Applied Statistics and Probability for Engineers

11-5 HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION 385

where we have assumed a two-sided alternative. Since the errors iare NID(0, ^2 ), it follows

directly that the observations Yiare NID( 0  1 xi, ^2 ). Now is a linear combination of

independent normal random variables, and consequently, is N( 1 , ^2 Sxx), using the bias

and variance properties of the slope discussed in Section 11-3. In addition, has

a chi-square distribution with n 2 degrees of freedom, and is independent of. As a

result of those properties, the statistic

(11-19)

follows the tdistribution with n 2 degrees of freedom under H 0 :  1 1,0. We would reject

H 0 :  1 1,0if

(11-20)

where t 0 is computed from Equation 11-19. The denominator of Equation 11-19 is the standard

error of the slope, so we could write the test statistic as

A similar procedure can be used to test hypotheses about the intercept. To test

(11-21)

we would use the statistic

(11-22)

and reject the null hypothesis if the computed value of this test statistic, t 0 , is such that

. Note that the denominator of the test statistic in Equation 11-22 is just the stan-
dard error of the intercept.
A very important special case of the hypotheses of Equation 11-18 is

(11-23)

These hypotheses relate to the significance of regression.Failure to reject H 0 :  1 0 is

equivalent to concluding that there is no linear relationship between xand Y. This situation is

illustrated in Fig. 11-5. Note that this may imply either that xis of little value in explaining the

variation in Yand that the best estimator of Yfor any xis (Fig. 11-5a) or that the true

relationship between xand Yis not linear (Fig. 11-5b). Alternatively, if H 0 :  1 0 is rejected,

this implies that xis of value in explaining the variability in Y(see Fig. 11-6). Rejecting H 0 :

 1 0 could mean either that the straight-line model is adequate (Fig. 11-6a) or that,

yˆY

H 1 :  10

H 0 :  1  0

0 t 00

 t 2,n     2

T 0 

ˆ 0    0,0

B

ˆ^2 c

1

n

x^2

Sxx

d



ˆ 0    0,0

se 1 ˆ 02

H 1 :  0 0,0

H 0 :  0 0,0

T 0 

ˆ 1    1,0

se 1 ˆ 12

0 t 00

 t 2,n     2

T 0 

ˆ 1    1,0

2 ˆ^2 Sxx

ˆ 1 ˆ^2

1 n 22 ˆ^2 ^2

ˆ 1

ˆ 1

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