(Hard.) Suppose Yi = a + bXi + i for i = 1,...,n, the i being IID with mean 0 and variance σ2, independent of the Xi. (Reminder: IID stands for “independent and identically distributed.”) Equation (2.5)

expressed a,ˆ bˆ in terms of five summary statistics: two means, two SDs, and r. Derive the formulas for a,ˆ bˆ from equation (6) in this chapter.

Show also that, conditionally on X, SE aˆ = σ

√n 1 +

X2 var(X) , SE bˆ = σ

sX

√n , where X = 1 n

n i=1 Xi, var(X) = 1 n

n i=1

(Xi − X)2, s2 X = var(X).

Hints. The design matrix M will be n×2. What is the first column? the second? Find M

M. Show that det(M

M) = n2var(X). Find (M

M)−1 and M

Y .