3. Suppose X1, . ., X, are N (, 2) random variables, and Y,..., Yn are N (2, ) ran- dom variables. Note that only the means are different. Furthermore, assume that the pairs (X1,Y),..., (X, Y) are independent. This problem compares two confidence intervals for - 2, namely the "pooled interval" (Corollary A on p. 423 of Rice) and the "paired interval" (p. 446 of Rice). (a) Show that if the true correlation p = Corr[Xi, Yi] a "paired interval has length zero. == 1, then with probability one (b) Assume now that the observations are such that the sample correlation coefficient is exactly equal to zero. Which confidence interval would be shorter, the "paired interval" or the "pooled interval"? In your answer determine the ratio of the two interval lengths.