©The Open University of Hong Kong

STAT S151F


Examination Specimen


PROBABILITY & DISTRIBUTIONS

Time Allowed: 2 hours

Examination Number


Student Number



THIS PAPER MUST BE RETURNED .

Admissible/Inadmissible materials in this examination:
Material/Stationery
provided to candidates
(standard items)
Material to be returned
to the invigilator at the
end of the examination
1. Calculators are allowed. (Please refer to the OUHK
Approved List.)
2. Dictionaries are NOT allowed.
Violation of the above may lead to disqualification from
the examination.
1. 1 Examination Paper
2. 1 Answer Book(s)

1. Examination Paper
2. Answer Book(s)


Instructions:

1. Answer this examination paper in English.
2. Read the rubric(s) in the examination paper carefully and write your answers in the answer book as
specified. Answers recorded elsewhere will not be marked. Begin each question on a new page and
write the question number at the top of each page you have worked on.
3. Write any rough work in the answer book(s) or this examination paper and cross it through
afterwards. Rough work will not be marked. No part of the answer book should be torn off.
4. Write clearly. It may not be possible to award marks where the writing is very difficult to read.
5. After the invigilator has announced that the examination has started, write your examination
number, student number and course code on the cover of the examination paper and answer book as
distributed by the invigilator. Failure to do so will mean that your work cannot be identified.
6. At the end of the examination, hand over the examination paper and answer book(s) to the
invigilator.
7. Do NOT open this examination paper until you are told to do so, otherwise you may be
disqualified.

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This examination consists of TWO parts. You should attempt ALL
questions in Part I and TWO questions in Part II.

Part I
Attempt all six questions in this part. This part carries 60% of the total
marks of this examination
Question 1 [10 marks]
(a) Explain the difference between a census and a sampling and describe the
advantages and disadvantages of each. [4]

(b) Determine whether the “the number of seats in a movie theater” is a qualitative
or quantitative data. Briefly explain. [2]

(c) A market researcher randomly selects 200 drivers under 55 years of age and
200 drivers over 55 years of age. State the sampling technique which is used in
this case. [1]

(d) Identify whether the statement describes inferential statistics or descriptive
statistics: “The average age of the students in a statistics class is 19 years.” [1]

(e) State the level of measurement for the following situations respectively:
 The nationalities listed in a recent survey (for example, Asian, European,
American etc).
 The ratings of a movie ranging from "poor" to "good" to "excellent". [2]

Question 2 [10 marks]
(a) The cholesterol levels (in milligrams per deciliter) of 30 adults are listed below. Draw
a box-and-whisker plot that represents the data.

154 156 165 165 170 171 172 180 184 185
189 189 190 192 195 198 198 200 200 200
205 205 211 215 220 220 225 238 255 265
[4]

(b) The numbers of home runs that Barry Bonds hit in the first 18 years of his major
league baseball career are listed below. Make a stem-and-leaf plot for this data. What
can you conclude about the data?

16 25 24 19 33 25 34 46 37
33 42 40 37 34 49 73 46 45
[4]
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(c) Heights of adult women have a mean of 63.6 in. and a standard deviation of 2.5 in.
Apply Chebyshev's Theorem to the data using k = 3. Interpret the results. [2]


Question 3 [10 marks]
(a) Use a standard normal table to find the z-score that corresponds to the cumulative area
of 0.01. [3]

(b) For the standard normal curve, find the z-score that corresponds to the 30th percentile. [3]

(c) Find the z-scores for which 98% of the distribution's area lies between -z and z. [4]


Question 4 [10 marks]
(a) A mathematics professor gives two different tests to two sections of his college
algebra courses. The first class has a mean of 56 with a standard deviation of 9 while
the second class has a mean of 75 with a standard deviation of 15. A student from the
first class scores a 62 on the test while a student from the second class scores an 83 on
the test. Which student perform better ? Explain. [6]

(b) Assume that the heights of men are normally distributed with a mean of 70.6 inches
and a standard deviation of 2.2 inches. If the top 5 percent and bottom 5 percent are
excluded for an experiment, what are the cutoff heights to be eligible for this
experiment? [4]


Question 5 [10 marks]
A sample of 500 households was selected in a large metropolitan area to determine
various information concerning consumer behaviours. Among the questions asked was
“Do you enjoy shopping for clothing?” Out of 240 males, 136 answered Yes. Out of 260
females, 224 answered Yes.

(a) State the complement of “enjoy shopping for clothing”. [2]

(b) Find the probability that a respondent enjoys shopping for clothing. [2]

(c) Find the probability that a respondent is a female and enjoys shopping for clothing. [2]

(d) Find the probability that a respondent is a male or does not enjoy shopping for
clothing. [2]

(e) Find the probability that a respondent is a male given the respondent does not enjoy
shopping for clothing. [2]

STAT S151F Page 4 of 13
Question 6 [10 marks]
A baseball player gets four hits and a sports announcer claims that getting four or
more hits are not ‘unusual’. The corresponding frequency distribution is listed as
below.

Hits 0 1 2 3 4 5 6 7
# of Players 7 9 7 4 1 1 2 1

(a) Define the term ‘unusual’ by making use of the statistical term ‘standard deviation’. [2]

(b) Estimate the probability of falling in the ‘unusual’ situation using Chebyshev’s
theorem. [3]

(c) Based on the results of “4 or more hits”, determine whether the sports announcer’s
claims follows the Chebyshev’s theorem. (i.e. Is 4 hits usual ?; Is 5 hits usual ?; etc)
Hints: Calculate the probabilities (empirical) of getting four or more hits. [5]




[End of Part I]
STAT S151F Page 5 of 13
Part II
Answer any TWO questions in this part of the examination. Each question
carries 20 marks. This part carries 40% of the total marks of the
examination.

Question 7 [20 marks]
You work at a bank and are asked to recommend the amount of cash to put in an ATM
each day. You do not want to put too much (security) or too little (customer irritation).
Here are the daily withdrawals (in 100s of dollars) for 30 days.
72 84 61 76 104 76 86 92 80 88
98 76 97 82 84 67 70 81 82 89
74 73 86 81 85 78 82 80 91 83

(a) Construct a relative frequency table for the data using 8 classes. [8]

(b) Construct a relative frequency histogram using the result in part (a). [4]

(c) If you put $9000 in the ATM each day, what percentage of the days in a month should
you expect to run out of cash? Explain. [3]

(d) If you are willing to run out of cash for 10% of the days, how much cash should you
put in the ATM each day? Explain. [3]

(e) Suggest a continuous probability distribution to model the amount of cash to out in an
ATM. Briefly explain your reasoning. [2]


Question 8 [20 marks]
You are dealt a hand of five cards from a standard deck of playing cards. Find the
probability of being dealt a hand consisting of:
(a) Four-of-a-kind. [5]

(b) A full house, which consists of three of one kind and two of another kind. [5]

(c) Three-of-a-kind (the other two cards are different from each other). [5]

(d) Two clubs and one of each of the other three suits. [5]
STAT S151F Page 6 of 13
Question 9 [20 marks]
A company assumes that 0.5% of the pay-checks for a year were calculated incorrectly.
The company has 500 employees and examines the payroll records from January to
March. Let x be the number of pay-check records examined by the company until finding
the first incorrect check.

(a) State the type of distribution used to model the random variable x. Explain. [2]

(b) Find the mean, variance, and standard deviation of x. [6]

(c) Estimate the number of pay-check records (on average) would you expect to examine
before finding one with an error. [2]

The following parts are not related to part (a) to (c).

It is estimated that sharks kill 5 people on average each year worldwide. Let y be the
random variable representing the number of people killed in a year.

(d) State the type of distribution used to model the random variable y. Explain. [2]

(e) Find the probability that at least 3 people are killed by sharks this year. [8]






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Formulae for Examination



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Statistical tables for Examination

Table - Standard normal distribution (1)


STAT S151F Page 13 of 13
Table - Standard normal distribution (2)



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