Section 8.5 Determining Which Method to Use 395 weights of two-month-old baby girls is the same as that of two-month-old baby boys? Explain. Yes 16. Eat your cereal: Boxes of cereal are labeled as containing 14 ounces. Following are the weights of a sample of 12 boxes. Assume that the population is normally distributed. 14.02 13.97 14.11 14.12 14.10 14.02 14.15 13.97 14.05 14.04 14.11 14.12 a. Find the sample standard deviation s. 0.0614 b. Construct a 98% confidence interval for the population standard deviation ��. (0.04, 0.12) c. The goal of the quality control manager is for the population standard deviation of the weights to be less than 0.03. Based on the confidence interval, is it reasonable to believe that the goal has been met? No 17. Eat your spinach: Six measurements were made of the mineral content (in percent) of spinach, with the following results. Assume that the population is normally distributed. 19.1 20.8 20.8 21.4 20.5 19.7 a. Find the sample standard deviation s. 0.838 b. Construct a 99% confidence interval for the population standard deviation ��. (0.46, 2.92) c. Based on the confidence interval, is it reasonable to believe that the population standard deviation might be less than 1.5? Explain. Yes Source: Journal of Nutrition 66:55–66 18. Mortgage rates: Following are interest rates (annual percentage rates) for a 30-year fixed rate mortgage from a sample of lenders in Macon, Georgia on June 20, 2013. It is reasonable to assume that the population is approximately normal. 4.750 4.375 4.176 4.679 4.426 4.227 4.125 4.250 3.950 4.191 4.299 4.415 Source: www.bankrate.com a. Find the sample standard deviation s. 0.2263 b. Construct a 95% confidence interval for population standard deviation ��. (0.16, 0.38) c. Based on the confidence interval, is it reasonable to believe that the population standard deviation is greater than 0.5? Explain. No Extending the Concepts The chi-square distribution is skewed, but as the number of degrees of freedom becomes large, the skewness diminishes. If the number of degrees of freedom, k, is large enough, the chi-square distribution is reasonably well approximated by a normal distribution with mean k and variance 2k. 19. Exact confidence interval: A sample of size 101 from a normal population has sample standard deviation s = 40. Use Table A.4 to find the exact critical values ��2 0.025 and ��2 0.975 for a 95% confidence interval, and construct a 95% confidence interval for ��. ��2 0.025 = 74.22, ��2 0.975 = 129.56; 95% CI: (35.14, 46.43) 20. Using the normal approximation: Refer to Exercise 19. Use the normal approximation to estimate the critical values ��2 0.025 and ��2 0.975 for a 95% confidence interval, and construct a 95% confidence interval for ��. ��2 0.025 = 72.28, ��2 0.975 = 127.72; 95% CI: (35.39, 47.05) 21. Comparing results: How close is the confidence interval based on the normal approximation constructed in Exercise 20 to the exact confidence interval constructed in Exercise 19? A more accurate normal approximation to ��2 �� is given by ��2 �� ≈ 0.5 ( z�� + √ 2k − 1 )2 , where z�� is the z-score that has area �� to its right. 22. More accuracy: Refer to Exercise 19. Use the more accurate normal approximation to estimate the critical values ��2 0.025 and ��2 0.975 for a 95% confidence interval, and construct a 95% confidence interval for ��. ��2 0.025 = 73.77, ��2 0.975 = 129.07; 95% CI: (35.21, 46.57) 23. Comparing results: How close is the confidence interval based on the more accurate normal approximation to the exact one? Answers to Check Your Understanding Exercises for Section 8.4 1. ��2 0.975 = 8.231, ��2 0.025 = 31.526 2. ��2 0.995 = 10.520, ��2 0.005 = 46.928 3. 4.13 < �� < 10.95 4. 8.60 < �� < 19.15 SECTION 8.5 Determining Which Method to Use Objectives 1. Determine which method to use when constructing a confidence interval Objective 1 Determine which method to use when constructing a confidence interval One of the challenges in constructing a confidence interval is to determine which method to use. The first step is to determine which type of parameter we are estimating. There are three types of parameters for which we have learned to construct confidence intervals: ∙ Population mean �� ∙ Population proportion p ∙ Population standard deviation �� or variance ��2
navidi_monk_elementary_statistics_2e_ch7-9
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