Suppose that we have 9 independent observations from a normal distribution with standard deviation 10. we wish to test

h_0 : mu = 150 vs. h_a : mu plusminus 150

the glrt with level alpha = 0.05 uses the test statistic

t_1 = |x - 150|

and has a critical value of c = 6.53. the test rejects the null hypothesis when t > c.

a. calculate the power of this test against the alternative mu = 151.
b. calculate the power of this test against the alternative mu = 149.
c. alternatively, consider the test statistic t_2 = |x - 152| and critical value c = 7.55

i. calculate the level of this second test.
ii. calculate the power against the alternatives mu = 151.
iii. calculate the power of this test against the alternative mu = 149.

d. which of the two tests do you think is better? why?