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respectively, where 2 is computed from Equation 11-13. The Minitab computer output in Table 11-2 reports the estimated standard errors of the slope and intercept under the column heading SE coeff.
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11-4 11-5
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SOME COMMENTS ON USES OF REGRESSION (CD ONLY) HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION
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An important part of assessing the adequacy of a linear regression model is testing statistical hypotheses about the model parameters and constructing certain con dence intervals. Hypothesis testing in simple linear regression is discussed in this section, and Section 11-6 presents methods for constructing con dence 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 normally and independently distributed with mean zero and variance 2, abbreviated NID(0, 2).
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11-5.1
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Use of t-Tests
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Suppose we wish to test the hypothesis that the slope equals a constant, say, priate hypotheses are H0: H1:
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(11-18)
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11-5 HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION
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where we have assumed a two-sided alternative. Since the errors i are NID(0, 2), it follows 2 directly that the observations Yi are NID( 0 ). Now 1 is a linear combination of 1xi, independent normal random variables, and consequently, 1 is N( 1, 2 Sxx), using the bias and variance properties of the slope discussed in Section 11-3. In addition, 1n 22 2 2 has a chi-square distribution with n 2 degrees of freedom, and 1 is independent of 2 . As a result of those properties, the statistic
T0 follows the t distribution with n H0: 1 1,0 if
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would reject
(11-20)
where t0 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
se1 1 2
A similar procedure can be used to test hypotheses about the intercept. To test H0: H1: we would use the statistic T0 B
0 0 0,0 0,0
(11-21)
1 cn
x d Sxx
se1 0 2
(11-22)
and reject the null hypothesis if the computed value of this test statistic, t0, is such that 0 t0 0 t 2,n 2. Note that the denominator of the test statistic in Equation 11-22 is just the standard error of the intercept. A very important special case of the hypotheses of Equation 11-18 is H0: H1:
0 0 (11-23)
These hypotheses relate to the signi cance of regression. Failure to reject H0: 1 0 is equivalent to concluding that there is no linear relationship between x and Y. This situation is illustrated in Fig. 11-5. Note that this may imply either that x is of little value in explaining the variation in Y and that the best estimator of Y for any x is y Y (Fig. 11-5a) or that the true relationship between x and Y is not linear (Fig. 11-5b). Alternatively, if H0: 1 0 is rejected, this implies that x is of value in explaining the variability in Y (see Fig. 11-6). Rejecting H0: 0 could mean either that the straight-line model is adequate (Fig. 11-6a) or that, 1
CHAPTER 11 SIMPLE LINEAR REGRESSION AND CORRELATION
Figure 11-5 The hypothesis H0: 1 0 is not rejected.
although there is a linear effect of x, better results could be obtained with the addition of higher order polynomial terms in x (Fig. 11-6b). EXAMPLE 11-2 We will test for signi cance of regression using the model for the oxygen purity data from Example 11-1. The hypotheses are H0: H1: and we will use
0.01. From Example 11-1 and Table 11-2 we have
14.97 2
20, Sxx
0.68088,
so the t-statistic in Equation 10-20 becomes t0
1 2
se1 1 2
21.18 0.68088 14.947
Since the reference value of t is t0.005,18 2.88, the value of the test statistic is very far into the critical region, implying that H0: 1 0 should be rejected. The P-value for this test is P 1.23 10 9 . This was obtained manually with a calculator. Table 11-2 presents the Minitab output for this problem. Notice that the t-statistic value for the slope is computed as 11.35 and that the reported P-value is P 0.000. Minitab also reports the t-statistic for testing the hypothesis H0: 0 0. This statistic is computed from Equation 11-22, with 0,0 0, as t0 46.62. Clearly, then, the hypothesis that the intercept is zero is rejected.