9.66.* Refer to the formula a = ¯y−b¯x for the y-intercept.

(a) Show that substituting x = ¯x into the prediction equation ˆy = a + bx yields the predicted y-value of ˆy = ¯y.
Show that this means that the least squares prediction equation passes through the point with coordinates ( ¯x, ¯y), the center of gravity of the data.

(b) Show that an alternative way of expressing the regression model is as ( ˆy − ¯y) = b(x − ¯x).

(c) Let y = final exam score and x = midterm exam score. Suppose the correlation is 0.70 and the standard deviation is the same for each set of scores. Show that ( ˆy − ¯y) = 0.70(x − ¯x); that is, the predicted difference between your final exam grade and the class mean is 70%
of the difference between your midterm exam score and the class mean, so your score is predicted to regress toward the mean.