With a bit of algebra, we can show that

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?

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SSResid (1) (y-5) from which it follows that n 1 - Se n-2 Unless n is quite small, (n - 1)/(n-2) = 1, so Se