math1155c

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BSc (Hons) Actuarial Science

Cohort: BAS/11/FT

Examinations for 2011 - 2012 / Semester 2

MODULE: PROBABILITY AND STATISTICS 2

MODULE CODE: MATH1155C

Duration: 2 Hours

Instructions to Candidates: 1. Answer all FOUR (4) Questions.

2. Questions may be answered in any order but your answers must show

the question number clearly.

3. Give non-exact numerical answers correct to 3 significant figures.

4. Always start a new question on a fresh page.

5. All questions carry equal marks.

6. Statistical tables are appended at the end of the paper.

7. Total marks: 100.

This question paper contains 4 questions and 9 pages.

ANSWER ALL FOUR (4) QUESTIONS

Question 1: (25 Marks)

(a) The random variable X follows the binomial distribution B(10, 0.15).

Find the probability that the mean of a random sample of 50 observa-

tions of X is greater than 1.4.

[5 marks]

(b) The random variable X has density function

f(x) =

(k + 1)xk; 0 x 1,0; otherwise.

Given n independent observations x1; x2; :::; xn of X, nd the maximum

likelihood estimator of k.

[7 marks]

(c) The weights, x grams, of 30 randomly chosen rabbits are summarised

as: Xx = 31 500;

Xx2 = 33 141 816:

(i) Prove that the sample mean is an unbiased estimate of the popu-

lation mean.

(ii) Calculate unbiased estimates of the population mean and variance

of the weights.

A random sample of n rabbits is taken.

(iii) Estimate the value of n for which it is approximately 95% certain

that the sample mean weight does not dier from the population

mean weight by more than 6 grams.

(iv) Estimate the value of n that would be required in order for a 98%

condence interval for the mean weight to have width 20.

[4+3+3+3=13 marks]

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(a) An insurance company has a certain type of policy which can be claimed

only 3 or more years after the policy is bought. The policy expires once

the claim is made. It is proposed to model the duration, X years, of

such a policy by the distribution with density function

f(x) = e(3x); 3 x 0:

The durations of n such policies which have expired and which may

be assumed to be independent are denoted by x1; x2; :::; xn. Using the

method of moments, nd an estimate for .

[6 marks]

(b) A survey was carried out among a random sample of men and women to

know whether they wear a watch on the left wrist, on the right wrist or

do not wear a watch. The table shows the number of men and women

in each category.

left wrist right wrist do not wearmen 85 78 37women 118 61 25

Perform a 2 test, at 5% signicance level, to determine whether there

is association between men and women regarding the wearing of watch.

[9 marks]

(c) In the past, the number of daily sales of a store has been modelled

by a random variable with distribution Po(0.8). Following a publicity

campaign, the manager hopes that the mean number of daily sales will

increase. To test, at 5% signicance level, whether this is the case,

the total number of sales during the rst 3 days after the campaign is

noted.

(i) Given that the total number of sales during the rst 3 days after

the campaign is 5, carry out the test.

(ii) Explain what is meant by a Type I error in this context and nd

the probability it occurs.

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(iii) State what further information is required in order to nd the

probability of a Type II error.

[5+4+1=10 marks]


(a) The weights of 12 mothers, x kg, and of their eldest daughters, y kg,

are as shown below.

x 65 63 67 64 68 62 70 66 68 67 69 71y 68 66 68 65 69 66 68 65 71 67 68 70

(i) Perform a linear regression analysis to obtain the equation of the

least-squares line in the form y = a+ bx:

(ii) Calculate the correlation coecient and the standard error esti-

mate.

(iii) Test, at 5% signicance level, the null hypothesis that the regres-

sion coecient, b, is as low as 0.18.

(iv) Calculate a 95% condence interval for b.

[6+4+4+3=17 marks]

(b) The thickness of washers produced by a machine since it started oper-

ating is known to follow a normal distribution with mean 2 mm and

standard deviation 0.3 mm. Due to a recent mechanical problem, an

engineer claims that the mean thickness has changed. A random sam-

ple of 100 washers was taken and the mean thickness was found to be

1.9475 mm.

(i) Test, at 10% signicance level, the engineer's claim.

(ii) This result is signicant at % level. Find the smallest possible

integer value of .

[5+3=8 marks]

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(a) Some students were divided into four groups, and each group was sub-

jected to one of four teaching techniques A, B, C and D. Due to

dropouts, the number of students varied at the end of the teaching

period when an assessment was carried out. The marks of the students

in each group are as shown below.

A 65 87 73 79 81 69B 75 69 83 81 72 79 90C 59 78 67 62 83 76D 94 89 80 88

(i) Conduct an analysis of variance test, at 5% signicance level, to

determine whether the data indicate a signicant dierence in the

mean achievements of the four groups.

(ii) Calculate a 95% condence interval for the mean mark of group

A.

(iii) Calculate a 95% condence interval for the dierence between the

mean marks of groups A and D. In light of this interval containing

zero or not, comment on the mean marks of groups A and D.

[9+3+5=17 marks]

(b) 600 people having an infection are divided equally into 2 groups A and

B. A pill is given to group A, but not to group B; otherwise, the two

groups are treated identically. One week later, it is found that 225

people in group A and 195 people in group B have recovered from the

infection.

(i) Test, at 5% signicance level, the hypothesis that the pill helps to

cure the infection.

(ii) Find the P-value of the test.

[6+2=8 marks]

***END OF QUESTION PAPER***

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