Applied Statistics and Probability for Engineers

402 CHAPTER 11 SIMPLE LINEAR REGRESSION AND CORRELATION

We note that the estimators of the intercept and slope in Equations 11-41 and 11-42 are

identical to those given by the method of least squares in the case where Xwas assumed to be

a mathematical variable. That is, the regression model with Yand Xjointly normally distrib-

uted is equivalent to the model with Xconsidered as a mathematical variable. This follows

because the random variables Ygiven Xxare independently and normally distributed with

mean  0  1 xand constant variance These results will also hold for any joint distribu-

tion of Yand Xsuch that the conditional distribution of Ygiven Xis normal.

It is possible to draw inferences about the correlation coefficient in this model. The

estimator of is the sample correlation coefficient

(11-43)

Note that

(11-44)

so the slope is just the sample correlation coefficient Rmultiplied by a scale factor that is

the square root of the “spread” of the Yvalues divided by the “spread” of the Xvalues.Thus,

and Rare closely related, although they provide somewhat different information. The

sample correlation coefficient Rmeasures the linear association between Yand X, while

measures the predicted change in the mean of Yfor a unit change in X. In the case of a math-

ematical variable x, Rhas no meaning because the magnitude of Rdepends on the choice of

spacing of x. We may also write, from Equation 11-44,

which is just the coefficient of determination. That is, the coefficient of determination R^2 is

just the square of the correlation coefficient between Yand X.

It is often useful to test the hypotheses

(11-45)

The appropriate test statistic for these hypotheses is

H 1 : 0

H 0 :  0

R^2 ˆ^21

SXX

SYY



ˆ 1 SXY

SST



SSR

SST

ˆ 1

ˆ 1

ˆ 1

ˆ 1 a

SST

SXX

b

(^1)  2
R
R
a
n
i 1
Yi 1 XiX 2
ca
n
i 1
1 XiX (^22) a
n
i 1
1 YiY 22 d
(^1)  2 
SXY
1 SXXSST 21 ^2
^2 Y 0 x.
which has the tdistribution with n2 degrees of freedom if H 0 : 0 is true. Therefore, we
would reject the null hypothesis if t 0  t 2,n 2. This test is equivalent to the test of the
T 0  (11-46)
R 1 n 2
21 R^2
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