Let X G(1,) and Y G(2,) be two independent random variables, with unknown parameters 1, 2, and . Further, let X1,X2,...,Xn1 and Y1,Y2,...,Yn2 be two random samples generated by X and Y, respectively.

Using this information, please answer the following questions.

(a) (2 marks) Let S, and $, be two unbiased variance estimators of o' for random samples (X1, X2, ... Xn) and (Y1, Y2, ...,Ymz), respectively. Show that the pooled variance estimator $2 (n1 - 1) 57 + (n2 - 1) 53 n1 + n2 - 2 is an unbiased estimator. (b) (2 marks) Show that for S, in (a) it holds Var($2) - 0 as n = n1 + 12 -+ 00 Hint: Use without proof that Var( S-) = 204 and Var (S4) = 204 n1-1 n2-1(c) (4 marks) We assume that the iron intake of a given person depends on their income. To check this assumption, we take two samples from two populations A and B with different levels of income and observe these samples over some time period. Both populations are assumed to have the same (unknown) variance oz. For the sample from the population A we have the data summary n1 = 51, y1 = 12.5 mg per day, and the sample standard deviation s1 = 4.75 mg per day. For the sample from the population B we have the data summary n2 = 65, y2 = 10.5 mg per day, and the sample standard deviation of 5.56 mg. Using this data, check the assumption that the average iron consumption in both popu- lations A and B is the same. Justify your statement by means of an appropriate p-value as well as an appropriate confidence interval. (d) (4 marks) How will your results in (c) change if we assume that the variances in the considered two populations A and B are different