Prior to a recent marketing campaign launched by one of its competitors, the daily sales of bread at a particular supermarket have been observed to follow a near Normal distribution with a mean of 400 loaves and a standard deviation of 40 loaves. In the 35 days since the marketing campaign the mean daily sales has been 380 loaves.

1. (a) Propose and carry out a test to see whether there has been a decrease in the daily sales of bread since the competitor's marketing campaign, clearly stating your null and alternative hypotheses. Use a 5% significance level. Justify your choice of test and state your conclusion clearly.

(6 marks)

1. (b) Using the statistical test in part (a), what is the chance that the wrong conclusion is reached if the mean daily sales of bread has not changed since the marketing campaign? Justify your answer.

(2 marks)

1. (c) Calculate the power of the statistical test used in part (a) if the true mean daily sales after the marketing campaign were to drop by 10%. Sketch out the shape of the power curve marking those points known to you.

(8 marks)

1. (d) When the source of the data described above is looked at more closely, it turns out that the values of mean and standard deviation prior to the marketing campaign were in fact only calculated from the previous 3 month period, i.e. 90 days, and for the 35 days since the campaign the standard deviation of loaves sold was 44. Revise your proposed test accordingly and re-test for a decrease in sales. Justify your choice of test and state clearly any additional assumptions that you have made.