Question 1.
a) If you were constructing a 99% confidence interval of the population mean based on a sample of n where the standard deviation of the sample s = 0.05, what is the value of the critical value of t that you should use? Explain/show how you obtain your answer. (5 pts.)
b) A confidence interval was used to estimate the proportion of statistics students that are female. A random sample of 72 statistics students generated the following 90% confidence interval: (0.438, 0.642). Using the information above, what total size sample would be necessary if we wanted to estimate the true proportion to within ±0.08 using 95% confidence? Explain/show how you obtain your answer. (15 pts.)
Question 2. Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is put into a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of 8 private colleges in the United States revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95.4, and 220.0.
a) What value will be used as the point estimate for the mean endowment of all private colleges in the United States? Explain/show how you obtain your answer. (5 pts.)
b) Calculate a 95% confidence interval for the mean endowment of all the private colleges in the United States assuming a normal distribution for the endowments. Explain/show how you obtain your answer. (15 pts.)
Question 3. As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, = 19.8 and S2 = 25.
a) Estimate the mean number of admissions per 24-hour period with a 95% confidence interval. Explain/show how you obtain your answer. (10 pts.)
b) Using the sample standard deviation as an estimate for the population standard deviation, what size sample should the director choose if she wishes to estimate the mean number of admissions per 24-hour period to within 1 admission with 99% reliability? Explain/show how you obtain your answer.