Suppose that X and Y are two independent, normal random variables, and let f{X1, X2, … Xm} andf{Y1, Y2, … Yn} be random samples of size m and n respectively from the random variables X and Y . It is often the case that we would like to test if mu_X = mu_Y , that is, if X and Y have equal means. Also, upon occasion, we need to test if sigma^2X = sigma^2Y . Show the full derivations for all the questions.

The technique of likelihood ratios is used to develop tests for both of these problems. The sets of hypotheses for these tests are given below. The test for (I) will be a likelihood ratio test, but the test for (II) will only be an approximate likelihood ratio test.

(I) H0 : mu_X = mu_Y vs. H1 : mu_X not equall to mu_Y where sigma^2X sigma^2X are unknown, however, they are equal.

(II) H0 :sigma^2X =sigma^2Y vs. H1 :sigma^2X =sigma^2Y where the parameters mu_X and mu_Y are both unknown