444 CHAPTER 9 Inferences from Two Samples SOLUTION REQUIREMENT CHECK We first verify that the three necessary requirements are satisfied. (1) The description of the selection of study subjects and the random assignment to groups confirms that the two samples can be treated as simple random samples for the purposes of this analysis. (2) The two samples are independent because subjects in the samples are not matched or paired in any way. (3) Consider a “success” to be a smoker who is not smoking after 52 weeks. For the e-cigarette group, the number of successes is 79 and the number of failures is 438 - 79 = 359, so they are both at least 5. For the nicotine replacement group, the number of successes is 44 and the number of failures 446 - 44 = 402, so they are both at least 5. The requirements are satisfied. The following steps are from the P-value method of testing hypotheses, which is summarized in Figure 8-1 on page 376. Step1: The claim of no significance difference between the two treatment groups can be expressed as p1 = p2. Step2: If p1 = p2 is false, then p1 ≠ p2. Step3: Because the claim of p1 ≠ p2 does not contain equality, it becomes the alternative hypothesis. The null hypothesis is the statement of equality, so we have H0: p1 = p2 H1: p1 ≠ p2 Step4: The significance level was specified as a = 0.05, so we use a = 0.05. (Who are we to argue?) Step5: Using Technology Steps 5 and 6 can be skipped when using technology; see the accompanying Statdisk display showing the test statistic and P-value. Statdisk Manual Calculation If not using technology, we use the normal distribution (with the test statistic given in the Key Elements box) as an approximation to the binomial distribution. We estimate the common value of p1 and p2 with the pooled sample estimate p calculated as shown below, with extra decimal places used to minimize rounding errors in later calculations. p = x1 + x2 n1 + n2 = 79 + 44 438 + 446 = 0.13914027 With p = 0.13914027, it follows that q = 1 - 0.13914027 = 0.86085973. The Lead Margin of Error Authors Stephen Ansolabehere and Thomas Belin wrote in their article “Poll Faulting” (Chance magazine) that “our greatest criticism of the reporting of poll results is with the margin of error of a single proportion (usually {3,) when media attention is clearly drawn to the lead of one candidate.” They point out that the lead is really the difference between two proportions (p1 - p2) and go on to explain how they developed the following rule of thumb: The lead is approximately 23 times larger than the margin of error for any one proportion. For a typical pre-election poll, a reported {3, margin of error translates to about {5, for the lead of one candidate over the other. They write that the margin of error for the lead should be reported. A S A a B in “
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