Suppose that X1, X2,..., Xm and Y1, Y2,..., Yn are independent random samples, with the

Chapter 7, Problem 7.15

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Suppose that X1, X2,..., Xm and Y1, Y2,..., Yn are independent random samples, with the variables Xi normally distributed with mean 1 and variance 2 1 and the variables Yi normally distributed with mean 2 and variance 2 2 . The difference between the sample means, X Y , is then a linear combination of m + n normally distributed random variables and, by Theorem 6.3, is itself normally distributed. a Find E(X Y ). b Find V(X Y ). c Suppose that 2 1 = 2, 2 2 = 2.5, and m = n. Find the sample sizes so that (X Y ) will be within 1 unit of (1 2) with probability .95.