Chapter 12, Problem 81SE

Let s_x and s_y denote the sample standard deviations of the observed x’s and y’s, respectively [that is, $s_x^2=\sum {(x_i-\bar{x})}^2/(n-1)$ and similarly for $s_y^2$].

1. a. Show that an alternative expression for the estimated regression line $y={\hat{\beta }}_0+{\hat{\beta }}_{1}x$ is

   $y=\bar{y}+r\cdot \frac{s_y}{s_x}(x-\bar{x})$

2. b. This expression for the regression line can he interpreted as follows. Suppose r = .5. What then is the predicted y for an x that lies 1 SD (s_x units) above the mean of the x_i’s? If r were 1, the prediction would be for y to lie 1 SD above its mean $\bar{y}$, but since r = .5, we predict a y that is only .5 SD (.5s_y unit) above $\bar{y}$. Using the data in Exercise 64, when UV transparency index is 1 SD below the average in the sample, by how many standard deviations is the predicted maximum prevalence above or below its average for the sample?