Consider a random sample of size 20 from a normal distribution, Xi ~ N(mu, s^2), and suppose that x(bar) = 11 and s^2 = 16.

a) Assuming it is known that s^2 = 4, test Ho: mu > or equal to 12 versus Ha: mu < 12 at the significance level alpha = .01.

b) What is B = P[TII] if in fact mu = 10.5?

c) What sample size is needed for the power of the test to be .9 for the alternative value mu = 10.5?

d) Test the hypothesis of a) assuming s^2 unknown.

e) Test Ho: s^2 < or equal to 9 against Ha: s^2 > 9 with significance level alpha = .01.

f) What sample size is needed for the test of e) to be 90% certain of rejecting Ho when in fact s^2 = 18? What is B = P[TII] in this case?