81. Let sx and sy denote the sample standard deviations of the observed x’s and y’s, respectively [so s 2 x

(xi  x)

2

/(n  1)

and similarly for s 2 y ].

a. Show that an alternative expression for the estimated regression line y

ˆ

0
ˆ

1x is y
y r
(x  x)

b. This expression for the regression line can be interpreted as follows. Suppose r
.5. What then is the predicted y for an x that lies 1 SD (sx units) above the mean of the xis?

If r were 1, the prediction would be for y to lie 1 SD above its mean y, but since r
.5, we predict a y that is only .5 SD (.5sy unit) above y. Using the data in Exercise 64 for a patient whose age is 1 SD below the average age in the sample, by how many standard deviations is the patient’s predicted CBG above or below the average CBG for the sample?