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STAT1306

# STAT1306 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1306 Introductory Statistics December 29, 2005 Time: 2:30 p.m. - 4:30 p.m. Candidates taking examinations that permit the use of calculators may use any cal- culator which fulﬁls the following criteria: (a) it should be self-contained, silent, battery-operated and pocket-sized and (b) it should have numeral-display facilities only and should be used only for the purposes of calculation. It is the candidate’s responsibility to ensure that the calculator operates satisfactorily and the candidate must record the name and type of the calculator on the front page of the examination scripts. Lists of permitted/prohibited calculators will not be made available to candidates for reference, and the onus will be on the candidate to ensure that the calculator used will not be in violation of the criteria listed above. Answer ANY FOUR questions. The questions are of equal value. 1. The following data show the number of hours (X) of 10 randomly selected students who studied for a statistics test and their scores (Y) on the test: In IE! (a) Fit a simple linear regression model Y = ,60+;81\$+€ to the data. (b) Given that SSE -_- zyzlm — 2-)? = 234.3128 and 3:10;. — Xi)? = 376.0. 1. Estimate with 0.95 conﬁdence the score that an individual student will obtain in the test if he studies 28 hours. Is this estimate reliable? Why or why not? ii. Interpret 30. Do you think it is reasonable to test the hypotheses that Ho : [30 = 0.0 against H1 : ﬁg > 0.0? Do not attempt to carry out the test, but just explain brieﬂy. iii. Test H0 : 51 = 3.0 against H1 : [31 > 3.0 at a = 0.05. State the name of the test, the test statistic, the rejection region and your conclusion clearly. S&AS: STAT1306 Introductory Statistics 2 2. A Statistics course has 2 subclasses, namely A and B, that use identical hand- outs, assignments, class tests and examinations, but are taught by professors Audrey and Betty, respectively. Audrey has been the teacher of this course for years, and Betty is a new teacher. The Head of Department and the Chair Professor suspect that teacher Betty may not be a good teacher and hence her students may suffer and their ﬁnal results may not be as good as the students in class A. (a) To justify his suspicion, the Head of Department draws a random sample 10 students from each of the two classes with their ﬁnal marks recorded below. Carry out an appropriate test for the Department Head’s claim at the 0.05 level of signiﬁcance. State the appropriate null and alternative hy- potheses, the name of the test, the test statistic, the rejection region and your conclusion clearly. (b) The Chair Professor studies this problem from another perspective by comparing the failure rates of the two subclasses. He randomly selects 200 students from the 2 classes and observes 12 failures among the 120 students from class A and 10 failures among the 80 students from class B. 1. Construct a 90% conﬁdence interval for the overall failure rate of the course by combining the two subclasses. ii. Test the hypothesis that the failure rate of class B is higher than that of class A at the 0.05 level of signiﬁcance. State the appropriate null and alternative hypotheses, the name of the test, the test statistic, the rejection region and your conclusion clearly. iii. Are your conclusions in (a) and (b)(ii) consistent? Which test do you think is more appropriate to detect a difference in the teaching skills between a new teacher and an experienced teacher? Brieﬂy describe in not more than 100 words. S&AS: STAT1306 Introductory Statistics 3 3. From long experience with a process of manufacturing gunpowder, it is known that the resulting muzzle velocity, Y, is normally distributed with a mean 1000 m/ sec and a standard deviation of 100 In / sec. A proposal for modiﬁcation is received for which it is claimed that the new gunpowder will result in a faster muzzle velocity, but leaving the standard deviation unchanged. (a) Formulate the null and alternative hypotheses for the above claim. (b) The new gunpowder is to be tested in n = 15 shells with mean muzzle velocity 17. Let a = P(Committing a type I error) which is now ﬁxed at the 0.025 level. Construct a rejection region so that the null hypothesis is rejected if 17 falls in this region. (c) If the true average muzzle velocity for the new gunpowder is ,u = 1080 m/sec, with the conditions in (b) remained unchanged, ﬁnd [3 where )6 = P(Committing a type II error). (d) What is the smallest value of n required in order to make 0: = 0.025 and ﬂ S 0.025 for the above hypothesis testing procedure? 4. The probability distribution of X, the daily number of passengers on a heli~ copter shuttle run from Hong Kong to Macau, is given as follows: I“ I Assume that X is independent from day to day. (a) What is the probability that the ﬁrst day having 3 or more passengers will occur on the fourth day starting from today? (b) What is the probability that in a particular week (7 days), the helicopter shuttle will have three or more passengers in fewer than two days? (c) What is the approximate probability that in a particular year (365 days), the helicopter shuttle will have at most one passenger in fewer than 150 days? ((1) What is the probability that the total number of passengers in two con- secutive days is equal to 3? (e) It is known that the total number of passengers in two consecutive days equals 3. What is the probability that there is exactly one of these days with no passenger? (f) Find the mean and the variance of the daily number of passengers on the helicopter shuttle. (g) In a random sample of n = 100 days, the daily number of passengers is recorded. What is the approximate probability that the sample mean exceeds 2.0? S&AS: STAT1306 Introductory Statistics 4 5. (a) Two scales, A and B, are used in a laboratory to weigh the rock specimens (in grams). A random sample of 10 rock specimens was selected and the weight of each rock specimen was obtained from each of the two scales. The following data were obtained in an experiment to check whether there is a systematic difference in the weights (in grams) obtained from the two scales A and B: Rock Specimen Scale A (X) VOOONCDCﬂibOOMl—‘I 10 Deﬁne Di = X,- — Yi, the difference of the weights obtained from scales A and B. i. Test HOZUD=0.25 VS H110D<0.25 at the 0.05 level of signiﬁcance where an is the population standard ‘ deviation of the difference Di. Write down clearly the test statistic, the rejection rule and your conclusion. Also state the assumption(s) necessary for the test to be valid. ii. With the conclusion and assumption(s) in (i), construct a 95% con- ﬁdence interval for up, the mean of the differences of the weights obtained from the two scales. Is there evidence to indicate that the weights obtained with the two scales are signiﬁcantly different? (b) Let X be a continuous random variable with probability density function 0 (1+¢)2, where C is an appropriate constant. i. Find C. ii. Deﬁne a new variable Y = X / (1 + X Show that the distribution function of Y is F(y) = y for 0 < y < l and hence ﬁnd P(Y < 2/5). 0<x<oo; f(\$) = elsewhere. S&AS: STAT1306 Introductory Statistics 5 A LIST OF STATISTICAL FORMULAE 2. Chebyshev’s Theorem: P( |X — u| < k0) _>_ 1 — £2— for k > 1 3. P(A u B) = P(A) + P(B) - P(A n B) 4. P(A n B) = P(B) - P(AIB) = P(A) . P(B|A) 5. Bayes Theorem I P(AJ'IB) = i=1 2 . i 7- Va?"(Y) = E[(Y - W] = 2(31— u)2p(y) = E(Y2) - #2 = 02 9' X N :10 = _p)n—ZI forx=0)1)"'7n; = np,Var(X) = np(1 .._p); MX(t) 2 (p ct +q)n 10. X N Geometric(p), P(X = x) = p (1 _ p)z—1’ for x = 1, 2, _ _ .; E(X) = 1/p,var<x> = (1 -P)/P2;Mx(t) = 13:6) 11- XNNeQBin(T,P), P(X=95) =( )Pr (1—p)\$_', forx=r,r+1,r+2,-~- r—l M) = r/p, Vamx) = «1 —p)/p2;MX(t) = [ p at 12. X N P0isson(A), P(X = x) = %e"‘ for x = 0,1,- -- ; E(X) = Var(X) = A Y— f — 13. Y ~ N(u,a2), z = w = Y " ~ N(0,1) Var(Y) 0 14. E(X) = p, Var(X) =0; X—u 0/\/7-1 15. Central Limit Theorem : For n being large, Z = xv mo, 1) _X-M 16.Z a/ﬁ ~ N(0,1); Xiza/za/Jﬁ S&AS: STAT1306 Introductory Statistics 6 X—u _ - = N n—; Xitn— a 17 T S/ﬂ t 1 1,/2S/\/ﬁ 15-1) . A 15(1-25) 18. Z=——-—————~N(0,1); pizaz \F—pu—pvn / n X_' _ _ 19_ Z=L__}_/)_2_ﬂ£2___mil~]v(0,1) 3+1)”. ’nx ny (nx — 1) + (ﬂy - 1) (nx + ny -— 2) X—Y’ —— — — — 1 1 21- T = (——S—)-—M ~ tnx+ny—2; (X “ Y) itnx+ny—2,a/2 SP —— + —— P L L nx 7;,» "X "Y 20. Pooled Variance: \$12, = + A 22_ Z = W ,;, N(0,1); 151(1 ' 151) + 152(1 -132) A _ A i z _ [111(1—1712 + 7120—112) (p1 p2) a/z n1 n2 n1 n2 2 __ (n -— 1)S2 2 . (n —- 1)S'2 (n —1)S2 X — 02 X(df=n—1)7 ﬂ ' 24. = n nz2i=1— (21:1 12:1 n V n 21:1 Xi — ( i=1 Xi)2V ” 22:1 Y? ‘ (21:1 YDZ [Bx/n — 2 25. T = ——;- ~ td.f..—_n—2 \/1 - p2 A 2711(Xi—XXYE—Y) A — A - 26. = L———-—_-——' = —— ‘31 E?=1(Xi- J02 ’ ° Y 51" m a SSE 27. SSE: Yi—Yi2; 2: . 1 X2 A 2 28.ﬁ~N(ﬂ,a§(—+—n-———)); ~N( __"e__.._. 0 0 n 21:1(371 "’ X)2 ’81 ’81, 22:1(372' —‘ XV 3:2) I}: tn_2,a/2 X = [130); ?o It tn_2,a/2 X 3.6.(Y2'0lX = \$0). ...
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STAT1306 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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