For the following problem of deciding between two simple hyp

For the following problem of deciding between two simple hypotheses (normal with unknown mean and known variance) by a sample of size 1 (single observation), H0 : Xi: N(0, 1) H0 : Xi: N(2, 1) use the test statistic X and determine its critical region for Classical test with significance level alpha = 0.05 (also, what is the power?)

Solution

test statistic Z= u1-u2/std

= 0-2/1

= -2

critical value = +/- 2.243

the critical region is   Z < -2.243 and z > 2.243

power = 1- alpha = 0.95