7-1 Estimating a Population Proportion 313 The sample proportion pn is the best point estimate of the population proportion p. Unbiased Estimator We use pn as the point estimate of p because it is unbiased and it is the most consistent of the estimators that could be used. (An unbiased estimator is a statistic that targets the value of the corresponding population parameter in the sense that the sampling distribution of the statistic has a mean that is equal to the corresponding population parameter. The statistic pn targets the population proportion p.) The sample proportion pn is the most consistent estimator of p in the sense that the standard deviation of sample proportions tends to be smaller than the standard deviation of other unbiased estimators of p. CP EXAMPLE 1 Online Courses The Chapter Problem included reference to a Sallie Mae survey of 950 undergraduate college students, and 53% of them said that they take online courses. Based on that result, find the best point estimate of the proportion of all undergraduate college students who take online courses. SOLUTION Because the sample proportion is the best point estimate of the population proportion, we conclude that the best point estimate of p is 0.53. (If using the sample results to estimate the percentage of all undergraduate college students who take online courses, the best point estimate is 53%.) YOUR TURN. Find the point estimate in Exercise 13 “Tennis Challenges.” Confidence Interval Why Do We Need Confidence Intervals? In Example 1 we saw that 0.53 is our best point estimate of the population proportion p, but we have no indication of how good that “best” estimate is. By giving us a range of values associated with a probability, a confidence interval gives us a much better sense of how good an estimate is. DEFINITION A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI. Here’s an example of a confidence interval illustrated later in Example 2: The 0.95 (or 95%) confidence interval estimate of a population proportion p is 0.499 * p * 0.562. Two key elements of a confidence interval are (1) the confidence level and the associated critical value, and (2) the margin of error. These key elements are described as follows. d Push Polling Push polling is the practice of political campaigning under the guise of a poll. Its name is derived from its objective of pushing voters away from opposition candidates by asking loaded questions designed to discredit them. This survey question was used in one campaign: “Please tell me if you would be more likely or less likely to vote for Roy Romer if you knew that Governor Romer appoints a parole board which has granted early release to an average of four convicted felons per day every day since Romer took office.” The National Council on Public Polls says that push polls are unethical. Reputable pollsters do not approve of push polling. DEFINITION The confidence level is the probability 1 - a (such as 0.95, or 95%) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. (The confidence level is also called the degree of confidence, or the confidence coefficient.) Confidence Level
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