Exam 8: Confidence Interval Estimation

An economist is interested in studying the incomes of consumers in a particular country. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in a mean income of $15,000. What is the upper end point in a 99% confidence interval for the average income?

TABLE 8-10 The president of a university is concerned that illicit drug use on campus is higher than the 5% targeted level. A random sample of 250 students from a population of 2,000 revealed that 7 of them had used illicit drugs during the last 12 months. -Referring to Table 8-10, what is the critical value for the 90% one-sided confidence interval for the proportion of students who had used illicit drugs during the last 12 months?

The confidence interval estimate of the population proportion is constructed around the sample proportion.

A prison official wants to estimate the proportion of cases of recidivism. Examining the records of 250 convicts, the official determines that there are 65 cases of recidivism. A 99% confidence interval for the proportion of cases of recidivism would go from ________ to ________.

TABLE 8-2 A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired mean 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. -Referring to Table 8-2, suppose the engineer had decided to estimate the mean length to within 0.03 with 99% confidence. Then the sample size would be ________.

TABLE 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77, 11.90, 11.64, 11.84, 12.13, 11.99, and 11.77. -Referring to Table 8-4, the critical value for a 99% confidence interval for this sample is ________.

Given a sample mean of 2.1 and a sample standard deviation of 0.7 from a sample of 10 data points, a 90% confidence interval will have a width of 2.36.

Holding the width of a confidence interval fixed, increasing the level of confidence can be achieved with a lower sample size.

A confidence interval was used to estimate the proportion of statistics students who 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?

TABLE 8-9 The superintendent of a unified school district of a small town wants to make sure that no more than 5% of the students skip more than 10 days of school in a year. A random sample of 145 students from a population of 800 showed that 12 students skipped more than 10 days of school last year. -Referring to Table 8-9, what is the upper bound of the 95% one-sided confidence interval for the proportion of students who skipped more than 10 days of school last year?

TABLE 8-7 A hotel chain wants to estimate the mean number of rooms rented daily in a given month. The population of rooms rented daily is assumed to be normally distributed for each month with a standard deviation of 24 rooms. During February, a sample of 25 days has a sample mean of 37 rooms. -Referring to Table 8-7, we are 99% confident that between 24.64% and 49.36% of the rooms will be rented daily in a given month.

TABLE 8-8 The president of a university would like to estimate the proportion of the student population that owns a personal computer. In a sample of 500 students, 417 own a personal computer. -Referring to Table 8-8, it is possible that the 99% confidence interval calculated from the data will not contain the proportion of the student population who own a personal computer.

TABLE 8-3 To become an actuary, it is necessary to pass a series of 10 exams, including the most important one, an exam in probability and statistics. An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program. They take a sample of 8 actuarial students in this program and determine that their scores are: 2, 5, 8, 8, 7, 6, 5, and 7. This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program. -Referring to Table 8-3, for the confidence interval to be valid, it is necessary that test scores of students in the special study program on the actuarial exam be normally distributed.

TABLE 8-7 A hotel chain wants to estimate the mean number of rooms rented daily in a given month. The population of rooms rented daily is assumed to be normally distributed for each month with a standard deviation of 24 rooms. During February, a sample of 25 days has a sample mean of 37 rooms. -Referring to Table 8-7, the critical value for a 99% confidence interval for this sample is ________.

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, X = 19.8 and S² = 25. Estimate the mean number of admissions per 24-hour period with a 95% confidence interval.

TABLE 8-9 The superintendent of a unified school district of a small town wants to make sure that no more than 5% of the students skip more than 10 days of school in a year. A random sample of 145 students from a population of 800 showed that 12 students skipped more than 10 days of school last year. -Referring to Table 8-9, what is the critical value for the 95% one-sided confidence interval for the proportion of students who skipped more than 10 days of school last year?

The t distribution allows the calculation of confidence intervals for means when the actual standard deviation is not known.

Other things being equal, the confidence interval for the mean will be wider for 95% confidence than for 90% confidence.

The t distribution allows the calculation of confidence intervals for means for small samples when the population variance is not known, regardless of the shape of the distribution in the population.

TABLE 8-5 A sample of salary offers (in thousands of dollars) given to management majors is: 48, 51, 46, 52, 47, 48, 47, 50, 51, and 59. Using this data to obtain a 95% confidence interval resulted in an interval from 47.19 to 52.61. -Referring to Table 8-5, it is possible that the mean of the population is between 47.19 and 52.61.