Section 8.1 Confidence Intervals for a Population Mean, Standard Deviation Known 365 58. Which interval is which? Matilda has constructed three confidence intervals, all from the same random sample. The confidence levels are 95%, 98%, and 99.9%. The confidence intervals are 6.4 < �� < 12.3, 5.1 < �� < 13.6, and 6.8 < �� < 11.9. Unfortunately, Matilda has forgotten which confidence interval has which level. Match each confidence interval with its level. 95%: (6.8, 11.9), 98%: (6.4, 12.3), 99.9%: (5.1, 13.6) 59. Don’t construct a confidence interval: A psychology professor at a certain college gave a test to the students in her class. The test was designed to measure students’ attitudes toward school, with higher scores indicating a more positive attitude. There were 30 students in the class, and their mean score was 78. Scores on this test are known to be normally distributed with a standard deviation of 10. Explain why these data should not be used to construct a confidence interval for the mean score for all the students in the college. 60. Don’t construct a confidence interval: A college alumni organization sent a survey to all recent graduates to ask their annual income. Twenty percent of the alumni responded, and their mean annual income was $40,000. Assume the population standard deviation is �� = $10,000. Explain why these data should not be used to construct a confidence interval for the mean annual income of all recent graduates. 61. Interpret a confidence interval: A dean at a certain college looked up the GPA for a random sample of 85 students. The sample mean GPA was 2.82, and a 95% confidence interval for the mean GPA of all students in the college was 2.76 < �� < 2.88. True or false, and explain: a. We are 95% confident that the mean GPA of all students in the college is between 2.76 and 2.88. True b. We are 95% confident that the mean GPA of all students in the sample is between 2.76 and 2.88. False c. The probability is 0.95 that the mean GPA of all students in the college is between 2.76 and 2.88. False d. 95% of the students in the sample had a GPA between 2.76 and 2.88. False 62. Interpret a confidence interval: A survey organization drew a simple random sample of 625 households from a city of 100,000 households. The sample mean number of people in the 625 households was 2.30, and a 95% confidence interval for the mean number of people in the 100,000 households was 2.16 < �� < 2.44. True or false, and explain: a. We are 95% confident that the mean number of people in the 625 households is between 2.16 and 2.44. False b. We are 95% confident that the mean number of people in the 100,000 households is between 2.16 and 2.44. True c. The probability is 0.95 that the mean number of people in the 100,000 households is between 2.16 and 2.44. False d. 95% of the households in the sample contain between 2.16 and 2.44 people. False 63. Interpret calculator display: The following display from a TI-84 Plus calculator presents a 95% confidence interval. a. Fill in the blanks: We are confident that the population mean is between and . 95%, 56.019, 60.881 b. Assume the population is not normally distributed. Is the confidence interval still valid? Explain. Yes 64. Interpret calculator display: The following display from a TI-84 Plus calculator presents a 99% confidence interval. a. Fill in the blanks: We are confident that the population mean is between and . 99%, 17.012, 20.048 b. Assume the population is not normally distributed. Is the confidence interval still valid? Explain. No 65. Interpret computer output: The following MINITAB output presents a 95% confidence interval. ������ �������������� ���������� = �� �������� ���������������� �� �� ���� �������� ���� ������ ���� �������� �� �������� ����~ ���� .�� ��������, ���� ��������/ a. Fill in the blanks: We are confident that the population mean is between and . 95%, 9.6956, 15.0084 b. Use the appropriate critical value along with the information in the computer output to construct a 99% confidence interval. (8.861, 15.843) c. Find the sample size needed so that the 95% confidence interval will have a margin of error of 1.5. 73 d. Find the sample size needed so that the 99% confidence interval will have a margin of error of 1.5. 125 66. Interpret computer output: The following MINITAB output presents a 98% confidence interval. ������ �������������� ���������� = �� �������� ���������������� �� �� ���� �������� �� ������ ���� �������� �� �������� ����~ ���� .�� ��������, �� ��������/ a. Fill in the blanks: We are confident that the population mean is between and . 98%, 0.2133, 5.1007 b. Use the appropriate critical value along with the information in the computer output to construct a 95% confidence interval. (0.598, 4.716) c. Find the sample size needed so that the 98% confidence interval will have a margin of error of 1.0. 347 d. Find the sample size needed so that the 95% confidence interval will have a margin of error of 1.0. 246 Extending the Concepts One-sided confidence intervals: A confidence interval provides likely minimum and maximum values for a parameter. In some cases, we are interested only in a maximum or only in a minimum. In these cases, we construct a one-sided confidence interval. A one-sided confidence interval can be an upper confidence bound,
navidi_monk_elementary_statistics_2e_ch7-9
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