Please help me solve these problems

1. Suppose we have two normally distributed populations with equal (unknown) variances and unknown means. Let U2 be the common variance, and pl, #2 the two means. Suppose we take samples from each population of size 'n, from each and want to test the null hypothesis H0 that ,u.1 = #2 against the alternative hypothesis pal 7E #2. Let Ehyg, Sf, 822 be the sample means and variances from the respective populations. Using the likelihood ratio test yields the statistic: A (Y1 - 72) e = (3,2452? 1'1. a critical region of the form: I0 I > 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 fl = 38, 3% = 18, 52 = 50, 8% = 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 #1 1.734) and P(T 1.330)