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.