1 A 95% con dence interval for the mean income of shop assistants in a certain city

is found to be ( 12,000, 15,000). Say in one sentence what this means. Would a

99% con dence interval be better than a 95% one? (Say why/why not.)

2. A charity believes that when it puts out an appeal for charitable donations the

donations it receives will be normally distributed with a mean of 50 and a standard

deviation of 6.

a) Find the probability that the rst donation it receives will be less than 40.

b) Find the value x such that 5% of donations are more than x.

3. A consultant for Dell was investigating computer usage among students at a

particular university. 200 undergraduates and 100 postgraduates were chosen at

random and asked if they owned a laptop. It was found that 81 of the

undergraduates and 63 of the postgraduates owned a laptop. The consultant

calculated that 48% (144 out of 300) of the students interviewed owned a laptop.

Explain, with reasons, whether the gure of 48% will be a good estimate of the

proportion of all students who own a laptop.

4. For a certain variable, the standard deviation in a large population is equal to 12.5.

How big a sample is needed to be 95% sure that the sample mean is within 1.5 units

of the population mean?

(5 marks)

5. a) What conclusions would you draw from a test which is signi cant at the 1%

level?

b) What conclusions would you draw from a test which is signi cant at the 10%

level, but not the 5% level?

c) An accounting rm wishes to test the claim that no more than 5% of a large

number of transactions contains errors. In order to test this claim, they examine

a random sample of 225 transactions and nd that exactly 20 of these are in

error. What conclusion should the rm draw? Use a 5% signi cance level.

6. A pro t-maximising retailer can obtain cameras from the manufacturer at a cost of

50 per camera. The retailer has been selling the cameras at a price of 80, and

at this price consumers have been buying 40 cameras per month. The retailer is

planning to lower the price to stimulate sales and knows that for each 5 reduction

in the price, 10 more cameras will be sold each month. Assuming price is a multiple

of 5, what price should the retailer charge and what will the monthly pro ts be?

7. Explain brie y the purpose of

a) sampling

b) model building.

Discuss any advantages and limitations.

8. The prospective operator of a shoe store has the opportunity to locate in an

established and successful shopping centre. Alternatively, at lower cost, he can

locate in a new centre, whose development has recently been completed. If the new

centre turns out to be very successful, it is expected that annual store pro ts from

location in it would be 130,000. If the centre is only moderately successful, annual

pro ts would be 60,000. If the new centre is unsuccessful, an annual loss of 10,000

would be expected. The pro ts to be expected from location in the established

centre will also depend to some extent on the degree of success of the new centre,

as potential customers may be drawn to it. If the new centre was unsuccessful,

annual pro ts for the shoe store located in the established centre would be expected

to be 90,000. However, if the new centre was moderately successful, the expected

pro ts would be 70,000, while they would be only 30,000 if the new centre turned

out to be very successful. All pro ts are inclusive of location cost. The probability

that the new shopping centre will be very successful is 0.4 and the probability it will

be moderately successful is also 0.4.

a) Draw the decision tree for this problem.

b) According to the expected monetary value criterion, where should the shoe

store be located? Assume a risk-neutral decision-maker.

c) Without calculating or drawing anything, explain brie y how a perfect forecast

of shopping centre success changes the decision tree in a) .