Introduction to Probability and Statistics for Engineers and Scientists

*9.10Multiple Linear Regression 397

It is now easy to see that the matrix equation

X′XB=X′Y

is equivalent to the set of normal Equations 9.10.1. Assuming that (X′X)−^1 exists, which
is usually the case, we obtain, upon multiplying it by both sides of the foregoing, that
the least squares estimators are given by

B=(X′X)−^1 X′Y (9.10.3)

Program 9.10 computes the least squares estimates, the inverse matrix (X′X)−^1 ,
andSSR.

EXAMPLE 9.10a The data in Table 9.4 relate the suicide rate to the population size and the
divorce rate at eight different locations.

TABLE 9.4

Population Divorce Rate Suicide Rate
Location in Thousands per 100,000 per 100,000

Akron, Ohio 679 30.4 11.6
Anaheim, Ca. 1,420 34.1 16.1
Buffalo, N.Y. 1,349 17.2 9.3
Austin, Texas 296 26.8 9.1
Chicago, IL. 6,975 29.1 8.4
Columbia, S.C. 323 18.7 7.7
Detroit, Mich. 4,200 32.6 11.3
Gary, Indiana 633 32.5 8.4

Fit a multiple linear regression model to these data. That is, fit a model of the form

Y=β 0 +β 1 x 1 +β 2 x 2 +e

whereYis the suicide rate,x 1 is the population, andx 2 is the divorce rate.

SOLUTION We run Program 9.10, and results are shown in Figures 9.15, 9.16, and 9.17.
Thus the estimated regression line is

Y=3.5073−.0002x 1 +.2609x 2

The value ofβ 1 indicates that the population does not play a major role in predicting the
suicide rate (at least when the divorce rate is also given). Perhaps the population density,
rather than the actual population, would have been more useful. ■