7-1 Estimating a Population Proportion 315 DEFINITION When using a sample statistic to estimate a population parameter, the margin of error, denoted by E, is the maximum likely amount of error (the amount by which the sample statistic misses the population parameter). When using a sample proportion pn to estimate a population proportion p, the difference between pn and the actual value of p is an error, and the margin of error is the maximum likely amount of that error. The margin of error E can be found by multiplying the critical value and the estimated standard deviation of sample proportions, as shown in Formula 7-1. (This estimate of a standard deviation of a sampling distribution of proportions is called a standard error of the sample proportions.) FORMULA 7-1 MARGIN OF ERROR E FOR PROPORTIONS E = za>2Bpnqn n c c Critical value Estimated standard deviation of sample proportions 4 Wald Confidence Interval When constructing a confidence interval using the margin of error E given in Formula 7-1, the result is called a Wald confidence interval. Part 2 of this section will discuss other methods for constructing a confidence interval. Interpreting a Confidence Interval We must be careful to interpret confidence intervals correctly. There is a correct interpretation and many different and creative incorrect interpretations of the confidence interval 0.499 6 p 6 0.562. Correct: “We are 95% confident that the interval from 0.499 to 0.562 actually does contain the true value of the population proportion p.” This is a short and acceptable way of saying that if we were to select many different random samples of size 950 and construct the corresponding confidence intervals, 95% of them would contain the population proportion p. In this correct interpretation, the confidence level of 95% refers to the success rate of the process used to estimate the population proportion. Wrong: “There is a 95% chance that the true value of p will fall between 0.499 and 0.562.” This is wrong because p is a population parameter with a fixed value; it is not a random variable with values that vary. Wrong: “95% of sample proportions will fall between 0.499 and 0.562.” This is wrong because the values of 0.499 and 0.562 result from one sample; they are not parameters describing the behavior of all samples. Confidence Level: The Process Success Rate A confidence level of 95% tells us that the process we are using should, in the long run, result in confidence interval limits that contain the true population proportion 95% of the time. Suppose that the true Shakespeare’s Vocabulary According to Bradley Efron and Ronald Thisted, Shakespeare’s writings included 31,534 different words. They used probability theory to conclude that Shakespeare probably knew at least another 35,000 words that he didn’t use in his writings. The problem of estimating the size of a population is an important problem often encountered in ecology studies, but the result given here is another interesting application. (See “Estimating the Number of Unseen Species: How Many Words Did Shakespeare Know?”; in Biometrika, Vol. 63, No. 3.) Th d b Margin of Error We now formally define the margin of error E that we have all encountered so often in media reports.
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