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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1306 INTRODUCTORY STATISTICS December 18, 2006 Time: 2:30 p.m.  4:30 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 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 questions. Marks are shown in square brackets. 1. (a) A uniform sixsided die is thrown six times. What is the probability of
obtaining at least one six? (b) How many times must a fair coin be tossed in order that the probability of observing at least one head is greater than 0.99?
[Total: 10 marks] 2. Consider a matching problem with the case of'N = 5; and let M5 equal the
number of correct matches out of 5 objects. (a) What is the probability distribution of M5 if the matching is made ran
domly?. . (b) What is the expected value and the variance of M5? (c) How many correct matches give signiﬁcant evidence that they are not made randomly? Justify your choice.
[Totak 10 marks] S&AS: STAT1306 Introductory Statistics 3. Assume that injurycausing accidents occurring in a large factory follow a Pois
son process with a rate of A = 0.3 per week and that the factory operates six
days per week. The occurrences of these accidents are observed from the start
of work on Monday of a certain week. (a) Compute the probability that the ﬁrst week is free of such accidents. (b) What is the probability that the ﬁrst accident occurs on Friday of the ﬁrst
week? (c) What is the probability that the ﬁrst accident occurs on Tuesday of the
second week? ((1) What is the expected number of working days lapsed before the ﬁrst acci dent occurs?
[Total: 10 marks] 4. The distribution of the demand (in number of units per unit time) for a product
can often be approximated by a normal probability distribution. For example,
a bakery has determined that the daily demand of the number of loaves of its
White bread has a normal distribution with mean 7,200 loaves and standard
deviation 300 loaves. Based on cost considerations the company has decided
that its best strategy is to produce a sufﬁcient number of loaves so that it will fully supply demand on 95% of all days.
(a) How many loaves of bread should the company produce? (b) Based on the production in part (a), on what percentage of days will the
company be left with more than 500 loaves of unsold bread? [Total: 15 marks] S&AS: STAT1306 Introductory Statistics 5. Telemarketers obtain names and telephone numbers from several sources. To
determine whether one particular source is better than a second, random sam
ples of 400 and 420 names and numbers were obtained from the two sources A
and B, respectively of which 52 and 46 from A and B made a purchase. (a) Construct a 99% conﬁdence interval for pA — p3, the difference in the pro—
portions of customers making a purchase from the two sources, state your
assumptions clearly. (b) Test the null hypothesis, H0 : pA 2 p3 against the alternative H1 : pA >
p 3. Give a value of the observed level of signiﬁcance. (c) Comment the results in (a) and [Total: 15 marks] 6. The Enviroumental Protection Agency sets a limit of 5 parts per million on
PCB (a dangerous substance) in water. A major manufacturing ﬁrm producing
PCB for electrical insulation discharges small amounts from the plant. The
company management, attempting to control the PCB in its discharge, has
given instructions to halt production if the mean amount of PCB in the efﬂuent
exceeds 3 parts per million. A random sample of 30 water specimens produced
the following statistics: i = 3.2 parts per million and s = 0.5 part per million. (a) Do these statistics provide suﬁicient evidence to halt the production pro
cess? Conduct a hypothesis test on the mean amount of PCB in the efﬂuent
with the level of signiﬁcance of 0.01. State clearly the null and alternative
hypotheses and your assumptions. (b) If you were the plant manager, would you want to use a large or a small
value for the level of signiﬁcance, at for the test in part (a)? Justify your
choice. (0) Calculate ﬂ, the type II error for the test described in part (a) assuming
that the true mean is p = 3.3 parts per million. ((1) What is the power of the test to detect the efﬂuent’s departure from the
standard of 3.0 parts per million when the mean is 3.3 parts per million? (e) Repeat parts (c) and ((1) assuming that the true mean is 3.4 parts per
million. What happens to the power of the test as the mean PCB of the
manufacturing ﬁrm departs further from the standard? [Total: 20 marks] S&AS: STAT1306 Introductory Statistics 7. A large supermarket chain has its own store brand for many grocery items.
These tend to be priced lower than other brands. For a particular item, the
chain wants to study the effect of varying the price for the major competing
brand on the sales of the store brand item, while the prices for the store brand
and all other brands are held ﬁxed. The experiment is conducted at one of the
chain’s stores over a sevenweek period, and the results are shown in the table. Week Competitor’s price Store brand sales
:1: (cents) y (a) Find the least squares line relating store brand sales y to major competitor’s
price :5. Interpret the results. (b) Plot the data and graph the line as a check on your calculations. (c) Using an appropriate test for testing the signiﬁcance of the slope parameter
of the least square line of the result in (a)? (d) Find a 90% conﬁdence interval for mean store brand sales when the
competitor’s price is 33 cents. (e) Suppose you were to set the competitor’s price at 33 cents. Find a 90%
prediction interval for the expected sales.
[Total: 20 marks] ************ END OF PAPER ***********V* lCS STAT1306 Introductory Statist' S&AS . ml: I
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