54. (i) Let (Xl ' Yl ) , ... , (Xn , y") be a sample from a bivariate normal distribution, and let -I ( E(X; - x)( Y; - Y) ) p=C , - VE(X; - X)2E(Y; _ y)2 where C(p) is determined such that ( [(X;-X)(y;-y) ) Pp ~C{p) /[(X; - X)2[(y; _ y)2 = 1-

a. 353 Then p is a lower confidence limit for the population correlation coefficient pat confidence level 1 - a; it is uniformly most accurate invariant with respect to the group of transformations X; = aX; +

b, Y;' = cY; +

d, with ac > 0, - 00 <

b, d < 00 . (ii) Determine p at level 1 - a = .95 when the observations are (12.9, .56), (9.8, .92), (13.1, .42), (12.5,1.01), (8.7,.63), (10.7,.58), (9.3,.72), (11.4, .64).