SECTION 6.4 Confidence Intervals for Variance and Standard Deviation 333 Study Tip When you construct a confidence interval for a population variance or standard deviation, the general round-off rule is to round off to the same number of decimal places as the sample variance or standard deviation. Constructing Confidence Intervals You randomly select and weigh 30 samples of an allergy medicine. The sample standard deviation is 1.20 milligrams. Assuming the weights are normally distributed, construct 99% confidence intervals for the population variance and standard deviation. SOLUTION The area to the right of x 2 R is Area to the right of x 2 R = 1 - c 2 = 1 - 0.99 2 = 0.005 and the area to the right of x 2 L is Area to the right of x 2 L = 1 + c 2 = 1 + 0.99 2 = 0.995. Using the values n = 30, d.f. = 29, and c = 0.99, the critical values x 2 R and x 2 L are x 2 R = 52.336 and x 2 L = 13.121. Using these critical values and s = 1.20, the confidence interval for s 2 is Left Endpoint Right Endpoint 1n - 12s2 x 2 R = 130 - 1211.2022 52.336 1n - 12s2 x 2 L = 130 - 1211.2022 13.121 ≈ 0.80 ≈ 3.18 0.80 6 s 2 6 3.18. The confidence interval for s is Left Endpoint Right Endpoint B130 - 1211.2022 52.336 6 s 6 B130 - 1211.2022 13.121 0.89 6 s 6 1.78. You can check your answer using technology, as shown below using Minitab. MINITAB Test and CI for One Variance 99% Cl for S 99% Cl for S 2 N StDev Variance using Chi-Square using Chi-Square 30 1.20 1.44 (0.89, 1.78) (0.80, 3.18) Interpretation With 99% confidence, you can say that the population variance is between 0.80 and 3.18, and the population standard deviation is between 0.89 and 1.78 milligrams. TRY IT YOURSELF 2 Construct the 90% and 95% confidence intervals for the population variance and standard deviation of the medicine weights. Answer: Page A40 Note in Example 2 that the confidence interval for the population standard deviation cannot be written as s { E because the confidence interval does not have s as its center. (The same is true for the population variance.) EXAMPLE 2
RkJQdWJsaXNoZXIy NjM5ODQ=