Introduction to Probability and Statistics for Engineers and Scientists

*9.10Multiple Linear Regression 405

Since the value of the preceding statistic is (

√
10/19.262)(.075)/.616=.088, thep-value
of the test of the hypothesis thatβ 2 =0is

p-value=P{|T 10 |>.088}

= 2 P{T 10 >.088}

=.9316 by Program 5.8.2.A

Hence, the hypothesis is accepted (and, in fact, would be accepted at any significance level
less than .9316). ■

REMARK

The quantity

R^2 = 1 −

SSR

∑

i

(Yi−Y)^2

which measures the amount of reduction in the sum of squares of the residuals when using
the model

Y=β 0 +β 1 x 1 +···+βnxn+e

as opposed to the model

Y=β 0 +e

is called thecoefficient of multiple determination.

9.10.1 Predicting Future Responses

Let us now suppose that a series of experiments is to be performed using the input levels
x 1 ,...,xk. Based on our data, consisting of the prior responsesY 1 ,...,Yn, suppose we
would like to estimate the mean response. Since the mean response is

E[Y|x]=β 0 +β 1 x 1 +···+βkxk

a point estimate of it is simply

∑k

i= 0 Bixiwherex^0 ≡1.

To determine a confidence interval estimator, we need the distribution of

∑k
i= 0 Bixi.
Because it can be expressed as a linear combination of the independent normal random
variablesYi,i=1,...,n, it follows that it is also normally distributed. Its mean and
variance are obtained as follows:

E





∑k

i= 0

xiBi



=

∑k

i= 0

xiE[Bi] (9.10.8)

=

∑k

i= 0

xiβi sinceE[Bi]=βi