Introduction to Probability and Statistics for Engineers and Scientists

378 Chapter 9: Regression

near+1 indicated that largexvalues were strongly associated with largeY values and
smallxvalues were strongly associated with smallYvalues, whereas a value near−1 indi-
cated that largexvalues were strongly associated with smallYvalues and smallxvalues
with largeYvalues.
In the notation of this chapter,

r=

SxY

√

SxxSYY

Upon using identity (9.3.4):

SSR=

SxxSYY−S^2 xY

Sxx

we see that

r^2 =

SxY^2

SxxSYY

=

SxxSYY−SSRSxx

SxxSYY

= 1 −

SSR

SYY

=R^2

That is,

|r|=

√

R^2

and so, except for its sign indicating whether it is positive or negative, the sample correla-
tion coefficient is equal to the square root of the coefficient of determination. The sign of
ris the same as that ofB.
The above gives additional meaning to the sample correlation coefficient. For instance,
if a data set has its sample correlation coefficientrequal to .9, then this implies that a
simple linear regression model for these data explains 81 percent (sinceR^2 =.9^2 =.81)
of the variation in the response values. That is, 81 percent of the variation in the response
values is explained by the different input values.

9.6Analysis of Residuals: Assessing the Model

The initial step for ascertaining whether or not the simple linear regression model

Y=α+βx+e, e∼N(0,σ^2 )