9-1 Two Proportions 443 Equivalent Methods When testing a claim about two population proportions: ■ The P-value method and the critical value method are equivalent. ■ The confidence interval method is not equivalent to the P-value method or the critical value method. Recommendation: If you want to test a claim about two population proportions, use the P-value method or critical value method; if you want to estimate the difference between two population proportions, use the confidence interval method. Hypothesis Tests For tests of hypotheses made about two population proportions, we consider only tests having a null hypothesis of p1 = p2 (so the null hypothesis is H0: p1 = p2). With the assumption that p1 = p2, the estimates of p n 1 and p n 2 are combined to provide the best estimate of the common value of pn 1 and p n 2, and that combined value is the pooled sample proportion p given in the preceding Key Elements box. The following example will help clarify the roles of x1, n1, p n 1, p, and so on. Note that with the assumption of equal population proportions, the best estimate of the common population proportion is obtained by pooling both samples into one big sample, so that p is the estimator of the common population proportion. P-Value Method P-Value: P-values are automatically provided by technology. If technology is not available, use Table A-2 (standard normal distribution) and find the P-value using the procedure given in Figure 8-3 on page 380. Critical Values: Use Table A-2. (Based on the significance level a, find critical values by using the same procedures introduced in Section 8-1.) Confidence Interval Estimate of p1 − p2 The confidence interval estimate of the difference p1 - p2 is 1pn 1 - p n 22 - E 6 1p1 - p22 6 p n 1 - p n 22 + E where the margin of error E is given by E = za>2Apn 1q n 1 n1 + pn 2q n 2 n2 . Rounding: Round the confidence interval limits to three significant digits. Table 9-1 in the Chapter Problem includes these two sample proportions of success for the e-cigarette treatment group and the nicotine replacement treatment group. Proportion of Success (not smoking after 52 weeks) E-Cigarette Group: pn 1 = 79>438 = 0.180 Nicotine Replacement Group: pn 2 = 44>446 = 0.099 Use a 0.05 significance level and the P-value method to test the claim that there is no difference in success rates between the two treatment groups. CP EXAMPLE 1 Is There a Difference Between Success Rates of E-Cigarettes and Nicotine Replacement for Smokers Who Try to Stop Smoking? continued Is the Cure Worse Than the Disease? This chapter includes methods showing that for the accompanying Chapter Problem, e-cigarettes are more effective than nicotine replacement treatments for smokers trying to stop smoking. However, critical thinking requires that we should look at the larger picture and ask whether it is wise to use e-cigarettes as a treatment. Both regular cigarettes and e-cigarettes contain nicotine, which is an addictive toxic substance that raises blood pressure and increases heart rate. As this is being written, there have been reports of deaths and sickness that appear to be related to the use of e-cigarettes. Massachusetts has placed a temporary ban on the sale of e-cigarettes. Based on the evidence available at the time of this writing, it appears that the use of e-cigarettes should not be recommended.
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