# Question

Let X-bar, Y-bar, S2X, and S2Y 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(μX, σ2X) and N(μY, σ2Y), where μX, μY, σ2X, and σ2Y are unknown. If σ2X/σ2Y = d, a known constant,

(c) Argue that the two random variables in (a) and (b) are independent.

(d) With these results, construct a random variable (not depending upon σ2Y) that has a t distribution and that can be used to construct a confidence interval for μX − μY.

(c) Argue that the two random variables in (a) and (b) are independent.

(d) With these results, construct a random variable (not depending upon σ2Y) that has a t distribution and that can be used to construct a confidence interval for μX − μY.

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