5.43 With a bit of algebra, we can show that SSResid 5 s1 2 r2d os y 2 yd2 from which it follows that se 5În 2 1 n 2 2

Ï1 2 r2 sy Unless n is quite small, (n 2 1)/(n 2 2) < 1, so se < Ï1 2 r2 sy

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

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 < 2.5 inches.

What would se be for the least-squares line used to predict 18-year-old height from 6-year-old height?

d. Referring to Part (c), suppose that you wanted to predict the past value of 6-year-old height from knowledge of 18-year-old height. Find the equation for the appropriate least-squares line. What is the corresponding value of se?