Exam 8: Confidence Interval Estimation

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, a 90% confidence interval calculated from the same data would be narrower than a 99% confidence interval.

Given a sample mean of 2.1 and a sample standard deviation of 0.7 from a sample of10 data points, a 90% confidence interval will have a width of 2.36.

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 8-1 The managers of a company are worried about the morale of their employees.To determine if a problem in this area exists, they decide to evaluate the attitudes of their employees with a standardized test.They select the Fortunato test of job satisfaction, which has a known standard deviation of 24 points. -Referring to Scenario 8-1, due to financial limitations, the managers decide to take a sample of 45 employees.This yields a mean score of 88.0 points.A 90% confidence interval would go fromto _.

The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day.The assistant provides the following interval estimate: from 740 to 920 books per day.If the head librarian knows that the population standard deviation is150 books checked out per day, and she asked her assistant to use 25 days of data to construct the interval estimate, what confidence level can she attach to the interval estimate?

A major department store chain is interested in estimating the mean amount its credit card customers spent on their first visit to the chain's new store in the mall.Fifteen credit card accounts wererandomly sampled and analyzed with the following results: $\bar { X } = \$ 50.50$ and S= 20 , S being thestandard deviation.Construct a 95% confidence interval for the mean amount its credit card customers spent on their first visit to the chain's new store in the mall if the amount spent follows a normal distribution.

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 assumption that the distribution of the number of people recognizing the product has a normal distribution.

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, a 95% confidence interval for the mean number of rooms rented daily in a given month is narrower than a 99% confidence interval.

A prison official wants to estimate the proportion of cases of recidivism.Examining the records of250 convicts, the official determines that there are 65 cases of recidivism.A confidence interval will be obtained for the proportion of cases of recidivism.Part of this calculation includes the estimated standard error of the sample proportion.For this sample, the estimated standard error is _.

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. -The head of a computer science department is interested in estimating the proportion of students entering the department who will choose the new computer engineering option.Suppose there is no information about the proportion of students who might choose the option.What size sample should the department head take if she wants to be 95% confident that the estimate is within 0.10 of the true proportion?

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, the critical value used in constructing a 90% confidence interval is.

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, determine the sample size needed to estimate the proportion of managers who have caught salespeople working a second job on company time to within ±0.02 with95% confidence.

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.

Suppose a 95% confidence interval for $\mu$ 0 turns out to be (1,000, 2,100).Give a definition of what it means to be "95% confident" as an inference.

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, we are 95% confident that the population mean numberof managers who have caught salespeople cheating on an expense report is between 0.5116 to 0.6484.

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, it is possible that the true proportion of people that recognize the product is between 0.36 and 0.54.

The t distribution

A major department store chain is interested in estimating the mean 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: X= $50.50 and S= 20 , S being the standard deviation.Assuming the distribution of the amount spent on their first visit is normal, what is the shape of the sampling distribution of the sample mean that will be used to create the desired confidence interval for $\mu$ ?

SCENARIO 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 Scenario 8-4, a confidence interval for this sample would be based on the t distribution with degrees of freedom.