394 Chapter 8 Confidence Intervals Through algebraic manipulation, we can solve for ��2, to obtain a level 100(1 − ��)% confidence interval for ��2: (n − 1)s2 ��2 ��∕2 < ��2 < (n − 1)s2 ��2 1−��∕2 USING TECHNOLOGY We use Example 8.20 to illustrate the technology steps. MINITAB Constructing a confidence interval for �� Step 1. Enter the data in Column C1. For Example 8.20, we use 1989.9, 1993.8, 2074.5, 2070.5, 2070.9, 2033.6, 1939.6. Step 2. Click on Stat, then Basic Statistics, then Graphical Summary. Step 3. Enter C1 in the Variables field and enter the confidence level in the Confidence Level field. For Example 8.20, we use 95. Step 4. Click OK (Figure A). Figure A SECTION 8.4 Exercises Exercises 1– 4 are the Check Your Understanding exercises located within the section. Understanding the Concepts In Exercises 5 and 6, fill in each blank with the appropriate word or phrase. 5. To find a confidence interval for a standard deviation from a sample of size 15, we use a chi-square distribution with degrees of freedom. 14 6. The method described for finding confidence intervals should be used only when the distribution of the population is almost exactly . normal In Exercises 7 and 8, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement. 7. When constructing a confidence interval for a standard deviation, we must find two critical values. True 8. If we have a confidence interval for a variance, we can obtain a confidence interval for the standard deviation by squaring the confidence bounds. False Practicing the Skills 9. Find the critical values for a 95% confidence interval using the chi-square distribution with 15 degrees of freedom. 6.262, 27.488 10. Find the critical values for a 99% confidence interval using the chi-square distribution with 5 degrees of freedom. 0.412, 16.750 11. Construct a 95% confidence interval for the population standard deviation �� if a sample of size 25 has standard deviation s = 15. (11.71, 20.87) 12. Construct a 99% confidence interval for the population standard deviation �� if a sample of size 8 has standard deviation s = 7.5. (4.41, 19.95) Working with the Concepts 13. SAT scores: Scores on the math SAT are normally distributed. A sample of 20 SAT scores had standard deviation s = 87. a. Construct a 98% confidence interval for the population standard deviation ��. (63.04, 137.26) b. Someone says that the scoring system for the SAT is designed so that the population standard deviation will be �� = 100. Does this confidence interval contradict this claim? Explain. No 14. IQ scores: Scores on an IQ test are normally distributed. A sample of 25 IQ scores had standard deviation s = 8. a. Construct a 95% confidence interval for the population standard deviation ��. (6.25, 11.13) b. The developer of the test claims that the population standard deviation is �� = 15. Does this confidence interval contradict this claim? Explain. Yes 15. Baby weights: Following are weights of 12 two-month-old baby girls. Assume that the population is normally distributed. 12.23 12.32 11.87 12.34 11.48 12.66 8.51 14.13 12.95 10.30 9.34 8.63 a. Find the sample standard deviation s. 1.798 b. Construct a 95% confidence interval for the population standard deviation ��. (1.27, 3.05) c. According to the National Health Statistics Reports, the standard deviation of the weight of two-month-old baby boys is 2.7 pounds. Based on the confidence interval, is it reasonable to believe that the standard deviation of the
navidi_monk_elementary_statistics_2e_ch7-9
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