5.41 Treatment and control groups. The previous exercise illustrates a common setting for statistical inference. This exercise gives the general form of the sampling distribution needed in this setting. We have a sample of n observations from a treatment group and an independent sample of m observations from a control group. Suppose that the response to the treatment has the N(μ_X, σ_X) distribution and that the response of control subjects has the N(μ_Y, σ_Y) distribution. Inference about the difference μ_Y − μ_X between the population means is based on the difference between the sample means in the two groups.

1. (a) Under the assumptions given, what is the distribution of ? Of ?

2. (b) What is the distribution of ?