7-3 Estimating a Population Standard Deviation or Variance 351 The two preceding inequalities can be combined into one inequality to get the format of the confidence interval used in this section: 1n - 12s2 x2 R 6 s 2 6 1n - 12s2 x2 L Determining Sample Size The procedures for finding the sample size necessary to estimate s are much more complex than the procedures given earlier for means and proportions. For normally distributed populations, Table 7-2 or the formula given in Exercise 24 “Finding Sample Size” on page 355 can be used. Statdisk also provides sample sizes. With Statdisk, select Analysis, Sample Size Determination, and then Estimate Standard Deviation. Minitab, Excel, StatCrunch, and the TI-83>84 Plus calculator do not provide such sample sizes. TABLE7-2 Finding Sample Size S To be 95% confident that s is within . . . of the value of S, the sample size n should be at least 1% 19,205 5% 768 10% 192 20% 48 30% 21 40% 12 50% 8 To be 99% confident that s is within . . . of the value of S, the sample size n should be at least 1% 33,218 5% 1,336 10% 336 20% 85 30% 38 40% 22 50% 14 EXAMPLE 3 Finding Sample Size for Estimating S We want to estimate the standard deviation s of all pulse rates of adult women. We want to be 99% confident that our estimate is within 5% of the true value of s. How large should the sample be? Assume that the population is normally distributed. SOLUTION From Table 7-2, we can see that 99% confidence and an error of 5% for s correspond to a sample of size 1336. We should obtain a simple random sample of 1336 pulse rates of adult women. YOUR TURN. Do Exercise 17 “IQ of Data Scientists.” Confidence Interval Estimate for Standard Deviation or Variance Access tech supplements, videos, and data sets at www.TriolaStats.com TECH CENTER Confidence Interval 1. Verify the distribution is normal using Data—Normality Assessment. 2. Click Analysis in the top menu. 3. Select Confidence Intervals from the dropdown menu and select Standard Deviation One Sample from the submenu. 4. Using Summary Statistics: Select the Use Summary Statistics tab and enter the desired confidence level, sample size, and sample standard deviation. Using Sample Data: Select the Use Data tab, enter the desired confidence level, and select the desired data column. 5. Click Evaluate. Statdisk Confidence Interval 1. Click Stat in the top menu. 2. Select Basic Statistics from the dropdown menu and select 1 Variance from the submenu. 3. Using Summary Statistics: Select Sample standard deviation or Sample Variance from the dropdown menu and enter sample size and sample standard deviation or sample variance. Using Sample Data: Select One or more samples, each in a column from the dropdown menu and select the desired data column(s). 4. Confirm Perform hypothesis test is not checked. 5. Click the Options button, enter the desired confidence level, and for Alternative Hypothesis select 3. 6. Click OK twice. The results include a confidence interval for the standard deviation and a confidence interval for the variance. Minitab Neither Excel nor XLSTAT has a function for generating a confidence interval estimate of standard deviation or variance. Excel continued
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