Consider the least squares fit of the linear model E(yi) = ????0 + ????1xi.

a. Show that ????̂

1 = [

∑

i

(xi − x̄)(yi − ȳ)]∕[

∑

i

(xi − x̄)

2].

b. Derive var(????̂
1). State the estimated standard error of ????̂
1, and discuss how its magnitude is affected by (i) n, (ii) the variability around the fitted line, (iii)
the sample variance of x. In experiments with control over setting values of x, what does (iii) suggest about the optimal way to do this?