Question

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 µ = 206mg/10ml and the standard deviation is σ = 36mg/100ml. Suppose, however, that you do not know the true population mean; instead, you hypothesize that µ 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 α = 0.05 level of significance is appropriate.

a. How could you increase the power?

b. You wish to test the null hypothesis H0: µ ≥ 230mg/100ml against the alternative HA: µ < 230mg/100ml at the alpha = 0.05 level of significance. If the true population mean is as low as 206mg/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?

Answer 1

GIVEN:
The experiment is to compare the mean serum cholesterol level of males who 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 µ = 206mg/10ml = 2.06mg/ml and the standard deviation is σ = 36mg/100ml = mg/ml

TO FIND:

1) Increase the power of the test:
The power of the test can be increased by:
a. increasing the level of significance ( but it increases the type 1 error)
b. increase the sample size
c. test for the one-sided hypothesis.
d. as the standard deviation of the observations decreases, the power of the test can be improved.

2)  To obtain a sample size for testing  H0: µ ≥ 230mg/100ml against HA: µ < 230mg/100ml at the alpha = 0.05 level of significance and  risk only a 5% chance of failing to reject H0

where D is the risk of failing to reject false H0

The required sample size is 7.

3) The sample size change required for 10% chance of failing to reject a false null hypothesis:

The required sample size is 3