304 CHAPTER 6 Confidence Intervals Sample Size For the same sample statistics, as the level of confidence increases, the confidence interval widens. As the confidence interval widens, the precision of the estimate decreases. One way to improve the precision of an estimate without decreasing the level of confidence is to increase the sample size. But how large a sample size is needed to guarantee a certain level of confidence for a given margin of error? By using the formula for the margin of error E = zc s1 n a formula can be derived (see Exercise 59) to find the minimum sample size n, as shown in the next definition. Given a c@confidence level and a margin of error E, the minimum sample size n needed to estimate the population mean m is n = a zc s E b 2 . If n is not a whole number, then round n up to the next whole number (see Example 6). Also, when s is unknown, you can estimate it using s, provided you have a preliminary random sample with at least 30 members. Finding a Minimum Sample Size to Estimate M Determining a Minimum Sample Size The researcher in Example 1 wants to estimate the mean number of hours spent per week on required athletic activities by all student-athletes in the conference. How many student-athletes must be included in the sample to be 95% confident that the sample mean is within 0.5 hour of the population mean? SOLUTION Using c = 0.95, zc = 1.96, s = 2.4 (from Example 2), and E = 0.5, you can solve for the minimum sample size n. n = a zc s E b 2 = a 1.96# 2.4 0.5 b 2 ≈ 88.51 Because n is not a whole number, round up to 89. So, the researcher needs at least 89 student-athletes in the sample. Interpretation The researcher already has 40 student-athletes, so the sample needs 49 more members. Note that 89 is the minimum number of student-athletes to include in the sample. The researcher could include more, if desired. TRY IT YOURSELF 6 In Example 6, how many student-athletes must the researcher include in the sample to be 95% confident that the sample mean is within 0.75 hour of the population mean? Compare your answer with Example 6. Answer: Page A40 EXAMPLE 6
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