Exam 8: Confidence Interval Estimation

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 8-5 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 Scenario 8-5, it is possible that the mean of the population is not between 47.19 and 52.61.

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?

SCENARIO 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 Scenario 8-2, the confidence interval goes from to .

It is desired to estimate the mean total compensation of CEOs in the Service industry.Data were randomly collected from 18 CEOs and the 95% confidence interval was calculated to be ($2,181,260,$5,836,180).Based on the interval above, do you believe the mean total compensation of CEOs in theService industry is more than $3,000,000?

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, we are 99% confident that between 246.36% and 493.64%of the rooms will be rented daily in a given month.

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, this confidence interval is only valid if the scores on theFortunato test are normally distributed.

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.If the dean wanted to estimate the proportion of all students receiving financial aid to within 3% with 99% reliability, how many students would need to be sampled?

SCENARIO 8-12 The Three Brothers Energy Drink Company bottles and distributes a popular drink for athletes and exercise enthusiasts.Because of its marketing successes the company has installed an additional filling machine and the managers are eager to use it in daily operations.The machine is set to fill bottles at 16 oz. However, we know there is inherent machine variability and quality control has determined through testing a mean of 16.2 oz.and a standard deviation of 0.3 oz.using a 100 bottle sample. -Refer to Scenario 8-12.Suppose you want to estimate the population mean fill with 95%confidence and 0.04 oz.sampling error rate.What should be your sample size?

Holding the sample size fixed, increasing the level of confidence in a confidence interval will necessarily lead to a wider confidence interval.

A sample of 100 fuses from a very large shipment is found to have 10 that are defective.The 95% confidence interval would indicate that, for this shipment, the proportion of defective fuses is between 0 and 0.28.

Sampling error equals half the width of a confidence interval.

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

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

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

For a t distribution with 12 degrees of freedom, the area between - 2.6810 and 2.1788 is 0.980.

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, approximately how large a sample did her assistant use to determine the interval estimate?

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, if we use the same sample information to obtain a 95% confidence interval, the resulting interval would be narrower than the one obtained here with 90% confidence.

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.

The t distribution is used to develop a confidence interval estimate of the population proportion when the population standard deviation is unknown.