Please help me with these problems

1. Suppose we have two normally distributed populations with equal (unknown) variances and unknown means. Let 02 be the common variance, and p.1, #2 the two means. Suppose we take samples from each p0pulation of size n from each and want to test the null hypothesis H0 that 111 = #2 against the alternative hypothesis #1 7E #2. Let 21,72, Sf, 822 be the sample means and variances from the reSpective p0pulations. Using the likelihood ratio test yields the statistic: a critical region of the form: || > K. (a) Show that Q is a random variable with the tdistribution with 2(n 1) degrees of freedom. (b) Suppose we take a sample of size n = 10 from each population and obtain sample means and variances El 2 38, a? = 18, 52 = 50, 5% = 20. Using a test as above with a = 10% should we reject the null hypothesis in this case? (c) Suppose that we obtain the same data as above, but are trying to test the hypothesis in 1.734) and P(T 1.330)