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University of the Free StateDepartment of Mathematical Statistics and Actuarial Science

STS618 / STS718

First Semester Test: 3 March 2014

Total marks: 50

Time: 90 min

1. Question 1 [10]

A study was carried out in July 2010 to investigate the profile of soccer enthusiasm

in Bloemfontein. A random sample of 1000 adults in Bloemfontein was drawn.

Among other questions, participants were asked whether or not they had tickets

for at least one of the world cup matches. Participants were cross-classified

according to gender, and whether or not they had at least one world cup ticket.

Of the 329 males in the sample, 159 had a word cup ticket. Of the females, 210had a world cup ticket.

Summarize the data in a 2 2 table and then answer the following questions

about soccer enthusiasm in Bloemfontein:

(a) Estimate the probability that a male has a world cup ticket. Is this a

conditional, marginal or joint probability? [2]

(b) Estimate the probability that somebody who has no ticket is a female. Is

this a conditional, marginal or joint probability? [2](c) Estimate the probability that an adult in Bloemfontein has a world cup

ticket. Is this a conditional, marginal or joint probability? [2]

(d) Estimate the probability of being female and having a world cup ticket. Is

this a conditional, marginal or joint probability? [2]

(e) Who was more likely to have a word cup ticket, males or females? Motivate

your answer. [2]

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2. Question 2 [7]

Consider the following Model 2 2 table.

Table 2.1: Model 2 2 Table

Characteristic B

Characteristic A Present Absent Total

Present n11 n12 n1+

Absent n21 n22 n2+

Total n+1 n+2 n

(a) Under cross-sectional sampling, specify the distribution of the cell frequen-

cies n11, n12, n21and n22. [1]

(b) Which total counts are considered fixed under cross-sectional sampling? [1]

(c) Under prospective sampling, specify the distribution of the cell frequency

n21 [1]

(d) Which total counts are considered fixed under prospective sampling? [1]

(e) Under retrospective sampling, specify the distribution of the cell frequency

n12 [1]

(f) Which total counts are considered fixed under retrospective sampling? [1]

(g) Specify the distribution of the cell frequency n21 when all marginal totals

are fixed. [1]

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3. Question 3 [4]

Before and during the 2010 World Cup the most convenient way to acquire tickets

for a World Cup match was via the internet, and credit card payment. There-

fore, a study was carried out on 27 June 2010 to investigate whether credit card

ownership was associated with having a ticket for Germanys famous World Cup

victory over England (4:1 !!) in Bloemfontein. That is, the research question

was the following: Is an owner of a credit card more likely to have a ticket and

attend the Germany-England match than somebody who did not own a credit

card. In the stadium during the match, a sample of 100 adult spectators were

asked whether or not they owned a credit card. At the some time, but outside

the stadium in the Waterfront shopping centre, a control sample of 100 adults

were asked the same question.

Was this a prospective, a retrospective or a cross-sectional study. Motivate your

answer. [4]

4. Question 4 [13]

Let be the probability that a student registered for STK114 attends class.

A random sample of size n = 120 of students registered for STK114 is taken,

and the random variable Xdenotes the number of students in the sample who

are actually found to be in class . We assume that X follows the binomial

distribution. The probability that exactly X= n1 members of the sample are

observed to be in class is denoted by Bin(n1,n ,), and is given by the probability

function

Prob(X= n1) = Bin(n1,n ,) =

n

n1

n1(1 )nn1

= n!

n1!(n n1)!n1(1 )nn1

Of the n= 120 students in the sample, n1= 85 were found to be in class.

(a) Derive the maximum likelihood estimate for . [5]

(b) We want to test the two-sided null-hypothesis H0 : = 0.8 against the

alternative HA: = 0.8. As test statistic we use the number n1of students

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in the sample who were in class. State the exact distribution ofn1 under

the null-hypothesis [2].

(c) In order to test the above null hypothesis, derive the exact P-value, that is,

the probability, given the null-hypothesis, of observing X = n1, or a more

extreme outcome than n1. (State the results in terms of the formula for the

probability function of distribution ofn1under the null-hypothesis; there is

no need to work out the actual P-value). [6]

5. Question 5 [16]

A cross-sectional study was carried out to investigate the attitude of students at

the UFS to compulsory class attendance. A random sample of students was drawn

from the total student population, and cross-classified according to whether or

not they were postgraduates, and whether or not they approved of compulsory

class attendance. The data for the sub-sample of students who study Actuarial

Science is as follows:

Table 5.1: Student seniority and approval of compulsory classes

Approve compulsory class attendance

Seniority Yes No Total

Postgraduate student 5 2 7

Undergraduate student 2 8 10

Total 7 10 17

We want to test the null hypothesis H0 No association between the variables

Seniorityand Approve compulsory class attendance. As test statistic wechoose the frequency n11.

(a) What is the exact distribution ofn11 under the null-hypothesis? [2]

(b) Why is it a good idea to carry out Fishers exact test (instead of a chi-square

test)? Motivate your answer. [3]

(c) Specify the 5 easy steps to test a null-hypothesis for Fishers exact test

for a 2 2 table. [5]

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(d) Carrying out Fishers exact test for Table 5.1 above, the table probabilities

are as follows:

Table 5.2: Table Probabilities

Table Cell Probability

n11 n12 n21 n22

7 0 0 10 0.0001

6 1 1 9 0.0036

5 2 2 8 0.0486

4 3 3 7 0.21603 4 4 6 0.3779

2 5 5 5 0.2721

1 6 6 4 0.0756

0 7 7 3 0.0062

Calculate the exact two-sided P-value and the exact one-sided P-value for

testing the null-hypothesis. [3]

(e) What can you conclude from the data in Table 5.1 and from the result ofthe hypothesis test? [3]

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