A researcher presents you with the following data Yi, i = 1,2,...29, which is independent and identically normally distributed with a mean and variance 2.

1. Set up the rejection region for testing the hypotheses H0 : = 20 vs H1 : 6= 20, based on Y1,...,Y29 and = 0.05. Assume is unknown.

2. Calculate the power of your test for the following values of the true parameter : 19.9, 19.95, 19.99, 20, 20.05, 20.1, 20.15, 20.2, 20.25, 20.3, 20.4, 20.5. The researcher provides the estimate, = .43. This value of should just be used in the power calculation and not in the actual testing procedure. Use your results to sketch a power curve for your test. Be sure to label your axes clearly.

3. Determine the necessary sample size so that the power at = 20.15 is at least 0.80 for a level = .05 test of H0 : 20 vs H1 : > 20.