Introduction to Probability and Statistics for Engineers and Scientists

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)