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? r = 0, y^ = 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 = 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? y^ = 7.92 + .544x, se = 1.02

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