Exam 8: Confidence Interval Estimation

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

The width of a confidence interval equals twice the sampling error.

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

TABLE 8-11 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 Table 8-11,the parameter of interest is the proportion of the student population who feel comfortable reporting cheating by their fellow students.

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

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

The difference between the lower limit of a confidence interval and the point estimate used in constructing the confidence interval is called the sampling error.

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,a 95% confidence interval for the mean voltage of the power pack is wider than a 99% confidence interval.

TABLE 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 Table 8-6,it is possible that the true proportion of people that recognize the product is between 0.36 and 0.54.

TABLE 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 Table 8-6,the parameter of interest is 54/120 = 0.45.

TABLE 8-13 A sales and marketing management magazine conducted a survey of salespeople cheating on their expense reports and other unethical conduct.In the survey of 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 Table 8-13,a 95% confidence interval will contain 95% of the population proportion of managers who have caught salespeople cheating on an expense report.

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 8-3 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,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.

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,we are 99% confident that the mean numbers of student population who own a personal computer is between 0.7911 and 0.8769.

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 is 150 books checked out per day,and she asked her assistant for a 95% confidence interval,approximately how large a sample did her assistant use to determine the interval estimate?

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?

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 8-5,95% of the time,the sample mean salary offer to management majors will fall between 47.19 and 52.61.

TABLE 8-14 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 Table 8-14,the report contains all the essential components for an ethical reporting of poll results.

TABLE 8-13 A sales and marketing management magazine conducted a survey of salespeople cheating on their expense reports and other unethical conduct.In the survey of 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 Table 8-13,construct a 95% confidence interval estimate of the population proportion of managers who have caught salespeople giving a kickback to a customer.

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 8-5,95% of the salary offers are between 47.19 and 52.61.