Exam 8: Confidence Interval Estimation

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, the standard error is 3%.

The t distribution is used to construct confidence intervals for the population mean when the population standard deviation is unknown.

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.

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?

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, the parameter of interest is the proportion of the student population who own a personal computer.

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, suppose the engineer had decided to estimate the mean length to within0.03 with 99% confidence.Then the sample size would be _.

The county clerk wants to estimate the proportion of voters who will need special election facilities.No information is available from past data.The clerk wants to construct a 95% confidence interval for the population proportion which extends at most 0.07 to either side of the sample proportion.How large a sample must be taken to assure these conditions are met?

Suppose a department store wants to estimate the mean age of the customers of its contemporary apparel department, correct to within 2 years, with level of confidence equal to 95%.Management believes that the standard deviation is 8 years.The sample size they should take 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. -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 8-4 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 95% confidence interval for the mean voltage of the power pack is wider than a 99% confidence interval.

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=0 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 8-11 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 do not have the information on the 49% in the interviews who said that they would purchase the new game?

The standardized normal distribution is used to develop a confidence interval estimate of the population proportion regardless of whether the population standard deviation is known.

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 foreach.For this sample, $\bar { X } = 396$ and S = 100.Which of the following assumptions is necessary for aconfidence interval to be valid?

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, the parameter of interest is the proportion of the rooms rented daily in a given month.

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.In calculating a 90% confidence interval for the population what would be the sampling error?

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 8-7 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 the average number of rooms rented daily in a given month is somewhere between 246.36 and 493.64.

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.

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 confidence interval will be based on degrees of freedom.