Introduction to Probability and Statistics for Engineers and Scientists

9.4Statistical Inferences About the Regression Parameters 371

Confidence Interval Estimator ofα

The 100(1−a) percent confidence interval forαis the interval

A±

√√

√√

∑

i

x^2 iSSR

n(n−2)Sxx

ta/2,n− 2

Hypothesis tests concerningαare easily obtained from Equation 9.4.3, and their
development is left as an exercise.

9.4.3 Inferences Concerning the Mean Responseα+βx 0

It is often of interest to use the data pairs (xi,Yi),i=1,...,n, to estimateα+βx 0 , the
mean response for a given input levelx 0. If it is a point estimator that is desired, then the
natural estimator isA+Bx 0 , which is an unbiased estimator since

E[A+Bx 0 ]=E[A]+x 0 E[B]=α+βx 0

However, if we desire a confidence interval, or are interested in testing some hypothesis
about this mean response, then it is necessary to first determine the probability distribution
of the estimatorA+Bx 0. We now do so.
Using the expression forBgiven by Equation 9.3.1 yields that

B=c

∑n

i= 1

(xi−x)Yi

where

c=

1

∑n

i= 1

xi^2 −nx^2

=

1

Sxx

Since

A=Y−Bx

we see that

A+Bx 0 =

∑n

i= 1

Yi

n

−B(x−x 0 )

=

∑n

i= 1

Yi

[

1

n

−c(xi−x)(x−x 0 )

]

Since theYi are independent normal random variables, the foregoing equation shows
thatA+Bx 0 can be expressed as a linear combination of independent normal random