SECTION 9.1 Correlation 479 The t-Test for a Correlation Coefficient In Example 4, you used 10 pairs of data to find r ≈ 0.874. Test the significance of this correlation coefficient. Use a = 0.05. SOLUTION The null and alternative hypotheses are H0: r = 0 (no correlation) and Ha: r ≠ 0 (significant correlation). Because there are 10 pairs of data in the sample, there are 10 - 2 = 8 degrees of freedom. Because the test is a two-tailed test, a = 0.05, and d.f. = 8, the critical values are -t0 = -2.306 and t0 = 2.306. The rejection regions are t 6 -2.306 and t 7 2.306. Using the t@test, the standardized test statistic is t = r B1 - r2 n - 2 Use the t@test for r. ≈ 0.874 B1 - 10.87422 10 - 2 Substitute 0.874 for r and 10 for n. ≈ 5.087. Round to three decimal places. You can check this result using technology. For instance, using a TI-84 Plus, you can find the standardized test statistic, as shown at the left. (Note that the result differs slightly due to rounding.) The figure below shows the location of the rejection regions and the standardized test statistic. D= 0.025 1 2 D= 0.025 1 2 0 1 2 3 −1 −2 −3 4 5 6 t ≈ 5.087 −t0 = −2.306 t0 = 2.306 t Because t is in the rejection region, you reject the null hypothesis. Interpretation There is enough evidence at the 5% level of significance to conclude that there is a significant linear correlation between gross domestic products and carbon dioxide emissions. TRY IT YOURSELF 7 In Try It Yourself 5, you calculated the correlation coefficient of the salaries and average home game attendances for the teams in Major League Baseball to be r ≈ 0.792. Test the significance of this correlation coefficient. Use a = 0.01. Answer: Page A42 In Example 7, you can use Table 11 in Appendix B to test the population correlation coefficient r. Given n = 10 and a = 0.05, the critical value from Table 11 is 0.632. Because 0 r0 ≈ 0.874 7 0.632, the correlation is significant. Note that this is the same result you obtained using a t-test for the population correlation coefficient r. EXAMPLE 7 Study Tip Be sure you see in Example 7 that rejecting the null hypothesis means that there is enough evidence that the correlation is significant. TI-84 PLUS y=ax+b ß≠0 and o≠0 t=5.099174604 p=9.3079858E-4 df=8 a=44.66291987 LinRegTTest
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