Introduction to Probability and Statistics for Engineers and Scientists

9.4Statistical Inferences About the Regression Parameters 361

In addition, if we let

SSR=

∑

i

(Yi−A−Bxi)^2

denote the sum of squares of the residuals, then

SSR

σ^2

∼χn^2 − 2

andSSRis independent of the least squares estimatorsAandB. Also,SSRcan be computed
from

SSR=

SxxSYY−(SxY)^2

Sxx

Program 9.2 will compute the least squares estimatorsAandBas well asx,

∑

ix

2
i,
Sxx,SxY,SYY, andSSR.

EXAMPLE 9.3a The following data relatex, the moisture of a wet mix of a certain product,
toY, the density of the finished product.

xi yi

57.4

69.3

7 10. 6

10 15. 4

xi yi

12 18. 1

15 22. 2

18 24. 1

20 24. 8

Fit a linear curve to these data. Also determineSSR.

SOLUTION A plot of the data and the estimated regression line is shown in Figure 9.4.
To solve the foregoing, run Program 9.2; results are shown in Figure 9.5. ■

9.4Statistical Inferences about the Regression Parameters

Using Proposition 9.3.1, it is a simple matter to devise hypothesis tests and confidence
intervals for the regression parameters.