Econ 15B S07 FINAL - PROBABILITY & STATISTICS (test form A)

Econ 15B S07 FINAL - PROBABILITY & STATISTICS (test form A)

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Unformatted text preview: Page I of“) Econ 15B: Probability and Statistics Final, 10:30 — 12:30, Thursday, June 14, 2007 Test Form: A Words of Wisdom: "In tfiefitture, I pfan on tanng more qfan active rot}: in the decisions I mafia ” — Paris Hilton (6/07) FIGURE A (Critical bound indicated by the vertical line) 1. We commonly set on = .05: a. For mathematical reasons that go beyond the scope of this course. 13. For a (somewhat complicated) reason that we proved in class. c. For no mathematical reason at all. d, Because that guarantees that we won’t need an overly large sample size. I None of the above. Page 3 ofIO 7. Up to the rounding error of the computing software, the function (1 —— @qnorm(.4)) calculates the following (assume that X has the appropriate distribution): a. The p such that pr(X S p) = .4 b. The p such that pr (X S .4) = p c. The p such that flp) ; .4, where f is the pdf of X. (1. The p such that fi.4) = p, where f is the pdf of X. None of the above. 8. As the size n of a random sample gets very large, the difference between the normal distribution N(0, 1) and the T distribution T(n — 1) becomes: a. Small. b. Very small. c. Insaner small. d. Freakin’ tiny! All of the above. [Hint e is the correct answer.] 9. @qtdist(4, 4) = a. 4 b. 8 @00 hard to be determined (by an intro statistics student) without a computer . Undefined e. None of the above 10. Suppose qN is the .975 quantile of N(0, l), and qT is the .975 quantile of the T distribution with n — 1 degrees of freedom. Suppose also that [a, b] is the confidence interval derived from the Normal distribution, and [c, d] is the confidence interval derived from the T distribution. (Both confidence intervals were derived from the same data set, and let us b—a_q_N d—c qr also assume 0 = 5.) Now consider the claim: a. This claim is true. b. This claim is false, because the first ratio also depends on the size of )7 . g @This claim is false, because (b — a) depends on X]— , but (d — c) depends on H y . n —l d. b and c e. None of the above. 11. If you calculate that a 95% confidence interval for the mean u of a population is given by (231.7, 278.5), then y0u can conclude: @ pr(231.7 s a s 278.5) = .95 b. pr(23l.7 < p < 278.5) = .95 c. pr(231.7 2 [J 2 278.5) = .95 d. a and b 6. None of the above. Page5 of“) 17. You are planning a clinical trial to see if you should reject the null hypothesis that a given drug has no effect (against the alternative that it does some good). During this planning phase, you discover that the drug is much more harmless than you had originally thought, and potentially has quite helpful effects. Using the methods discussed in this course, you might reflect this new information by deliberately doing what (even though this choice might v o .n - effects): .” ecreasmg B 6. None of the above. 18. Ultimately, T distributions are: a. Probability distributions that reflect the deep underlying distribution of many phenomena of actual interest. b. Very special forms of the normal distributions 0. Mostly just a statistical ‘trick’ for addressing certain questions by ‘reorganizing’ or summarizing data so that its new form comes from approximately a T- distribution. d. a and b @ None of the above. 19. To use Eviews to calculate a 95% confidence interval using the student’s T distribution and a set of 17 observations, your formula would include the subformula: a. @qtdist(.025, 17) b. @ctdist(.05, 17) ‘I c. @qtdist(.025, .975, 17) d. Cannot tell; need information about the (estimated) standard deviation of the data set ® None of the above. 20. If the random variable X (using a random sample of size n) is an unbiased estimator of the ulation parameter 6, then: @(X) = e . lim n ——> 00 X = 6 c. For the given data set, X = (a, b), the confidence interval at the given confidence ' level which is always built into X (1. For the given data set, X = (a, b), the confidence interval at the some selected confidence level, which is not necessarily built into X. e. None of the above. 21. 13(82) = 02 a. True. 1). False; E(SZ) > 0'2. <95 @ False; E(SZ) < 02. d. False; sometimes E(Sz) < 02, and sometimes E(Sz) > 0'2. e. None of the above. Page 7 ofIO 27. The distinctive structure of a two-sided hypothesis test (as opposed to a one-sided test) re ects the fact that: ® You want to show that 2? lies in one of the extreme ‘tails’ of It; ’3 distribution. vb. You want to show that 35 lies in one of the extreme ‘tails’ of X’s distribution. 0. You cannot assume that u S pm) or that tt 2 um. (1. You do in fact know either that u 5 pm or that u .2 pm. e. None of the above. 28. If c is your critical bound for a one-sided hypothesis test, then if you switch to a two- sided test (leaving everything else the same), your critical bound will always be: a. c . ,1, _. ,. _ ..._. .c-W‘ “' " ""' "‘“h-w—L. e. None of the above. 29. Compared to two—sided hypothesis tests, one—sided tests: a Are more powerful . Are less powerful Q. Have larger a-levels t d. a and c 6. None of the above. 30. Let fix) be the pdf of the distribution NQJ, e2). 1 = dx 1 (x — m2 a. ex ——Hw—— J27rcr p[ 20'2 _(x-#) “(x-fly I b. M03 exp1: 202 ] c_ (it—m mpg] VZEJZ 20' x (x - m2 d. — ex — — J27r0'2 p[ 202 ] None of the above. 31. If it is extremely important that you don’t incorrectly reject the null hypothesis using a give; data set, you would do the following: Make a very small, although that will mean increasing the size of [3. . _ Make ct very small, although that will mean decreasing the size of B. 0. Make B very large, although that will mean increasing the size of 0L. (1. Make B very small, although that will mean increasing the size of 0L. e. None of the above. Page 9 of“) Scratch Paper Page 60f10 22. Suppose the variance of a population is known to be 3, and for your desired confidence level 0, @qnorrn(c) = b. Then if your sample is of size 75, the length of your confidence intervalis: a. 1b 75 b. ——6—b 75 c. 2b 5 d. 3b 5 © None of the above. 23. Using the definitions of 0t and [3 as we have done in this course, the probability of a Type Ierroris: @ or b. (lea) c. B d. (1—5) 6. (17B 24. When T = t, the p—value for a one-sided test using a T-distribution with k degrees of freedom is always given by (up to rounding error of the software): @gtdistfi, k.)____ ' tdist(t, k) c. ist(t, k—l) d. @qtdist(t,k-1) ,le. None of the above 25. Using the definitions of a and B as we have done in this course, the probability of a Type Herroris: a. on b. (1—00 ® B i (1-l3) e. a—B 26. Using the definitions of on and [3 as we have done in this course, the probability of a correct rejection of the null hypothesis is: a. Ci. b. (1—41) B C. (6 (1 4%) e. on—B Page 4 of“) 12. The primary advantage to empirical science of a T—distribution over a normal distribution in hypothesis testing is that T-distributions a. Account for the unknown variance of the underlying population. b. Account for the degrees of freedom in the sample. a and'b. d. T-distributions are easier to work with. 6. None of the above. 13. To shorten the length of a confidence interval, you could @ Increase the size of your sample. . Decrease the size of your confidence level. c. Decrease the size of the T-distribution d. a and b e. a, b, and c 14. Let it) be the pdf of N(0, l) and let f be the pdf of a T-distribution with k degrees of fieedom. Then for all numbers x: a. ¢(x) (fix). b. ¢(x) =flx). y. C- W) >flxl- @Cannot be determined: it depends on the size of k. e. None of the above. 15. Using the distribution T(5), the quantile of O is: a. 0 e 5 c. 1 d, — 00 e. Too hard to be determined (by an intro statistics student) without a computer 16. Suppose that on a two—tailed T-test, your value I exceeds only the rightmost critical boundary appropriate to this test. Since the test is two-tailed, you lmow that your p-Value will be larger than it would be if you had merely performed the corresponding one-tailed T-test. Nonetheless, it is absolutely gyaranteed that your resulting p—value will be below your oc-level for the test that you did in fact perform. a. True. b. False; it depends on how close t is to the (Jr-level. c. False; it depends on how many degrees of freedom you have. cl. b and c. ’ "N. Islone of the above. “5134‘s?- I 2. 9 l7 4. l5 5. EE 6. % Page20f10 Referring to Figure A at the beginning of this exam, (1 — B) is given by the area of the region(s) (regions are the shapes bounded by black lines): a. m and n (9 j and k c. l, m, and n d. 1 and m 6. None of the above Referring to Figure A at the beginning of this exam, on is given by the area of the region(s) (regions are the shapes bounded by black lines): a. m b. l and m c. l, m, and n @k e. None of the above If r (t < 0) is your “studentized” score from your data, and H0 is the null hypothesis, then when you (appropriately) reject H0 in a one-tailed composite hypothesis test at the on level of significance, this means that: a. pr(Ho l t) = or pr(TSt]H0)<oc c. pr(Ho|Tsr)<ot d. Cannot tell fiom the information given. e. None of the above. Let f be the pdf of the distribution NQJ, 0'2). flu) = 74 when (and only when): 0’2 = V 27: - 74 74 d. 0' = w 4221' {L e.) None of the above. a a. Roughly speaking, the Central Limit Theorem is crucial to hypothesis testing because: a. It ensures that X is (approximately) normally distributed. Q It ensures that X is (approximately) normally distributed. c. It enables us to form the null and alternative hypotheses. d. It is what makes the (statistical) power relevant to the hypothesis test. 6. It is what makes 0t relevant to the hypothesis test. ...
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Econ 15B S07 FINAL - PROBABILITY & STATISTICS (test form A)

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