Let X1,X2,...,XnXiid N(X,2X) and Y1,Y2,...,YnYiid N(Y,2Y) with the Xs and Ys being independent. In what follows assume that 2X = 2Y and that 2X and 2Y are both unknown. Consider the hypotheses H0:XY=d0 and H1:XY /= d0.

a. Derive an appropriate test statistic for this test.

b. What is the distribution of the test statistic when H0 is true?

c. Define the rejection region for an level test.

d. What is the type I error for the level test that you defined in (c)?

e. How can a 100(1)% confidence interval be used to carry out this test? Prove it.