Questions :

1 A woodcutter has to cut 100 fence posts of a standard length and he has a metal bar of

the required length to act as the standard. The woodcutter decides to vary his

procedure from post to post he cuts the first post using the metal standard, then uses

this post as his standard for the cut of the next post. He continues in a similar manner,

each time using the most recently cut post as the standard for the next cut.

Each time the woodcutter cuts a post there is an error in the length cut relative to the

standard being employed for that cut you should assume that the errors are

independent observations of a random variable with mean 0 and standard deviation

3mm.

Calculate, approximately, the probability that the length of the final post differs from

the length of the original metal standard by more than 15mm.

[5]

2 A researcher wishes to investigate whether a coin is balanced or not, that is if

P heads ( ) 0.5 = . She throws the coin four times and decides to accept the hypothesis

0 H P heads : ( ) 0.5 = in a test against the alternative 1 H P heads : ( ) 0.5 , if the number

of times that the coin lands "heads" is 1, 2, or 3.

(i) Calculate the probability of the type I error of this test. [3]

(ii) Calculate the probability of the type II error of this test, if the true probability

that the coin lands "heads" is 0.7. [3]

[Total 6]

3 Pressure readings are taken regularly from a meter. It transpires that, in a random

sample of 100 such readings, 45 are less than 1, 35 are between 1 and 2, and 20 are

between 2 and 3.

Perform a 2

goodness of fit test of the model that states that the readings are

independent observations of a random variable that is uniformly distributed on (0, 3).

[5] CT3 A20085 PLEASE TURN OVER

4 In an investigation about the duration of insurance policies of a certain type, a sample

of n policies is studied. All n policies have been initiated at the same time, which is

also the time of the start of the investigation. For each policy, the time T (in months)

until the policy expires can be modelled as an exponential random variable with

parameter , independently of the times for all other policies.

(i) Suppose that the investigation is terminated as soon as k policies have expired,

where k is a known (predetermined) constant. The observed policy expiry

times are denoted by t1, t2, ..., tk with 0 < k n and t1< t2 <... < tk.

(a) Show that the probability that any randomly selected policy is still in

force at the time of the termination of the investigation is kt

e

.

(b) Show that the likelihood function of the parameter , using information

from all n policies, is given by

1 ( ) ( )

k

i

i k

t

k nk t L ee =

= .

Hence find the maximum likelihood estimate (MLE) of .

(c) Consider an investigation on 20 policies which is terminated when five

policies have expired, giving the following observed expiry times (in

months):

1.03 6.67 12.70 12.88 21.54

Calculate the MLE of based on this sample.

[9]

(ii) Suppose instead that the investigation is terminated after a fixed length of time

t0. The number of policies that have expired by time t0 is considered to be a

random variable, denoted by K.

(a) Explain clearly why the distribution of K is binomial and determine its

parameters.

(b) Hence find the MLE of in this case.

(c) Consider an investigation on 20 policies that is terminated after 24

months. By the time of termination five policies have expired.

Use this information to calculate the MLE of in this case.

[9]

[Total 18] CT3 A20086

5 The members of the computer games clubs of three neighbouring schools decide to

take part in a light-hearted competition. Each club selects five of its members at

random under a procedure agreed and supervised by the clubs. Each selected student

then plays a particular game at the end of which the score he/she has attained is

displayed and recorded - the standard set by the games designers is such that

reasonably competent players should score about 100.

The results are as follows:

School 1 105 134 96 147 116

School 2 103 81 91 100 110

School 3 137 115 105 123 149

(i) An analysis of variance is conducted on these results and gives the following

ANOVA table:

Source of variation d.f. SS MSS

Between schools 2 2,298 1,149

Residual 12 3,468 289

Total 14 5,766

(a) Test the hypothesis that there are no school effects against a general

alternative.

You should quote a narrow range of values within which the

probability-value of the data lies, and state your conclusion clearly.

(b) Calculate a 95% confidence interval for the underlying mean score for

club members in School 1, using the information available from all

three schools.