Let us take a sample from population X from a normal distribution with unknown mean pm and unknown variance 0:, and population Y from normal distribution with unknown mean pay and unknown variance 0:. Suppose that we want to nd the interval estimation of differences between two population means p51 pay. Let $1,32,~- ,a'nm denote the random sample of size a: form the population X and let y1, y2, - - - ,yny denote the random sample of size y form the population Y. Assume that the populations X and Y a: = a: = 02. We have used in our class the usual estimator of the common variance 0'2 is obtained by pooling the sample data to obtain the pooled variance estimator 32p: zzam a?)2 + Elisa W 32 = P n3+ny2 Let s: and a: be the sample variances of sample X and sample Y respectively. a) 2 Show that 3:\"; is an unbiased estimator for the common variance cr . b) Show that if HI = ny, 3%. is simply the average of s: and 3:. G) Show that if 111 a my and n1 > my, 3:\"; is the weighted average of 32 and 3:, with larger weight given to sample X