Exam 8: Confidence Interval Estimation

When determining the sample size necessary for estimating the true population mean, which factor isnot considered when sampling with replacement?

SCENARIO 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 Scenario 8-3, a 90% confidence interval for the mean score of actuarial students in the special program is from to .

The difference between the sample size and the population size is called the sampling error.

Other things being equal, as the confidence level for a confidence interval increases, the width of the interval increases.

In estimating the population mean with the population standard deviation unknown, if the sample size is 12, there will be 6 degrees of freedom.

SCENARIO 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 240 rooms.During February, a sample of 25 days has a sample mean of 370 rooms. -Referring to Scenario 8-7, it is possible that the 99% confidence interval calculated from the data will not contain the population mean number of rooms rented daily in a given month.

The difference between the sample proportion and the population proportion is called the sampling error.

SCENARIO 8-9 A university wanted to find out the percentage of students who felt comfortable reporting cheating by their fellow students.A survey of 2,800 students was conducted and the students were asked if they felt comfortable reporting cheating by their fellow students.The results were 1,344 answered "Yes" and 1,456 answered "no". -Referring to Scenario 8-9, the sampling error of a 99% confidence interval for the proportion of student population who feel comfortable reporting cheating by their fellow students is .

SCENARIO 8-11 A poll was conducted by the marketing department of a video game company to determine the popularity of a new game that was targeted to be launched in three months.Telephone interviews with 1,500 young adults were conducted which revealed that 49% said they would purchase the new game.The margin of error was ±3 percentage points. -Referring to Scenario 8-11, what is the needed sample size to obtain a 95% confidence interval estimate of the percentage of the targeted young adults who will purchase the new game by allowing the same level of margin of error?

A population parameter is used to estimate a confidence interval.

SCENARIO 8-9 A university wanted to find out the percentage of students who felt comfortable reporting cheating by their fellow students.A survey of 2,800 students was conducted and the students were asked if they felt comfortable reporting cheating by their fellow students.The results were 1,344 answered "Yes" and 1,456 answered "no". -Referring to Scenario 8-9, we are 99% confident that between 45.57% and 50.43% of the student population feel comfortable reporting cheating by their fellow students.

SCENARIO 8-10 A sales and marketing management magazine conducted a survey on salespeople cheating on their expense reports and other unethical conduct.In the survey on 200 managers, 58% of the managers have caught salespeople cheating on an expense report, 50% have caught salespeople working a second job on company time, 22% have caught salespeople listing a "strip bar" as a restaurant on an expense report, and 19% have caught salespeople giving a kickback to a customer. -Referring to Scenario 8-10, construct a 95% confidence interval estimate of the population proportion of managers who have caught salespeople cheating on an expense report.

SCENARIO 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 Scenario 8-8, a confidence interval estimate of the population proportion would only be valid if the distribution of the number of students who own a personal computer is normal.

SCENARIO 8-11 A poll was conducted by the marketing department of a video game company to determine the popularity of a new game that was targeted to be launched in three months.Telephone interviews with 1,500 young adults were conducted which revealed that 49% said they would purchase the new game.The margin of error was ±3 percentage points. -Referring to Scenario 8-11, what is the needed sample size to obtain a 95% confidence interval in estimating the percentage of the targeted young adults who will purchase the new game to within±5%?

If you were constructing a 99% confidence interval of the population mean based on a sample ofn=25 where the standard deviation of the sample S = 0.05, the critical value of t will be

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.Summary statistics yield $\bar { X } = 180.975$ 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 university system enrolling hundreds of thousands of students is considering a change in the way students pay for their education.Currently, the students pay $400 per credit hour.The university system administrators are contemplating charging each student a set fee of $7,000 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 provideda good estimate for the population standard deviation.How large a total sample would they need to take?

SCENARIO 8-6 After an extensive advertising campaign, the manager of a company wants to estimate the proportion of potential customers that recognize a new product.She samples 120 potential consumers and finds that 54 recognize this product.She uses this sample information to obtain a 95% confidence interval that goes from 0.36 to 0.54. -Referring to Scenario 8-6, this interval requires the use of the t distribution to obtain the confidence coefficient.

SCENARIO 8-10 A sales and marketing management magazine conducted a survey on salespeople cheating on their expense reports and other unethical conduct.In the survey on 200 managers, 58% of the managers have caught salespeople cheating on an expense report, 50% have caught salespeople working a second job on company time, 22% have caught salespeople listing a "strip bar" as a restaurant on an expense report, and 19% have caught salespeople giving a kickback to a customer. -Referring to Scenario 8-10, construct a 95% confidence interval estimate of the population proportion of managers who have caught salespeople working a second job on company time.

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