show work for all choice problems. 1. Suppose a 95% confidence interval for turns out to be (1,000, 2,100). To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. Which of the following will result in a reduced interval width? a) Increase the sample size. b) Decrease the confidence level. c) Both increase the sample size and decrease the confidence level. d) Both increase the confidence level and decrease the sample size. KEYWORDS: confidence interval, properties, width 2. Suppose a 95% confidence interval for has been constructed. If it is decided to take a larger sample and to decrease the confidence level of the interval, then the resulting interval width would . (Assume that the sample statistics gathered would not change very much for the new sample.) a) be larger than the current interval width b) be narrower than the current interval width c) be the same as the current interval width d) be unknown until actual sample sizes and reliability levels were determined KEYWORDS: confidence interval, properties, width 3. In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a interval. a) narrower b) wider c) less significant d) biased KEYWORDS: confidence interval, properties, width 4. A major department store chain is interested in estimating the average amount its credit card customers spent on their first visit to the chain's new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results: and X $50.50 s 2 400 . Assuming the distribution of the amount spent on their first visit is approximately normal, what is the shape of the sampling distribution of the sample mean that will be used to create the desired confidence interval for ? a) b) c) d) Approximately normal with a mean of $50.50 A standard normal distribution A t distribution with 15 degrees of freedom A t distribution with 14 degrees of freedom KEYWORDS: confidence interval, mean, t distribution 1 5. A major department store chain is interested in estimating the average amount its credit card customers spent on their first visit to the chain's new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results: and X $50.50 s 2 400 . Construct a 95% confidence interval for the average amount its credit card customers spent on their first visit to the chain's new store in the mall assuming that the amount spent follows a normal distribution. a) $50.50 $9.09 b) $50.50 c) $50.50 d) $50.50 $10.12 $11.00 $11.08 ANSWER: TYPE: MC DIFFICULTY: Easy KEYWORDS: confidence interval, mean, t distribution 6. 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. What value will be used as the point estimate for the mean endowment of all private colleges in the United States? a) $1,447.8 b) $180.975 c) $143.042 d) $8 ANSWER: TYPE: MC DIFFICULTY: Easy KEYWORDS: point estimate, mean 7. 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, 2 490.0, 122.6, 177.5, 95.4, and 220.0. Summary statistics yield X 180.975 and s 143.042 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. a) $180.975 $94.066 b) $180.975 $99.123 c) $180.975 $116.621 d) $180.975 $119.586 ANSWER: TYPE: MC DIFFICULTY: Moderate KEYWORDS: confidence interval, mean, t distribution 3 . 8. A university system enrolling hundreds of thousands of students is considering a change in the way students pay for their education. Presently the students pay $55 per credit hour. The university system administrators are contemplating charging each student a set fee of $750 per quarter, regardless of how many credit hours each takes. To see if this proposal would be economically feasible, the administrators would like to know how many credit hours, on the average, each student takes per quarter. A random sample of 250 students yields a mean of 14.1 credit hours per quarter and a standard deviation of 2.3 credit hours per quarter. Suppose the administration wanted to estimate the mean to within 0.1 hours at 95% reliability and assumed that the sample standard deviation provided a good estimate for the population standard deviation. How large a sample would they need to take? ANSWER: n= TYPE: PR DIFFICULTY: Easy KEYWORDS: mean, sample size determination 9. 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, and s2 = 25. Which of the following assumptions X 19.8 is necessary in order for a confidence interval to be valid? a) The population sampled from has an approximate normal distribution. b) The population sampled from has an approximate t distribution. c) The mean of the sample equals the mean of the population. d) None of these assumptions are necessary. ANSWER: 4 TYPE: MC DIFFICULTY: Moderate KEYWORDS: confidence interval, mean, t distribution 10. 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, and s2 = 25. Estimate the mean number of X 19.8 admissions per 24-hour period with a 95% confidence interval. ANSWER: 5 TYPE: PR DIFFICULTY: Moderate KEYWORDS: confidence interval, mean, t distribution 6 11. 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, and s2 = 25. If the director wishes to estimate the X 19.8 mean number of admissions per 24-hour period to within 1 admission with 99% reliability, what size sample should she choose? ANSWER: n= TYPE: PR DIFFICULTY: Moderate KEYWORDS: mean, sample size determination 7 12. A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. Use a 90% confidence interval to estimate the true proportion of students who receive financial aid. ANSWER: TYPE: PR DIFFICULTY: Moderate KEYWORDS: confidence interval, proportion 13. A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. The 95% confidence interval for p is 0.59 0.07. Interpret this interval. a) We are 95% confident that the true proportion of all students receiving financial aid is between 0.52 and 0.66. b) 95% of the students get between 52% and 66% of their tuition paid for by financial aid. c) We are 95% confident that between 52% and 66% of the sampled students receive some sort of financial aid. d) We are 95% confident that 59% of the students are on some sort of financial aid. ANSWER: TYPE: MC DIFFICULTY: Moderate 8 KEYWORDS: confidence interval, proportion, interpretation TABLE 8-3 A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired length of the insulation is 12 feet. It is known that the standard deviation in the cutting length is 0.15 feet. A sample of 70 cut sheets yields a mean length of 12.14 feet. This sample will be used to obtain a 99% confidence interval for the mean length cut by machine. 14. Referring to Table 8-3, the critical value to use in obtaining the confidence interval is ________. ANSWER: TYPE: FI DIFFICULTY: Easy KEYWORDS: confidence interval, mean, standardized normal distribution 15. Referring to Table 8-3, the confidence interval goes from ________ to ________. ANSWER: TYPE: FI DIFFICULTY: Moderate KEYWORDS: confidence interval, mean, standardized normal distribution 16. Referring to Table 8-3, the confidence interval indicates that the machine is _____ working properly. ANSWER: TYPE: TF DIFFICULTY: Moderate KEYWORDS: confidence interval, mean, standardized normal distribution, interpretation 9 10