594 CHAPTER 11 Goodness-of-Fit and Contingency Tables The null hypothesis and alternative hypothesis are as follows: H0: Autism is independent of the MMR vaccine. H1: Autism and the MMR vaccine are dependent. The significance level is a = 0.05. Because the data in Table 11-6 are in the form of a contingency table, we use the x2 distribution with this test statistic: x2 = a1O - E22 E = 125 - 18.34022 18.340 + g+ 11427 - 1420.34022 1420.340 = 3.198 P-Value from Technology If using technology, results typically include the x2 test statistic and the P-value. For example, see the accompanying TI-83>84 Plus display showing the test statistic is x2 = 3.198 and the P-value is 0.074. P-Value from Table A-4 If using Table A-4 instead of technology, first find the number of degrees of freedom: 1r - 121c - 12 = 12 - 1212 - 12 = 1 degree of freedom. Because the test statistic of x2 = 3.198 is less than 3.841 in Table A-4 for the row corresponding to 1 degree of freedom, we know that P@value 7 0.05. Because the P-value is greater than the significance level of 0.05, we fail to reject the null hypothesis of independence between autism and the vaccine. Critical Value If using the critical value method of hypothesis testing, the critical value of x2 = 3.841 is found from Table A-4, with a = 0.05 in the right tail and the number of degrees of freedom given by 1r - 121c - 12 = 12 - 1212 - 12 = 1. The test statistic of x2 = 3.198 does not fall in the critical region bounded by the critical value x2 = 3.841, so we fail to reject the null hypothesis of independence between autism and the vaccine. YOUR TURN. Do Exercise 5 “Lie Detector.” INTERPRETATION Based on the data in Table 11-1, it appears that autism is not linked to the MMR vaccine. The data support the conclusion of the study that “these findings indicate no harmful association between MMR vaccine receipt and ASD (autism spectrum disorder) even among children already at higher risk for ASD.” Within days of this writing, Peter Marks, director of the Center for Biologics Evaluation and Research for the FDA said that “vaccinating against measles, mumps and rubella not only protects us and our children, it protects people who can’t be vaccinated, including children with compromised immune systems due to illness and its treatment, such as cancer.” TI-83, 84 2 : 2 Contingency Table vs Two Proportions The data in Table 11-1 can be expressed as two proportions: ■ Unvaccinated Group: P(autism) = 25>387 ■ Vaccinated Group: P(autism) = 64>1491 Because we now have two proportions, why can’t we use the methods of Section 9-1 “Two Proportions” instead of the methods of this section? Well, we can! See Exercise 21 for verifying that results from the two different methods are equivalent.
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