Section 8.3 Confidence Intervals for a Population Proportion 385 EXAMPLE 8.18 Construct a confidence interval with a small sample In a random sample of 10 businesses in a certain city, 6 of them had more than 15 employees. Use the small-sample method to construct a 95% confidence interval for the proportion of businesses in this city that have more than 15 employees. Solution The adjusted sample proportion is ̃p = x + 2 n + 4 = 6 + 2 10 + 4 = 0.5714 The critical value is z��∕2 = 1.96. The 95% confidence interval is therefore ̃p − z��∕2 √ ̃p(1 − ̃p) n + 4 < p < ̃p + z��∕2 √ ̃p(1 − ̃p) n + 4 √ 0.5714(1 − 0.5714) 0.5714 − 1.96 10 + 4 √ 0.5714(1 − 0.5714) < p < 0.5714 + 1.96 10 + 4 0.312 < p < 0.831 Check Your Understanding 6. In a simple random sample of 15 seniors from a certain college, 8 of them had found jobs. Use the small-sample method to construct a 95% confidence interval for the proportion of seniors at that college who have found jobs. (0.302, 0.751) Answer is on page 390. Using technology to implement the small-sample method Because the only difference between the small-sample method and the traditional method is the use of ̃p rather than ̂p, a software package or calculator such as the TI-84 Plus that uses the traditional method can be made to produce a confidence interval using the small-sample method. Simply input x + 2 for the number of individuals in the category of interest, and n + 4 for the sample size. The small-sample method is better overall The small-sample method can be used for any sample size, and recent research has shown that it has two advantages over the traditional method. First, the margin of error is smaller, because we divide by n + 4 rather than n. Second, the actual probability that the small-sample confidence interval covers the true population proportion is almost always at least as great as, or greater than, that of the traditional method. This holds for confidence levels of 90% or more, which are the levels commonly used in practice. For more information on this method, see the article ‘‘Approximate is Better Than ‘Exact’ for Interval Estimation of Binomial Proportions’’ (A. Agresti and B. Coull, The American Statistician, 52:119–126).
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