7-1 Estimating a Population Proportion 319 4. Except for relatively rare cases, the quality of the poll results depends on the sampling method and the size of the sample, but the size of the population is usually not a factor. CAUTION Never think that poll results are unreliable if the sample size is a small percentage of the population size. The population size is usually not a factor in determining the reliability of a poll. Finding the Point Estimate and E from a Confidence Interval Sometimes we want to better understand a confidence interval that might have been obtained from a journal article or technology. If we already know the confidence interval limits, the sample proportion (or the best point estimate) pn and the margin of error E can be found as follows: Point estimate of p: pn = (upper confidence interval limit) + (lower confidence interval limit) 2 Margin of error: E = (upper confidence interval limit) - (lower confidence interval limit) 2 EXAMPLE 3 Finding the Sample Proportion and Margin of Error The article “High-Dose Nicotine Patch Therapy,” by Dale, Hurt, et al. (Journal of the American Medical Association, Vol. 274, No. 17) includes this statement: “Of the 71 subjects, 70% were abstinent from smoking at 8 weeks (95% confidence interval [CI], 58% to 81%).” Use that statement to find the point estimate pn and the margin of error E. SOLUTION We get the 95% confidence interval of 0.58 6 p 6 0.81 from the given statement of “58% to 81%.” The point estimate pn is the value midway between the upper and lower confidence interval limits, so we get pn = (upper confidence limit) + (lower confidence limit) 2 = 0.81 + 0.58 2 = 0.695 The margin of error can be found as follows: E = (upper confidence limit) - (lower confidence limit) 2 = 0.81 - 0.58 2 = 0.115 Using Confidence Intervals for Hypothesis Tests A confidence interval can be used to address some claim made about a population proportion. For example, if sample results consist of 70 heads in 100 tosses of a coin, the resulting 95% confidence interval of 0.610 6 p 6 0.790 is evidence supporting the claim that the proportion of heads is different from 50% (because 0.50 is not contained within the confidence interval). Author Testifying in Court The author testified as an expert witness in New York State Supreme Court. He testified on behalf of a former student who lost an election apparently due to a misleading ballot format in one district. The author used a confidence interval as one item of evidence. When questioned about the confidence interval, the attorney opposing the former student argued that for my confidence interval, the 95% confidence level corresponded to a 5% error rate, and if you add that 5% to the margin of error of 3 percentage points, you get a total of 8%, which is far too high. That argument made no sense because the “5% error rate” and the “3 percentage point margin of error” are two totally different measures and it would be absurd to add them. In this case, lack of basic knowledge of statistics severely damaged the opposing attorney’s arguments, and my former student won his case.
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