Introduction to Probability and Statistics for Engineers and Scientists

384 Chapter 9: Regression

TABLE 9.3

xPPPˆ −Pˆ

5 .061 .063 −.002

10 .113 .109 .040

20 .192 .193 −.001

30 .259 .269 −.010

40 .339 .339 .000

50 .401 .401 .000

60 .461 .458 .003

80 .551 .556 −.005

Transforming this back into the original variable gives that the estimates ofcanddare

ˆc=e−A=.9847

1 −dˆ=e−B=.9901

and so the estimated functional relationship is

Pˆ= 1 −.9847(.9901)x

The residualsP−Pˆare presented in Table 9.3. ■

9.8Weighted Least Squares

In the regression model

Y=α+βx+e

it often turns out that the variance of a response is not constant but rather depends on its
input level. If these variances are known — at least up to a proportionality constant —
then the regression parametersαandβshould be estimated by minimizing a weighted
sum of squares. Specifically, if

Var(Yi)=

σ^2

wi

then the estimatorsAandBshould be chosen to minimize

∑

i

[Yi−(A+Bxi)]^2

Var(Yi)

=

1

σ^2

∑

i

wi(Yi−A−Bxi)^2