Applied Statistics and Probability for Engineers

384 CHAPTER 11 SIMPLE LINEAR REGRESSION AND CORRELATION

Now consider the variance of. Since we have assumed that V(i)^2 , it follows that

V(Yi)^2 , and it can be shown that

(11-16)

For the intercept, we can show that

(11-17)

Thus, is an unbiased estimator of the intercept  0. The covariance of the random variables

and is not zero. It can be shown (see Exercise 11-69) that cov( )  ^2.

The estimate of ^2 could be used in Equations 11-16 and 11-17 to provide estimates of the

variance of the slope and the intercept. We call the square roots of the resulting variance esti-

mators the estimated standard errorsof the slope and intercept, respectively.

ˆ 0 ˆ 1 ˆ 0 , ˆ 1 x Sxx

ˆ 0

E 1 ˆ 02  0 and V 1 ˆ 02 ^2 c

1

n

x^2

Sxx

d

V 1 ˆ 12 

^2

Sxx

ˆ 1

In simple linear regression the estimated standard error of the slopeand the

estimated standard error of the interceptare

respectively, where ˆ^2 is computed from Equation 11-13.

se 1 ˆ 12 

B

ˆ^2

Sxx

and se 1 ˆ 02 

B

ˆ^2 c

1

n

x^2

Sxx

d

Definition

The Minitab computer output in Table 11-2 reports the estimated standard errors of the slope

and intercept under the column heading “SEcoeff.”

11-4 SOME COMMENTS ON USES OF REGRESSION (CD ONLY)

11-5 HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION

An important part of assessing the adequacy of a linear regression model is testing statistical hy-

potheses about the model parameters and constructing certain confidence intervals. Hypothesis

testing in simple linear regression is discussed in this section, and Section 11-6 presents meth-

ods for constructing confidence intervals. To test hypotheses about the slope and intercept of the

regression model, we must make the additional assumption that the error component in the

model, , is normally distributed. Thus, the complete assumptions are that the errors are nor-

mally and independently distributed with mean zero and variance ^2 , abbreviated NID(0, ^2 ).

11-5.1 Use of t-Tests

Suppose we wish to test the hypothesis that the slope equals a constant, say, 1,0. The appro-

priate hypotheses are

H 1 :  1 1,0 (11-18)

H 0 :  1 1,0

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