Let X1,,Xn be i.i.d. random variables from normal distribution with mean 1 and variance 2 and Y1,,Ym be i.i.d. random variables from normal distribution with mean 2 and variance 2. Find the likelihood ratio test for testing H0:1=2 against the alternative that H0:1=2. The two samples are independent. Show that the likelihood ratio test is the same with the one obtained using the distribution of XY and the distribution of 2(n+m2)Sp2, where Sp2 is the unbiased estimator of the common but unknown 2..