Applied Statistics and Probability for Engineers

11-6 CONFIDENCE INTERVALS 391

Now is an unbiased point estimator of Yx 0 , since and are unbiased estimators of

 0 and  1. The variance of is

This last result follows from the fact that cov (Refer to Exercise 11-71). Also,

is normally distributed, because 1 and 0 are normally distributed, and if we use as an

estimate of ^2 , it is easy to show that

has a tdistribution with n 2 degrees of freedom. This leads to the following confidence in-

terval definition.

ˆY 0 x 0   Y 0 x 0

B

ˆ^2 c

1

n

1 x 0    x 22

Sx x

d

ˆ ˆ ˆ^2

1 Y, ˆ 12  0 ˆY (^0) x 0
V 1 ˆY 0 x 02 ^2 c
1
n
1 x 0 x 22
Sx x
d
ˆY 0 x 0
ˆY 0 x 0 ˆ 0 ˆ 1
A 100(1 )% confidence intervalabout the mean responseat the value of
xx 0 , say , is given by
(11-31)
where ˆY 0 x 0 ˆ 0 ˆ 1 x 0 is computed from the fitted regression model.
Y 0 x 0 ˆY 0 x 0 t 2, n 2
B
ˆ^2 c
1
n
1 x 0 x 22
Sx x
d
ˆY 0 x 0 t 2, n 2
B
ˆ^2 c
1
n
1 x 0 x 22
Sx x
d
Y 0 x 0
Definition
Note that the width of the confidence interval for is a function of the value specified for
x 0. The interval width is a minimum for and widens as increases.
EXAMPLE 11-5 We will construct a 95% confidence interval about the mean response for the data in Example
11-1. The fitted model is and the 95% confidence interval on
is found from Equation 11-31 as
Suppose that we are interested in predicting mean oxygen purity when x 0 1.00%. Then
and the 95% confidence interval is
e89.232.101
B
1.18 c
1
20

1 1.00 1.1960 22
0.68088
df
ˆY 0 x1.0074.28314.947 1 1.00 2 89.23
ˆY 0 x 0 2.101
B
1.18c
1
20

1 x 0 1.1960 22
0.68088
d
Y 0 x 0
ˆY 0 x 0 74.28314.947x 0 ,
x 0 x 0 x 0 x 0
Y 0 x 0
c 11 .qxd 5/20/02 1:16 PM Page 391 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: