324 CHAPTER 6 Confidence Intervals Finding a Minimum Sample Size One way to increase the precision of a confidence interval without decreasing the level of confidence is to increase the sample size. Given a c@confidence level and a margin of error E, the minimum sample size n needed to estimate the population proportion p is n = npnqa zc Eb 2 . If n is not a whole number, then round n up to the next whole number (see Example 4). Also, note that this formula assumes that you have preliminary estimates of np and nq. If not, use np = 0.5 and nq = 0.5. Finding a Minimum Sample Size to Estimate p Determining a Minimum Sample Size You are running a political campaign and wish to estimate, with 95% confidence, the population proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the population proportion. Find the minimum sample size needed when (1) no preliminary estimate is available and (2) a preliminary estimate gives np = 0.31. Compare your results. SOLUTION 1. Because you do not have a preliminary estimate of np, use np = 0.5 and nq = 0.5. Using z c = 1.96 and E = 0.03, you can solve for n. n = npnqa zc Eb 2 = 10.5210.52a 1.96 0.03b 2 ≈ 1067.11 Because n is not a whole number, round up to the next whole number, 1068. 2. You have a preliminary estimate of np = 0.31. So, nq = 0.69. Using z c = 1.96 and E = 0.03, you can solve for n. n = npnqa zc Eb 2 = 10.31210.692a 1.96 0.03b 2 ≈ 913.02 Because n is not a whole number, round up to the next whole number, 914. Interpretation With no preliminary estimate, the minimum sample size should be at least 1068 registered voters. With a preliminary estimate of np = 0.31, the sample size should be at least 914 registered voters. So, you will need a larger sample size when no preliminary estimate is available. TRY IT YOURSELF 4 A researcher is estimating the population proportion of people in the United States who had private health care coverage over a recent 6-month period. The estimate must be accurate within 2% of the population proportion with 90% confidence. Find the minimum sample size needed when (1) no preliminary estimate is available and (2) a previous survey found that 62.2% of people in the United States had private health care coverage over a recent 6-month period. (Source: National Center for Health Statistics) Answer: Page A40 EXAMPLE 4 Study Tip The reason for using np = 0.5 and nq = 0.5 when no preliminary estimate is available is that these values yield the maximum value of the product npnq. (See Exercise 37.) In other words, without an estimate of np, you must use a larger sample.
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