408 Chapter 9: Regression
Multiple Linear Regression
Compute Inverse
Back 1 Step
1 1 1 1 1 1
.02
.03
.03
.04
.10
.15
1.05
1.20
1.25
1.30
1.30
1.00
0
1 2 3 4 5 6
ABC
FIGURE 9.18
To determine a prediction interval forY(x), note first that sinceB 0 ,...,Bkare based
on prior responses, it follows that they are independent ofY(x). Hence, it follows that
Y(x)−
∑k
i= 0 Bixiis normal with mean 0 and variance given by
Var
Y(x)−
∑k
i= 0
Bixi
=Var[Y(x)]+Var
∑k
i= 0
Bixi
by independence
=σ^2 +σ^2 x′(X′X)−^1 x from Equation 9.10.10
and so
Y(x)−
∑k
i= 0
Bixi
σ
√
1 +x′(X′X)−^1 x
∼N(0, 1)