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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 selfcontained, silent,
batteryoperated and pocketsized 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 minibuses, 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 minibus 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 (afterlessbefore) 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.
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