SECTION 6.1 Confidence Intervals for the Mean (s Known) 307 43. When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain. (a) Increase in the level of confidence (b) Increase in the sample size (c) Increase in the population standard deviation 44. Describe how you would construct a 90% confidence interval to estimate the population mean age for students at your school. Constructing Confidence Intervals In Exercises 45 and 46, use the information to construct 90% and 99% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. 45. One-Way Commute Times Researchers surveyed a random sample of 32 U.S. workers ages 16 years and over who did not work from home and asked how long (in minutes) it takes them to get from home to work. The responses are listed. 30 48 35 46 27 27 25 37 18 45 18 37 26 38 33 14 73 16 8 54 27 53 29 72 40 29 17 42 32 24 62 85 From past studies, the researchers assume that s is 14.9 minutes. (Adapted from U.S. Census Bureau) 46. Sodium Chloride Concentrations The sodium chloride concentrations (in grams per liter) for 36 randomly selected seawater samples are listed. Assume that s is 7.61 grams per liter. 30.63 33.47 26.76 15.23 13.21 10.57 16.57 27.32 27.06 15.07 28.98 34.66 10.22 22.43 17.33 28.40 35.70 14.09 11.77 33.60 27.09 26.78 22.39 30.35 11.83 13.05 22.22 13.45 18.86 24.92 32.86 31.10 18.84 10.86 15.69 22.35 47. Determining a Minimum Sample Size Determine the minimum sample size required when you want to be 95% confident that the sample mean is within one unit of the population mean and s = 4.8. Assume the population is normally distributed. 48. Determining a Minimum Sample Size Determine the minimum sample size required when you want to be 99% confident that the sample mean is within two units of the population mean and s = 1.4. Assume the population is normally distributed. 49. Cholesterol Contents of Cheese A cheese processing company wants to estimate the mean cholesterol content of all one-ounce servings of a type of cheese. The estimate must be within 0.75 milligram of the population mean. (a) Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 3.10 milligrams. (b) The sample mean is 29 milligrams. Using the minimum sample size with a 95% level of confidence, does it seem likely that the population mean could be within 3% of the sample mean? within 0.3% of the sample mean? Explain.
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