Suppose that the differential brightness B of a certain star has an unknown value, and that it is desired to test the following simple hypotheses:
The statistician knows that when he goes to the observatory at midnight to measure 0 there is probability 1/2 that the meteorological conditions will be good and he will be able to obtain a measurement X having a normal distribution with mean 0 and variance 1. He also knows that there is probability 1/2 that the meteorological conditions will be poor and he will obtain a measurement Y having a normal distribution with mean 0 and variance 100.
(a) Construct the most powerful test that has conditional size a = 0.05, given good meteorological conditions; and that has conditional size a = 0.05, given poor meteorological conditions.
(b) Construct the most powerful test that has conditional size a = 0.000001, given good meteorological conditions; and that has conditional size a = 0.099999, given poor meteorological conditions.
(c) Show that the overall size of both the test found in part (a) and the test found in part (b) is 0.05, and determine the power of each of these two tests.