Let X, Y, S2 X , and S2 Y be the respective sample means and unbiased estimates of the

QUESTION:

Let X, Y, S2 X , and S2 Y be the respective sample means and unbiased estimates of the variances obtained from independent samples of sizes n and m from the normal distributions N(X , 2 X ) and N(Y , 2 Y ), where X , Y , 2 X , and 2 Y are unknown. If 2 X /2 Y = d, a known constant, (a) Argue that (X Y) (X Y ) d2 Y /n + 2 Y /m is N(0, 1). (b) Argue that (n 1)S2 X d2 Y + (m 1)S2 Y 2 Y is 2(n+m2). (c) Argue that the two random variables in (a) and (b) are independent. (d) With these results, construct a random variable (not depending upon 2 Y ) that has a t distribution and that can be used to construct a confidence interval for X Y