Data from the Framingham Study allow us to compare the distributions of initial serum cholesterol levels for two populations of males: those who go on to develop coronary heart disease and those who do not. The mean serum cholesterol level of the population of men who do not develop heart disease is = 205mg/100ml and the standard deviation is = 36mg/100ml. Suppose, however, that you do not know the true population mean; instead, you hypothesize that is equal to 230mg/100ml. This is the mean initial serum cholesterol level of men who eventually develop the disease. Since it is believed that the mean serum cholesterol level for the men who do not develop heart disease cannot be higher than the mean level for men who do, a one-sided test conducted at the = 0.05 level of significance is appropriate.

a. How could you increase the power?

b. You wish to test the null hypothesis H0: 230mg/100ml against the alternative HA: < 230mg/100ml at the alpha = 0.05 level of significance. If the true population mean is as low as 205mg/100ml, you want to risk only a 5% chance of failing to reject H0. How large a sample would be required?

c. How would the sample size change if you were willing to risk a 10% chance of failing to reject a false null hypothesis?