322 CHAPTER 7 Estimating Parameters and Determining Sample Sizes DEFINITION The coverage probability of a confidence interval estimate of the population proportion p is the actual proportion of such confidence intervals that contain the true population proportion. If we select a specific confidence level, such as 0.95 (or 95%), we would like to get the actual coverage probability equal to our desired confidence level. Disadvantage of the Wald Confidence Interval (described in Part 1): Too “Liberal”: The coverage probability of a Wald confidence interval is usually less than or equal to the selected confidence level. For example, if we select a 95% confidence level, we usually get 95% or fewer of confidence intervals containing the population proportion p. For this reason, the Wald confidence interval is rarely used in professional applications and professional journals. Better-Performing Confidence Intervals Important note about exercises: Except for some Beyond the Basics exercises, the exercises for this Section 7-1 are based on the method for constructing a Wald confidence interval as described in Part 1, not the confidence intervals described here. It is recommended that students learn the methods presented earlier, but recognize that there are better methods available, and they can be used with suitable technology. Plus Four Method The plus four confidence interval performs better than the Wald confidence interval in the sense that its coverage probability is closer to the confidence level that is used. Procedure for the Plus Four Method 1. Add 2 to the number of successes x. 2. Add 2 to the number of failures (so the number of trials n is increased by 4). 3. Proceed to find the confidence interval using the same method described in Part 1. The plus four confidence interval has coverage probabilities similar to those for the Wilson score confidence interval that follows. Wilson Score Another confidence interval that performs better than the Wald CI is the Wilson score confidence interval. The upper and lower confidence interval limits are calculated using the following: pn + z2 a>2 2n { za>2B pnqn + z2 a>2 4n n 1 + z2 a>2 n The Wilson score confidence interval performs better than the Wald CI in the sense that the coverage probability is closer to the confidence level. Given its calculation complexity, it is easy to see why this superior Wilson score confidence interval is not used much in introductory statistics courses. The complexity of the above expression can be circumvented by using some technologies, such as Statdisk, that provide Wilson score confidence interval results. Real-World Margin of Error The New York Times reported on research showing that the margin of error in a typical survey is about twice as large as we might expect with our theoretical calculations. In a typical survey, we say that we have 95% confidence that the margin of error is three percentage points, but David Rothschild and Sharad Goel say that the real-world margin of error is about six percentage points. They base that conclusion on an analysis of 4221 surveys conducted late in election campaigns. They compared survey results to actual election results to identify the larger margin of error. Two explanations: (1) There is a discrepancy between those who are surveyed and those who actually vote; (2) There is a nonresponse error that occurs when the likelihood of responding to the survey is somehow related to how the survey questions are answered. They found that those who support a trailing candidate are less likely to respond to the survey. There is also bias created with the wording of survey questions. These are very real factors that could create a large discrepancy between a margin of error that we theoretically expect and the real-world margin of error that actually occurs.
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