7-1 Estimating a Population Proportion 321 SOLUTION a. With a 95% confidence level, we have a = 0.05, so za>2 = 1.96. Also, the margin of error is E = 0.03, which is the decimal equivalent of “three percentage points.” The prior survey suggests that pn = 0.79, so qn = 0.21 (found from qn = 1 - 0.79). Because we have an estimated value of pn, we use Formula 7-2 as follows: n = 3za>24 2 pnqn E2 = 31.9642 10.79210.212 0.032 = 708.135 = 709 1rounded up2 We must obtain a simple random sample that includes at least 709 adults. b. With no prior knowledge of pn (or qn), we use Formula 7-3 as follows: n = 3za>24 2 # 0.25 E2 = 31.9642 # 0.25 0.032 = 1067.11 = 1068 1rounded up2 We must obtain a simple random sample that includes at least 1068 adults. INTERPRETATION To be 95% confident that our sample percentage is within three percentage points of the true percentage for all adults, we should obtain a simple random sample of 1068 adults, assuming no prior knowledge. By comparing this result to the sample size of 709 found in part (a), we can see that if we have no knowledge of a prior study, a larger sample is required to achieve the same results as when the value of pn can be estimated. YOUR TURN. Do Exercise 31 “Wiggle Your Ears.” Role of the Population Size N Formulas 7-2 and 7-3 are remarkable because they show that the sample size does not depend on the size (N) of the population; the sample size depends on the desired confidence level, the desired margin of error, and sometimes the known estimate of pn. (See Exercise 39 “Finite Population Correction Factor” for dealing with cases in which a relatively large sample is selected without replacement from a finite population, so the sample size n does depend on the population size N.) PART 2 Better-Performing Confidence Intervals Disadvantage of Wald Confidence Interval Wald Coverage Probability Is Too Liberal A concept used to gauge the quality of a confidence interval is the coverage probability, defined as follows. CAUTION Try to avoid these three common errors when calculating sample size: 1. Don’t make the mistake of using E = 3 as the margin of error corresponding to “three percentage points.” If the margin of error is three percentage points, use E = 0.03. 2. Be sure to substitute the critical z score for za>2. For example, when working with 95% confidence, be sure to replace za>2 with 1.96. Don’t make the mistake of replacing za>2 with 0.95 or 0.05. 3. Be sure to round up to the next higher integer; don’t round off using the usual round-off rules. Round 708.135 to 709. Curbstoning The glossary for the Census defines curbstoning as “the practice by which a census enumerator fabricates a questionnaire for a residence without actually visiting it.” Curbstoning occurs when a census enumerator sits on a curbstone (or anywhere else) and fills out survey forms by making up responses. Because data from curbstoning are not real, they can affect the validity of the Census. The extent of curbstoning has been investigated in several studies, and one study showed that about 4% of Census enumerators practiced curbstoning at least some of the time. The methods of Section 7-1 assume that the sample data have been collected in an appropriate way, so if much of the sample data have been obtained through curbstoning, then the resulting confidence interval estimates might be very flawed.
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