With a bit of algebra, we can show that SSResid 5 11 2 r 2 2 g 1y 2 y2 2 from which it follows that se 5 Å

n 2 1 n 2 2

"1 2 r 2 sy Unless n is quite small, (n 1)/(n 2) 1, so se < "1 2 r 2 sy

a. For what value of r is se as large as sy? What is the least-squares line in this case? r 5 0, y^ 5 y

b. For what values of r will se be much smaller than sy?

c. A study by the Berkeley Institute of Human Development (see the book Statistics by Freedman et al., listed in the back of the book) reported the following summary data for a sample of n 5 66 California boys:
r .80 At age 6, average height 46 inches, standard deviation 1.7 inches.

At age 18, average height 70 inches, standard deviation