396 CHAPTER 8 Hypothesis Testing StatCrunch Step 4: The significance level is a = 0.05. Step 5: Because the claim involves the proportion p, the statistic relevant to this test is the sample proportion pn and the sampling distribution of sample proportions can be approximated by the normal distribution. Step 6: Technology If using technology, the test statistic and the P-value will be provided. See the accompanying results from StatCrunch showing that the test statistic is z = -2.41 (rounded) and the P@value = 0.008. YOUR TURN. Do Exercise 9 “Cursed Movie.” Table A-2 If technology is not available, proceed as follows to conduct the hypothesis test using the P-value method summarized in Figure 8-1 on page 376. The test statistic z = -2.41 is calculated as follows: z = pn - p Apq n = 0.292 - 0.30 B10.30210.702 19,136 = -2.41 Refer to Figure 8-3 on page 380 for the procedure for finding the P-value. For this left-tailed test, the P-value is the area to the left of the test statistic. Using Table A-2, we see that the area to the left of z = -2.41 is 0.0080, so the P-value is 0.0080. Step 7: Because the P-value of 0.0080 is less than or equal to the significance level of 0.05, we reject the null hypothesis. INTERPRETATION Because we reject the null hypothesis, we support the alternative hypothesis. We therefore conclude that there is sufficient evidence to support the claim that fewer than 30% of adults have sleepwalked. Critical Value Method If we were to repeat Example 1 using the critical value method of testing hypotheses, we would see that in Step 6 the critical value is z = -1.645, which can be found from technology or Table A-2. In Step 7 we would reject the null hypothesis because the test statistic of z = -2.41 would fall within the critical region bounded by z = -1.645. We would then reach the same conclusion given in Example 1. Confidence Interval Method If we were to repeat Example 1 using the confidence interval method, we would use a 90% confidence level because we have a left-tailed test. (See Table 8-1.) We get this 90% confidence interval: 0.287 6 p 6 0.297. Because the entire range of the confidence interval falls below 0.30, there is sufficient evidence to support the claim that fewer than 30% of adults have sleepwalked. Lie Detectors and the Law Why not simply require all criminal suspects to take polygraph (lie detector) tests and eliminate trials by jury? According to the Council of Scientific Affairs of the American Medical Association, when lie detectors are used to determine guilt, accuracy can range from 75% to 97%. However, a high accuracy rate of 97% can still result in a high percentage of false positives, so it is possible that 50% of innocent subjects incorrectly appear to be guilty. Such a high chance of false positives rules out the use of polygraph tests as the single criterion for determining guilt. W r c p p d a
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