Question 4 Hypothesis Testing (15 Marks)

(a) One factor in choosing a location for a new clothing store is the mean clothing expenditure per household in the surrounding area. Ideally, the shop wants to move into an area where the population spends an average of $500 per year on clothes. A random survey of 22 households in the area reveals that the mean and standard deviation of annual expenditure on clothes is $446 and $90, respectively. Can we conclude at the 5% level that this sample mean annual expenditure is statistically significantly less than the ideal population mean of $500? Why or why not? (5 marks)

(b) For the past few years, the number of customers of a drive-in bottle shop has averaged 22 per hour. This year, another bottle shop one kilometre away opened a drive-in window. The manager of the first shop is not sure if this will result in a decrease in customers. A random sample of 24 hours showed an average of 19.7 customers per hour with a standard deviation of 4.2 customers per hour. Can we conclude at the 5% level of significance that the mean from the sample 24 hours has resulted in a decrease from previous years? Why or why not? (10 marks)