1. A survey of the amount spent on a package holiday by a random sample of 65 customers of a travel agency shows that the average expenditure is 423 with a standard deviation of 115. (a) Estimate the mean amount spent by all customers, with 99 percent confidence. (b) If you wanted to obtain an estimate with a half-width (half the margin of error) of no more than 25, how many more people would you need to question?

2. A credit card company wants to determine the mean income of its cardholders. A random sample of 225 cardholders were drawn and the sample average income of 16450 with a standard deviation of 3675. (a) Construct a 99 percent confidence interval for the true mean income. (b) Management decided that the confidence interval in (a) was too large to be useful. In particular, the management wanted to estimate the mean income to be within 200, with a confidence level of 99 percent. How large a sample should be selected?

3. (a) Suppose you have taken a sample of 100 people from a population and found that the percentage of then with a certain characteristic is 10 percent. What would be the value of STEP and the width of the 95 percent confidence interval in this case? (b) Now suppose that you obtain the same figure of 10 percent, but from samples with 400, 900, and 1600 people. What are the corresponding values of STEP? (c) Using the values you have calculated, sketch a graph to show how step varies with samples size. What do you notice? (d) In the light of your graph, comment on the fact that many market research and political opinion surveys contain 500 to 1000 respondents. 4. The latest internal accounts for an off-license chain showed that the annual sales of wines and spirits had fallen by more than 30%. This fall has been blamed on the relaxation of the limits of duty-free goods that can be brought into Britain from EU countries. In order to test this theory, it was decided to ask a random sample of shoppers if they intend to travel to France this year. Of the 75 shoppers questioned, 27 were certain to go to France at least once. It was also decided to ask a random sample of 60 returning holidaymakers how much they had spent on duty-free alcohol. Of these 60, 8 refused to answer, and for the remaining 52 people, the average spend was found to be 37.26, with a standard deviation of 35.97. (a) What is the percentage of shoppers who said they were definitely making at least one trip to France this year? (b) Calculate the 95% confidence interval for the true percentage of shoppers who intend to travel to France this year. Interpret this interval. (c) Calculate the 95 percent confidence interval for the true mean amount spent on duty-free alcohol for all holidaymakers returning to France (d) Calculate the number of shoppers to be questioned so that the half-width of the confidence interval i.e. the x + Z (STEP) or x - Z(STEP) term, for the percentage of shoppers who intend to travel to France is no more than 3%. (e) The half-width of the confidence interval for the average spend on duty-free alcohol wants to be reduced to 5. How many holidaymakers need to be sampled? (f) What reservations (if any) do you have about this kind of survey?