Section 8.1 Confidence Intervals for a Population Mean, Standard Deviation Known 363 38. A sample of size n = 80 is drawn from a normal population whose standard deviation is �� = 6.8. The sample mean is ̄x = 40.41. a. Construct a 90% confidence interval for ��. (39.16, 41.66) b. If the population were not approximately normal, would the confidence interval constructed in part (a) be valid? Explain. Yes 39. A population has standard deviation �� = 21.3. a. How large a sample must be drawn so that a 99% confidence interval for �� will have a margin of error equal to 2.5? 482 b. If the required confidence level were 95%, would the necessary sample size be larger or smaller? Smaller 40. A population has standard deviation �� = 17.3. a. How large a sample must be drawn so that a 95% confidence interval for �� will have a margin of error equal to 1.4? 587 b. If the required confidence level were 98%, would the necessary sample size be larger or smaller? Larger 41. A population has standard deviation �� = 12.7. a. How large a sample must be drawn so that a 96% confidence interval for �� will have a margin of error equal to 2.5? 109 b. If the required margin of error were 1.5, would the necessary sample size be larger or smaller? Larger 42. A population has standard deviation �� = 9.2. a. How large a sample must be drawn so that a 92% confidence interval for �� will have a margin of error equal to 0.8? 406 b. If the required margin of error were 1.4, would the necessary sample size be larger or smaller? Smaller Working with the Concepts 43. SAT scores: A college admissions officer takes a simple random sample of 100 entering freshmen and computes their mean mathematics SAT score to be 458. Assume the population standard deviation is �� = 116. a. Construct a 99% confidence interval for the mean mathematics SAT score for the entering freshman class. (428, 488) b. If the sample size were 75 rather than 100, would the margin of error be larger or smaller than the result in part (a)? Explain. Larger c. If the confidence level were 95% rather than 99%, would the margin of error be larger or smaller than the result in part (a)? Explain. Smaller d. Based on the confidence interval constructed in part (a), is it likely that the mean mathematics SAT score for the entering freshman class is greater than 500? No 44. How many computers? In a simple random sample of 150 households, the sample mean number of personal computers was 1.32. Assume the population standard deviation is �� = 0.41. a. Construct a 95% confidence interval for the mean number of personal computers. (1.25, 1.39) b. If the sample size were 100 rather than 150, would the margin of error be larger or smaller than the result in part (a)? Explain. Larger c. If the confidence level were 98% rather than 95%, would the margin of error be larger or smaller than the result in part (a)? Explain. Larger d. Based on the confidence interval constructed in part (a), is it likely that the mean number of personal computers is greater than 1.25? Yes 45. Babies: According to the National Health Statistics Reports, a sample of 360 one-year-old baby boys in the United States had a mean weight of 25.5 pounds. Assume the population standard deviation is �� = 5.3 pounds. a. Construct a 95% confidence interval for the mean weight of all one-year-old baby boys in the United States (25.0, 26.0) b. Should this confidence interval be used to estimate the mean weight of all one-year-old babies in the United States? Explain. No c. Based on the confidence interval constructed in part (a), is it likely that the mean weight of all one-year-old boys is less than 28 pounds? Yes 46. Watch your cholesterol: A sample of 314 patients between the ages of 38 and 82 were given a combination of the drugs ezetimibe and simvastatin. They achieved a mean reduction in total cholesterol of 0.94 millimole per liter. Assume the population standard deviation is �� = 0.18. a. Construct a 98% confidence interval for the mean reduction in total cholesterol in patients who take this combination of drugs. (0.92, 0.96) b. Should this confidence interval be used to estimate the mean reduction in total cholesterol for patients over the age of 85? Explain. No c. Based on the confidence interval constructed in part (a), is it likely that the mean reduction in cholesterol level is less than 0.90? No Source: International Journal of Clinical Practice 58:653–658 47. How smart is your phone? A random sample of 11 Samsung Galaxy smartphones being sold over the Internet in 2013 had the following prices, in dollars: 199 169 385 329 269 149 135 249 349 299 249 Assume the population standard deviation is �� = 85. a. Explain why it is necessary to check whether the population is approximately normal before constructing a confidence interval. n ≤ 30 b. Following is a dotplot of these data. Is it reasonable to assume that the population is approximately normal? Yes 100 150 200 250 300 350 400 c. If appropriate, construct a 95% confidence interval for the mean price for all phones of this type being sold on the Internet in 2013. If not appropriate, explain why not. (202.6, 303.1)
navidi_monk_elementary_statistics_2e_ch7-9
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