348 CHAPTER 7 Estimating Parameters and Determining Sample Sizes Confidence Interval for Estimating a Population Standard Deviation or Variance Objective Construct a confidence interval estimate of a population standard deviation or variance. Notation s = population standard deviation s = sample standard deviation n = number of sample values x2 L = left-tailed critical value of x2 s 2 = population variance s2 = sample variance E = margin of error x2 R = right-tailed critical value of x2 Requirements 1. The sample is a simple random sample. 2. The population must have normally distributed values (even if the sample is large). The requirement of a normal distribution is much stricter here than in earlier sections, so large departures from normal distributions can result in large errors. (If this requirement is not satisfied, use the bootstrap method described in Section 7-4.) Confidence Interval Estimate of the Population Variance S2 1n - 12 s2 x2 R 6 s 2 6 1n - 12s2 x2 L Confidence Interval Estimate of the Population Standard Deviation S B1n - 12 s2 x2 R 6 s 6 B1n - 12 s2 x2 L Round-Off Rule 1. Original Data: When using the original set of data values, round the confidence interval limits to one more decimal place than is used for the original data. 2. Summary Statistics: When using the summary statistics (n, s), round the confidence interval limits to the same number of decimal places used for the sample standard deviation. KEY ELEMENTS Procedure for Constructing a Confidence Interval for S or S2 Confidence intervals can be easily constructed with technology or they can be constructed by using Table A-4 with the following procedure. 1. Verify that the two requirements are satisfied: The sample is a random sample from a normally distributed population. 2. Using n - 1 degrees of freedom, find the critical values x2 R and x2 L that correspond to the desired confidence level (as in Example 1). 3. To get a confidence interval estimate of s 2, use the following: 1n - 12s2 x2 R 6 s 2 6 1n - 12s2 x2 L
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