1 A 95% con?¦dence interval for the mean income of shop assistants in a certain cityis found to be (?12,000, ?15,000). Say in one sentence what this means. Would a99% 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 thedonations it receives will be normally distributed with a mean of ?50 and a standarddeviation 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 aparticular university. 200 undergraduates and 100 postgraduates were chosen atrandom and asked if they owned a laptop. It was found that 81 of theundergraduates and 63 of the postgraduates owned a laptop. The consultantcalculated 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 theproportion 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 unitsof 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 largenumber of transactions contains errors. In order to test this claim, they examinea random sample of 225 transactions and ?¦nd that exactly 20 of these are inerror. What conclusion should the ?¦rm draw? Use a 5% signi?¦cance level.5. 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, andat this price consumers have been buying 40 cameras per month. The retailer isplanning to lower the price to stimulate sales and knows that for each ?5 reductionin the price, 10 more cameras will be sold each month. Assuming price is a multipleof ?5, what price should the retailer charge and what will the monthly pro?¦ts be?6. Explain brie?¦?y the purpose ofa) samplingb) model building.Discuss any advantages and limitations.7. The prospective operator of a shoe store has the opportunity to locate in anestablished and successful shopping centre. Alternatively, at lower cost, he canlocate in a new centre, whose development has recently been completed. If the newcentre turns out to be very successful, it is expected that annual store pro?¦ts fromlocation in it would be ?130,000. If the centre is only moderately successful, annualpro?¦ts would be ?60,000. If the new centre is unsuccessful, an annual loss of ?10,000would be expected. The pro?¦ts to be expected from location in the establishedcentre 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 expectedto be ?90,000. However, if the new centre was moderately successful, the expectedpro?¦ts would be ?70,000, while they would be only ?30,000 if the new centre turnedout to be very successful. All pro?¦ts are inclusive of location cost. The probabilitythat the new shopping centre will be very successful is 0.4 and the probability it willbe 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 shoestore be located? Assume a risk-neutral decision-maker.c) Without calculating or drawing anything, explain brie?¦?y how a perfect forecastof shopping centre success changes the decision tree in a).