19. Draw a stem and leaf diagram for this data. Make
it perfect so you don‘t miss out on this point! 20. What is the shape of this distribution _
symmetric, skewed left, or skewed right? 21. Find 5. 22. Find Var(X). 23. You are not exoected to know the technical method for determining whether or not a particuiar
outcome is an outlier. However, you should understand the concept of an outlier weii enough to
answer this: Given that the above random sample contains exactiy one outlier, what is this outlier? 24. Which gives more—complete information, a point estimate or an interval estimate? 25. Suppose we are performing a hyeothesis test, and the following numbers are given or have been
computed: n, E, s, as, and [3. Using one or more of these numbers as appropriate, how should the
power of the test be calculated? Part IV. Free Resoonse (6 points)
Follow directions careﬁitly, and show all the steps neeéed to arrive at your sotution. 26. Let X be the random variable which measures the weight of a bag of potato chios. The target
standard deviation for X is o‘ = .2 ounces. It is feared that the standard deviation is higher than .2
ounces. A random sampte of n = 15 bags is taken and weighed, resulting in a sampte standard
deviation of s = .25 ounces. (It is reasonable to assume that X is normal.) (at) State the appropriate null and alternative hypotheses for this test. Good notation counts. H6: H}: (b) Find the P vaiue for the test. Be sure to show the set~ttp (exactly what probability you are ﬁnding), at Ieast one intermediate step, and a numerical answer, rounded to four decimal
places. Good notation counts. (c) At the signiﬁcance levet of a x .05, what is the appropriate conclusion? Do not state your
conclusion in terms of He or H1. Instead, state the conclusion that should be drawn about the standard deviation (of the weights of the bags of potato chips). Word your answer careﬁtlty to
state exactly what you mean. ...
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