When Does " T " Really Have a T -Distribution

Given a random sample of sizes n and m from independent normal distributions, when does

t = (xbar-ybar-(mu[x]-mu[y]))/(sqrt(((n-1)*s[x]^2+(...

have a t -distribution with n + m - 2 degrees of freedom?

1. Let n = 6, m = 18, sigma[x]^2 = 1 , sigma[y]^2 = 36 , mu[x] = 0 , and mu[y] = 0 . Solution

2. Let n = 6, m = 18, sigma[x]^2 = 16 , sigma[y]^2 = 16 , mu[x] = 0 , and mu[y] = 0 . Solution

3. Let n = 12, m = 12, sigma[x]^2 = 1 , sigma[y]^2 = 36 , mu[x] = 0 , and mu[y] = 0 . Solution

4. Let n = 18, m = 6, sigma[x]^2 = 1 , sigma[y]^2 = 36 , mu[x] = 0 , and mu[y] = 0 . Solution