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1301_0405sem1

# 1301_0405sem1 - 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 STAT1301 PROBABILITY AND STATISTICS I December 28, 2004 Time: 6:30 p.m. — 8:30 p.m. Candidates taking examinations that permit the use of calculators may use any calculator 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 satisfacto- rily 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 candi- date to ensure that the calculator used will not be in violation of the criteria listed above. Answer ALL SEVEN questions. Marks are shown in square brackets. 1. Data set 1, \$1,rg,\$3, has 331 + 1132 + x3 = 33 and x? + x3 + mg 2 390. Data set 2, r4,\$5,m6,x7, has mean = [1,2 = 12 and standard deviation = 02 = 2.5. Data set 3, x8,x9,x10,x11,x12, has mean 2 p3 = 15 and variance 2 = 03 = 15. (a) The three sets of data are mixed together. Find the mean and standard deviation of \$1,212, . .. ,rlz. (b) Let y, = 5 -— 2:13,, i = 1,2,... ,12. Find the mean and standard deviation Of 91,112, ' ' - iii/12' (e) Let 'LUz'j = \$.- + yj, Where i = 1,2,3, and j = 4,5,6,7. Find the mean and standard deviation 0f 11114, W15, 71116, 11117, U124, 1025, W26, W27, 11134, was, “’36: 1037. [Total: 15 marks] 2. Let X denote the delay, in hours, of a ﬂight from Airport H. The probability density function of X is k3—x, O_:z:__3, f(w)={( ) << 0, otherwise. S&AS: STAT1301 Probability and Statistics I Find (a) k, (b) P(1 s X s 2), and (c) the mean and variance of X. [Total: 15 marks] 3. Every morning, Miss Fung leaves home at 7:50 sharp. She goes to the bus station to catch a bus. Having taken the ride and left the bus, she further walks to her ofﬁce. Thus, her trip consists of three independent time intervals (in minutes): X = From home to bus : mean = 15, standard deviation = 2.5. (including waiting time) Y On the bus : mean = 30, standard deviation 2 6.0. T From bus stop to ofﬁce : mean = 12, standard deviation 2 2.0. (a) Find the mean and standard deviation of her Whole travelling time. (b) Suppose each of the three time intervals is distributed normally. Find the probability that she is late for work, if her ofﬁce starts business at 9:00 am. [Total: 10 marks] 4. (a) A station has two mini-buses, A and B, awaiting, each already having 13 seats occupied. Passengers come singly and independently, selecting A with probability 0.35, and selecting B with probability 0.65. Find the probability that A is ﬁlled up before B and moves out ﬁrst. (A mini-bus has 16 seats.) (b) If 60% of all persons ﬂying across the Atlantic Ocean experience the symp— toms of jet lag for at least 24 hours, What is the probability that among 95 persons ﬂying across the Atlantic Ocean, at most 50 will experience the symptoms of jet lag for at least 24 hours? (c) Events A and B are such that P(AﬂB’) = %, P(A’|B) = g, and P(A’lB’) = g. Find P(BIA) and P(A u B’). (d) An urn contains 5 red and 8 White balls. Four balls are randomly drawn, one by one, and without replacement. What is the probability that the ﬁrst and the last balls drawn are of different colours? [Total: 15 marks] S&AS: STAT1301 Probability and Statistics I 5. The following table gives information on the average saturated fat (:3, in grams) consumed per day and the cholesterol level (3/, in milligrams per hundred millilitres) for eight males: Fat consumption, 1:,- 55 65 50 34 43 58 69 36 Cholesterol level, y,- 180 210 195 165 170 205 235 150 (a) Fit a regression line, 37 = a + bx, to the data. (b) Calculate the correlation coefﬁcient between :3 and y. (c) A ninth man consumed 52 grams of fat per day. Predict his cholesterol level. (d) A survey on four other men yielded the following summary statistics: 21;,- = 200, 2x? = 11100, 2y,- = 750, Eyf = 145000, Em,- = 39500. Incorporating this additional information into the computation, update the regression line and the correlation coefﬁcient. [Total: 15 marks] 6. (a) In a cafeteria, baked beans are served either in ordinary portions or in children’s portions. The quantity given for an ordinary portion is a normal variable with mean 120 g and standard deviation 6 g and the quantity given for a children’s portion is a normal variable with mean 55 g and standard deviation 4 g. What is the probability that John, who has two children’s portions, is given more food than his father, who has an ordinary portion? (b) The records of six independent determinations of the boiling point of a par- ticular liquid are (in °C) 140.6, 140.8, 139.7, 140.7, 139.8, 141.1. Construct a 95% conﬁdence interval for the ‘true boiling point’, assuming that the experimental boiling point can be modelled by a normal distribution. (0) Twelve standard lengths of wool were extended over a given load before and after washing. The (after-less-before) differences (cm) in the extensions were 1.7, —0.1, 0.3, 0.9, —0.4, 2.1, 1.1, 0.5, 0.9, 0.7, —0.6, 1.3. Test, at the 0.01 signiﬁcance level, whether or not washing increases exten- sibility in general. Write down the null and alternative hypotheses ﬁrst. (d) The following are the numbers of sales which a random sample of nine salesmen of industrial chemicals in Guangzhou and a random sample of 3 S&AS: STAT1301 Probability and Statistics I . seven salesmen of industrial chemicals in Shanghai made over a ﬁxed period of time: Guangzhou, 23,-: 41, 47, 62, 39, 56, 64, 37, 61, 52 Shanghai, 31,-: 34, 63, 45, 44, 55, 24, 43 Assuming normal populations with equal variance, test whether or not Guangzhou salesmen are more efﬁcient in general. Write down the null and alternative hypotheses ﬁrst. [Total: 15 marks] In order to assess the probability, p, of a successful outcome, an experiment is independently performed 200 times, of which 72 are successful. Construct a 95% conﬁdence interval for p. According to a genetic theory, the blood groups of a certain tribe of inhab- itants should have the following percentages: O = 42%, B = 8%, A2 = 9%, A1 = 34%, A18 2 3%, and .423 = 4%. A blood test experiment on 500 inhabitants yielded the frequencies: 0 = 200, B = 55, A2 = 40, A1 = 190, A1B = 6, and A2B = 9. Test at the 1% signiﬁcance level Whether the theoretical model ﬁts the data. Explain your result. Analysis of the rate of turnover of employees by a personnel manager pro— duced the following table showing the length of stay of 200 people who left the company for other employment. Length of employment (years) Managerial 4 1 1 6 32 28 25 23 Skilled Unskilled Use a 0.01 level of signiﬁcance to test Whether length of employment is independent of grade. [Total: 15 marks] ************ END OF PAPER ************ STAT1301 Probability and Statistics I S&AS .rgIsafaIga IraIsEEEIia Iaﬁgﬂ I. E: I 5.5 I a I 5:“ I 5w Em I E. I gm 3 I Em 2m . l a H . a a H N H H .. H n E I a mmzls§w SISXMIEN im. a 3+: . , . . . E5 .8 x .55 39. Du :IuEIbu Z. TIIIIII? 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