Let X1, .... Xng ~ N(ux, 3') and Yl, .... Yay ~ N(my,5'). For py - AX, we've constructed a 1 - a confidence interval based on the t-distributed pivot U = Y - X - (my - #x) 5,\\/1y +1x where (ny - 1)5 + (nx - 1)8% ax + ny - 2 The CI is [Y - X +tnstay-2,0/25pV1y + 1x]. Here, we used S to estimate the shared variance o'. Now let's consider using S* to estimate o', where S* =,,LE(X -X). Show that VY- X - (MY - ux) Sx v1y + 1x is t-distributed and find its degrees of freedom. Construct a 1 - a CI based on this pivot V. Intuitively, the V-based CI is worse because we didn't use all information when estimating o'. Let's compare the CIs constructed based on U and V. Derive a formula for the expectation of each CI's length. If nx = 10, ny = 10, a = 0.05, what's the ratio between the two expected lengths? You can use the following fact in your calculation: . If W ~ X2, then EVW = Vale . r(t + 1) = tr(t). + 2 3- . to,0.025 = 2.26, t19,0.028 = 2.09