Applied Statistics and Probability for Engineers

438 CHAPTER 12 MULTIPLE LINEAR REGRESSION

12-3.2 Confidence Interval on the Mean Response

We may also obtain a confidence interval on the mean response at a particular point, say,

x 01 ,x 02 ,p, x 0 k. To estimate the mean response at this point, define the vector

The mean response at this point is which is estimated by

(12-35)

This estimator is unbiased, since and the variance of

is

(12-36)

A 100(1) % CI on can be constructed from the statistic

(12-37)

ˆY 0 x 0 Y 0 x 0

2 ˆ^2 ̨x¿ 0 1 X¿X 2  1 x 0

Y (^0) x 0
V 1 ˆY (^0) x 02 ^2 x¿ 01 X¿X 2 ^1 x 0
ˆY (^0) x 0
E 1 x¿ 0 ˆ 2 x 0 ¿ E 1 Y 0 x 02 Y 0 x 0
ˆY 0 x 0 x 0 ¿ˆ
E 1 Y 0 x 02 Y 0 x 0 x¿ 0 ,
x 0 
1
x 01
Ex 02 U
o
x 0 k
Equation 12-38 is a CI about the regression plane (or hyperplane). It is the multiple regression
generalization of Equation 11-31.
EXAMPLE 12-7 The engineer in Example 12-1 would like to construct a 95% CI on the mean pull strength for
a wire bond with wire length x 1 8 and die height x 2 275. Therefore,
x 0 £
1
8
275
§
For the multiple linear regression model, a 100(1)% confidence interval on the
mean responseat the point x 01 , x 02 ,..., x 0 kis

Y 0 x 0
ˆY 0 x 0 t2,np 2 ˆ^2 ̨x¿ 0 1 ¿ 2 ^1 x 0 (12-38)
ˆY 0 x 0 t2,np 2 ˆ^2 ̨x¿ 0 1 X¿X 2 ^1 x 0
Definition
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