362 Chapter 8 Confidence Intervals EXCEL Constructing a confidence interval for the mean when �� is known The CONFIDENCE.NORM command returns the margin of error for a confidence interval when the population standard deviation �� is known. Step 1. In an empty cell, select the Insert Function icon and highlight Statistical in the category field. Step 2. Click on the CONFIDENCE.NORM function and press OK. Step 3. Enter the value of �� (0.02) in the Alpha field, the population standard deviation (15) in the Standard dev field, and the sample size (100) in the Size field. Step 4. Click OK (Figure G) to obtain the margin of error m. The confidence interval is given by ̄x − m < �� < ̄x + m. Figure G SECTION 8.1 Exercises Exercises 1– 16 are the Check Your Understanding exercises located within the section. Understanding the Concepts In Exercises 17–20, fill in each blank with the appropriate word or phrase. 17. A single number that estimates the value of an unknown parameter is called a estimate. point 18. The margin of error is the product of the standard error and the . critical value 19. In the confidence interval 24.3 ± 1.2, the quantity 1.2 is called the . margin of error 20. If we increase the confidence level and keep the sample size the same, we the margin of error. increase In Exercises 21–24, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement. 21. The confidence level is the proportion of all possible samples for which the confidence interval will cover the true value. True 22. To construct a confidence interval for a population mean, we add and subtract the critical value from the point estimate. False 23. Increasing the sample size while keeping the confidence level the same will result in a narrower confidence interval. True 24. If a 95% confidence interval for a population mean is 1.7 < �� < 2.3, then the probability is 0.95 that the mean is between 1.7 and 2.3. False Practicing the Skills In Exercises 25–28, find the critical value z��∕2 needed to construct a confidence interval with the given level. 25. Level 95% 1.96 26. Level 85% 1.44 27. Level 96% 2.05 28. Level 99.7% 2.97 In Exercises 29–32, find the levels of the confidence intervals that have the given critical values. 29. 2.326 98% 30. 2.576 99% 31. 2.81 99.5% 32. 1.04 70% 33. A sample of size n = 49 is drawn from a population whose standard deviation is �� = 4.8. a. Find the margin of error for a 95% confidence interval for ��. 1.344 b. If the sample size were n = 60, would the margin of error be larger or smaller? Smaller 34. A sample of size n = 50 is drawn from a population whose standard deviation is �� = 26. a. Find the margin of error for a 90% confidence interval for ��. 6.049 Tech: 6.048 b. If the sample size were n = 40, would the margin of error be larger or smaller? Larger 35. A sample of size n = 32 is drawn from a population whose standard deviation is �� = 12.1. a. Find the margin of error for a 99% confidence interval for ��. 5.510 b. If the confidence level were 90%, would the margin of error be larger or smaller? Smaller 36. A sample of size n = 64 is drawn from a population whose standard deviation is �� = 24.18. a. Find the margin of error for a 95% confidence interval for ��. 5.924 b. If the confidence level were 98%, would the margin of error be larger or smaller? Larger 37. A sample of size n = 10 is drawn from a normal population whose standard deviation is �� = 2.5. The sample mean is ̄x = 7.92. a. Construct a 95% confidence interval for ��. (6.37, 9.47) b. If the population were not approximately normal, would the confidence interval constructed in part (a) be valid? Explain. No
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