Refer to the matched-pairs experiment and assume that the ith measurement (i = 1, 2), in

Chapter 12, Problem 12.17

(choose chapter or problem)

Refer to the matched-pairs experiment and assume that the ith measurement (i = 1, 2), in the jth pair, where j = 1, 2,..., n, is Yi j = i + Uj + i j, where i = expected response for population i, where i = 1, 2, Uj = a random variable that is uniformly distributed on the interval (1, +1), i j = random error associated with the ith measurement in the jth pair. Assume that the i js are independent normal random variables with E(i j) = 0 and V(i j) = 2, and that Uj and i j are independent. a Find E(Yi j). b Argue that the Y1 js, for j = 1, 2,..., n, are not normally distributed. (There is no need to actually find the distribution of the Y1-values.) c Show that Cov(Y1 j, Y2 j) = 1/3, for j = 1, 2,..., n. d Show that Dj = Y1 j Y2 j are independent, normally distributed random variables. e In parts (a)(d), you verified that the differences within each pair can be normally distributed even though the individual measurements within the pairs are not. Can you come up with another example that illustrates this same phenomenon?