382 Chapter 8 Confidence Intervals Explain It Again Computing the sample size: When computing the necessary sample size, use a value of p̂ from a previously drawn sample if one is available. Otherwise, use p̂ = 0.5. SUMMARY Let m be the desired margin of error, and let z��∕2 be the critical value. The sample size n needed so that a confidence interval for a proportion will have margin of error approximately equal to m is n = ̂p(1 − ̂p) (z��∕2 m )2 if a value for ̂p is available n = 0.25 (z��∕2 m )2 if no value for ̂p is available (This is equivalent to assuming that ̂p = 0.5.) If the value of n given by the formula is not a whole number, round it up to the nearest whole number. By rounding up, we can be sure that the margin of error is no greater than the desired value m. EXAMPLE 8.15 Find the necessary sample size Example 8.14 described a sample of 517 music teachers, 403 of whom believe that video games have a positive effect on music education. Estimate the sample size needed so that a 95% confidence interval will have a margin of error of 0.03. Solution The desired level is 95%. The critical value is therefore z��∕2 = 1.96. We now compute ̂p: ̂p = 403 517 = 0.779497 The desired margin of error is m = 0.03. Since we have a value for ̂p, we substitute ̂p = 0.779497, z��∕2 = 1.96, and m = 0.03 into the formula n = ̂p(1 − ̂p) (z��∕2 m )2 and obtain n = (0.779497)(1 − 0.779497) (1.96 0.03 )2 = 733.67 (round up to 734) We estimate that we need to sample 734 teachers to obtain a 95% confidence interval with a margin of error of 0.03. EXAMPLE 8.16 Find the necessary sample size We plan to sample music teachers in order to construct a 95% confidence interval for the proportion who believe that listening to hip-hop music has a positive effect on music education. We have no value of ̂p available. Estimate the sample size needed so that a 95% confidence interval will have a margin of error of 0.03. Explain why the estimated sample size in this example is larger than the one in Example 8.15. Solution The desired level is 95%. The critical value is therefore z��∕2 = 1.96. The desired margin of error is m = 0.03. Since we have no value of ̂p, we substitute the values z��∕2 = 1.96 and m = 0.03 into the formula n = 0.25 (z��∕2 m )2
navidi_monk_elementary_statistics_2e_ch7-9
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