a. What is the standard error of the mean, ...?
b. At 95% confidence, what is the margin of error?
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8.2 A simple random sample of 50 items from a population with σ = 6 resulted in a sample mean of 32.
a. Provide a 90% confidence interval for the population mean.
b. Provide a 95% confidence interval for the population mean.
c. Provide a 99% confidence interval for the population mean.
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8.3 A simple random sample of 60 items resulted in a sample mean of 80. The population standard deviation is σ = 15.
a. Compute the 95% confidence interval for the population mean.
b. Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean.
c. What is the effect of a larger sample size on the interval estimate?
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8.4 A 95% confidence interval for a population mean was reported to be 152 to 160. If σ = 15, what sample size was used in this study?
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8.5 In an effort to estimate the mean amount spent per customer for dinner at an Atlanta restaurant, data were collected for a sample of 49 customers. The data collected are shown in the CD file named Restaurant. Based upon past studies the population standard deviation is assumed known with σ = $5.
a. At 95% confidence, what is the margin of error?
b. Develop a 95% confidence interval estimate of the mean amount spent for dinner.
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8.6 Nielsen Media Research conducted a study of household television viewing times during the 8 p.m. to 11 p.m. time period. The data contained in the CD file named Nielsen are consistent with the findings reported (The World Almanac, 2003). Based upon past studies the population standard deviation is assumed known with a = 3.5 hours. Develop a 95% confidence interval estimate of the mean television viewing time per week during the 8 p.m. to 11 p.m. time period.
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8.7
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8.8 The National Quality Research Center at the University of Michigan provides a quarterly measure of consumer opinions about products and services (The Wall Street Journal, February 18, 2003). A survey of 10 restaurants in the Fast Food/Pizza group showed a sample mean customer satisfaction index of 71. Past data indicate that the population standard deviation of the index has been relatively stable with a = 5.
a. What assumption should the researcher be willing to make if a margin of error is desired?
b. Using 95% confidence, what is the margin of error?
c. What is the margin of error if 99% confidence is desired?
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8.9
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8.10 Playbill magazine reported that the mean annual household income of its readers is $119,155 (Playbill, January 2006). Assume this estimate of the mean annual household income is based on a sample of 80 households and, based on past studies, the population standard deviation is known to be a = $30,000.
a. Develop a 90% confidence interval estimate of the population mean.
b. Develop a 95% confidence interval estimate of the population mean.
c. Develop a 99% confidence interval estimate of the population mean.
d. Discuss what happens to the width of the confidence interval as the confidence level is increased. Does this result seem reasonable? Explain.
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8.11 For a t distribution with 16 degrees of freedom, find the area, or probability, in each region.
a. To the right of 2.120
b. To the left of 1.337
c. To the left of −1.746
d. To the right of 2.583
e. Between −2.120 and 2.120
f. Between −1.746 and 1.746
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8.12 Find the t value(s) for each of the following cases.
a. Upper tail area of .025 with 12 degrees of freedom
b. Lower tail area of .05 with 50 degrees of freedom
c. Upper tail area of .01 with 30 degrees of freedom
d. Where 90% of the area falls between these two t values with 25 degrees of freedom
e. Where 95% of the area falls between these two t values with 45 degrees of freedom
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8.13 The following sample data are from a normal population: 10, 8, 12, 15, 13, 11, 6, 5.
a. What is the point estimate of the population mean?
b. What is the point estimate of the population standard deviation?
c. With 95% confidence, what is the margin of error for the estimation of the population mean?
d. What is the 95% confidence interval for the population mean?
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8.14 A simple random sample with n = 54 provided a sample mean of 22.5 and a sample standard deviation of 4.4.
a. Develop a 90% confidence interval for the population mean.
b. Develop a 95% confidence interval for the population mean.
c. Develop a 99% confidence interval for the population mean.
d. What happens to the margin of error and the confidence interval as the confidence level is increased?
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8.15 Sales personnel for Skillings Distributors submit weekly reports listing the customer contacts made during the week. A sample of 65 weekly reports showed a sample mean of 19.5 customer contacts per week. The sample standard deviation was 5.2. Provide 90% and 95% confidence intervals for the population mean number of weekly customer contacts for the sales personnel.
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8.16 The mean number of hours of flying time for pilots at Continental Airlines is 49 hours per month (The Wall Street Journal, February 25, 2003). Assume that this mean was based on actual flying times for a sample of 100 Continental pilots and that the sample standard deviation was 8.5 hours.
a. At 95% confidence, what is the margin of error?
b. What is the 95% confidence interval estimate of the population mean flying time for the pilots?
c. The mean number of hours of flying time for pilots at United Airlines is 36 hours per month. Use your results from part (b) to discuss differences between the flying times for the pilots at the two airlines. The Wall Street Journal reported United Airlines as having the highest labor cost among all airlines. Does the information in this exercise provide insight as to why United Airlines might expect higher labor costs?
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8.17 The International Air Transport Association surveys business travelers to develop quality ratings for transatlantic gateway airports. The maximum possible rating is 10. Suppose a simple random sample of 50 business travelers is selected and each traveler is asked to provide a rating for the Miami International Airport. The ratings obtained from the sample of 50 business travelers follow....Develop a 95% confidence interval estimate of the population mean rating for Miami.
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8.18
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8.19
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8.20 Is your favorite TV program often interrupted by advertising? CNBC presented statistics on the average number of programming minutes in a half-hour sitcom (CNBC, February 23, 2006). The following data (in minutes) are representative of their findings....Assume the population is approximately normal. Provide a point estimate and a 95% confidence interval for the mean number of programming minutes during a half-hour television sitcom.
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8.21 Consumption of alcoholic beverages by young women of drinking age has been increasing in the United Kingdom, the United States, and Europe (The Wall Street Journal, February 15, 2006). Data (annual consumption in liters) consistent with the findings reported in The Wall Street Journal article are shown for a sample of 20 European young women....Assuming the population is roughly symmetric, construct a 95% confidence interval for the mean annual consumption of alcoholic beverages by European young women.
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8.22
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8.23 How large a sample should be selected to provide a 95% confidence interval with a margin of error of 10? Assume that the population standard deviation is 40.
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8.24 The range for a set of data is estimated to be 36.
a. What is the planning value for the population standard deviation?
b. At 95% confidence, how large a sample would provide a margin of error of 3?
c. At 95% confidence, how large a sample would provide a margin of error of 2?
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8.25 Refer to the Scheer Industries example in Section 8.2. Use 6.84 days as a planning value for the population standard deviation.
a. Assuming 95% confidence, what sample size would be required to obtain a margin of error of 1.5 days?
b. If the precision statement was made with 90% confidence, what sample size would be required to obtain a margin of error of 2 days?
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8.26 The average cost of a gallon of unleaded gasoline in Greater Cincinnati was reported to be $2.41 (The Cincinnati Enquirer, February 3, 2006). During periods of rapidly changing prices, the newspaper samples service stations and prepares reports on gasoline prices frequently. Assume the standard deviation is $.15 for the price of a gallon of unleaded regular gasoline, and recommend the appropriate sample size for the newspaper to use if they wish to report a margin of error at 95% confidence.
a. Suppose the desired margin of error is $.07.b. Suppose the desired margin of error is $.05.c. Suppose the desired margin of error is $.03.
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8.27 Annual starting salaries for college graduates with degrees in business administration are generally expected to be between $30,000 and $45,000. Assume that a 95% confidence interval estimate of the population mean annual starting salary is desired. What is the planning value for the population standard deviation? How large a sample should be taken if the desired margin of error is
a. $500?
b. $200?
c. $100?
d. Would you recommend trying to obtain the $100 margin of error? Explain.
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8.28
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8.29 The travel-to-work time for residents of the 15 largest cities in the United States is reported in the 2003 Information Please Almanac. Suppose that a preliminary simple random sample of residents of San Francisco is used to develop a planning value of 6.25 minutes for the population standard deviation.
a. If we want to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 2 minutes, what sample size should be used? Assume 95% confidence.
b. If we want to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 1 minute, what sample size should be used? Assume 95% confidence.
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8.30 During the first quarter of 2003, the price/earnings (P/E) ratio for stocks listed on the New York Stock Exchange generally ranged from 5 to 60 (The Wall Street Journal, March 7, 2003). Assume that we want to estimate the population mean P/E ratio for all stocks listed on the exchange. How many stocks should be included in the sample if we want a margin of error of 3? Use 95% confidence.
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8.31 A simple random sample of 400 individuals provides 100 Yes responses.
a. What is the point estimate of the proportion of the population that would provide Yes responses?
b. What is your estimate of the standard error of the proportion, ...?
c. Compute the 95% confidence interval for the population proportion.
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8.32 A simple random sample of 800 elements generates a sample proportion ....
a. Provide a 90% confidence interval for the population proportion.b. Provide a 95% confidence interval for the population proportion.
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8.33 In a survey, the planning value for the population proportion is p* = .35. How large a sample should be taken to provide a 95% confidence interval with a margin of error of .05?
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8.34 At 95% confidence, how large a sample should be taken to obtain a margin of error of .03 for the estimation of a population proportion? Assume that past data are not available for developing a planning value for p*.
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8.35
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8.36 According to statistics reported on CNBC, a surprising number of motor vehicles are not covered by insurance (CNBC, February 23, 2006). Sample results, consistent with the CNBC report, showed 46 of 200 vehicles were not covered by insurance.
a. What is the point estimate of the proportion of vehicles not covered by insurance?
b. Develop a 95% confidence interval for the population proportion.
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8.37 Towers Perrin, a New York human resources consulting firm, conducted a survey of 1100 employees at medium-sized and large companies to determine how dissatisfied employees were with their jobs (The Wall Street Journal, January 29, 2003). Representative data are shown in the file JobSatisfaction. A response of Yes indicates the employee strongly disliked the current work experience.
a. What is the point estimate of the proportion of the population of employees who strongly dislike their current work experience?
b. At 95% confidence, what is the margin of error?
c. What is the 95% confidence interval for the proportion of the population of employees who strongly dislike their current work experience?
d. Towers Perrin estimates that it costs employers one-third of an hourly employee’s annual salary to find a successor and as much as 1.5 times the annual salary to find a successor for a highly compensated employee. What message did this survey send to employers?
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8.38 According to Thomson Financial, through January 25, 2006, the majority of companies reporting profits had beaten estimates (BusinessWeek, February 6, 2006). A sample of 162 companies showed 104 beat estimates, 29 matched estimates, and 29 fell short.
a. What is the point estimate of the proportion that fell short of estimates?
b. Determine the margin of error and provide a 95% confidence interval for the proportion that beat estimates.
c. How large a sample is needed if the desired margin of error is .05?
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8.39 The percentage of people not covered by health care insurance in 2003 was 15.6% (Statistical Abstract of the United States, 2006). A congressional committee has been charged with conducting a sample survey to obtain more current information.
a. What sample size would you recommend if the committee’s goal is to estimate the current proportion of individuals without health care insurance with a margin of error of .03? Use a 95% confidence level.
b. Repeat part (a) using a 99% confidence level.
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8.40
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8.41 America’s young people are heavy Internet users; 87% of Americans ages 12 to 17 are Internet users (The Cincinnati Enquirer, February 7, 2006). MySpace was voted the most popular Web site by 9% in a sample survey of Internet users in this age group. Suppose 1400 youths participated in the survey. What is the margin of error, and what is the interval estimate of the population proportion for which MySpace is the most popular Web site? Use a 95% confidence level.
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8.42
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8.43 A Phoenix Wealth Management/Harris Interactive survey of 1500 individuals with net worth of $1 million or more provided a variety of statistics on wealthy people (Business Week, September 22, 2003). The previous three-year period had been bad for the stock market, which motivated some of the questions asked.
a. The survey reported that 53% of the respondents lost 25% or more of their portfolio value over the past three years. Develop a 95% confidence interval for the proportion of wealthy people who lost 25% or more of their portfolio value over the past three years.b. The survey reported that 31% of the respondents feel they have to save more for retirement to make up for what they lost. Develop a 95% confidence interval for the population proportion.c. Five percent of the respondents gave $25,000 or more to charity over the previous year. Develop a 95% confidence interval for the proportion who gave $25,000 or more to charity.d. Compare the margin of error for the interval estimates in parts (a), (b), and (c). How is the margin of error related to ...? When the same sample is being used to estimate a variety of proportions, which of the proportions should be used to choose the planning value p*? Why do you think p* = .50 is often used in these cases?
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8.44 A sample survey of 54 discount brokers showed that the mean price charged for a trade of 100 shares at $50 per share was $33.77 (AAII Journal, February 2006). The survey is conducted annually. With the historical data available, assume a known population standard deviation of $15.
a. Using the sample data, what is the margin of error associated with a 95% confidence interval?
b. Develop a 95% confidence interval for the mean price charged by discount brokers for a trade of 100 shares at $50 per share.
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8.45 A survey conducted by the American Automobile Association showed that a family of four spends an average of $215.60 per day while on vacation. Suppose a sample of 64 families of four vacationing at Niagara Falls resulted in a sample mean of $252.45 per day and a sample standard deviation of $74.50.
a. Develop a 95% confidence interval estimate of the mean amount spent per day by a family of four visiting Niagara Falls.
b. Based on the confidence interval from part (a), does it appear that the population mean amount spent per day by families visiting Niagara Falls differs from the mean reported by the American Automobile Association? Explain.
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8.46
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8.47 Many stock market observers say that when the P/E ratio for stocks gets over 20 the market is overvalued. The P/E ratio is the stock price divided by the most recent 12 months of earnings. Suppose you are interested in seeing whether the current market is overvalued and would also like to know the proportion of companies that pay dividends. A random sample of 30 companies listed on the New York Stock Exchange (NYSE) is provided (Barron’s, January 19, 2004).
a. What is a point estimate of the P/E ratio for the population of stocks listed on the New York Stock Exchange? Develop a 95% confidence interval.
b. Based on your answer to part (a), do you believe that the market is overvalued?
c. What is a point estimate of the proportion of companies on the NYSE that pay dividends? Is the sample size large enough to justify using the normal distribution to construct a confidence interval for this proportion? Why or why not?
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8.48 US Airways conducted a number of studies that indicated a substantial savings could be obtained by encouraging Dividend Miles frequent flyer customers to redeem miles and schedule award flights online( US Airways Attaché, February 2003). One study collected data on the amount of time required to redeem miles and schedule an award flight over the telephone. A sample showing the time in minutes required for each of 150 award flights scheduled by telephone is contained in the data set Flights. Use Excel to help answer the following questions.
a. What is the sample mean number of minutes required to schedule an award flight by telephone?
b. What is the 95% confidence interval for the population mean time to schedule an award flight by telephone?
c. Assume a telephone ticket agent works 7.5 hours per day. How many award flights can one ticket agent be expected to handle a day?
d. Discuss why this information supported US Airways’s plans to use an online system to reduce costs.
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8.49 A survey by Accountemps asked a sample of 200 executives to provide data on the number of minutes per day office workers waste trying to locate mislabeled, misfiled, or misplaced items. Data consistent with this survey are contained in the data set ActTemps.
a. Use ActTemps to develop a point estimate of the number of minutes per day office workers waste trying to locate mislabeled, misfiled, or misplaced items.
b. What is the sample standard deviation?
c. What is the 95% confidence interval for the mean number of minutes wasted per day?
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8.50 Mileage tests are conducted for a particular model of automobile. If a 98% confidence interval with a margin of error of 1 mile per gallon is desired, how many automobiles should be used in the test? Assume that preliminary mileage tests indicate the standard deviation is 2.6 miles per gallon.
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8.51 In developing patient appointment schedules, a medical center wants to estimate the mean time that a staff member spends with each patient. How large a sample should be taken if the desired margin of error is two minutes at a 95% level of confidence? How large a sample should be taken for a 99% level of confidence? Use a planning value for the population standard deviation of eight minutes.
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8.52 Annual salary plus bonus data for chief executive officers are presented in the BusinessWeek Annual Pay Survey. A preliminary sample showed that the standard deviation is $675 with data provided in thousands of dollars. How many chief executive officers should be in a sample if we want to estimate the population mean annual salary plus bonus with a margin of error of $100,000? (Note: The desired margin of error would be E = 100 if the data are in thousands of dollars.) Use 95% confidence.
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8.53 The National Center for Education Statistics reported that 47% of college students work to pay for tuition and living expenses. Assume that a sample of 450 college students was used in the study.
a. Provide a 95% confidence interval for the population proportion of college students who work to pay for tuition and living expenses.
b. Provide a 99% confidence interval for the population proportion of college students who work to pay for tuition and living expenses.
c. What happens to the margin of error as the confidence is increased from 95% to 99%?
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8.54 A USA Today/CNN/Gallup survey of 369 working parents found 200 who said they spend too little time with their children because of work commitments.
a. What is the point estimate of the proportion of the population of working parents who feel they spend too little time with their children because of work commitments?
b. At 95% confidence, what is the margin of error?
c. What is the 95% confidence interval estimate of the population proportion of working parents who feel they spend too little time with their children because of work commitments?
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8.55 Which would be hardest for you to give up: Your computer or your television? In a recent survey of 1677 U.S. Internet users, 74% of the young tech elite (average age of 22) say their computer would be very hard to give up (PC Magazine, February 3, 2004). Only 48% say their television would be very hard to give up.
a. Develop a 95% confidence interval for the proportion of the young tech elite that would find it very hard to give up their computer.
b. Develop a 99% confidence interval for the proportion of the young tech elite that would find it very hard to give up their television.
c. In which case, part (a) or part (b), is the margin of error larger? Explain why.
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8.56 Cincinnati/Northern Kentucky International Airport had the second highest on-time arrival rate for 2005 among the nation’s busiest airports (The Cincinnati Enquirer, February 3,2006). Assume the findings were based on 455 on-time arrivals out of a sample of 550 flights.
a. Develop a point estimate of the on-time arrival rate (proportion of flights arriving on time) for the airport.
b. Construct a 95% confidence interval for the on-time arrival rate of the population of all flights at the airport during 2005.
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8.57 The 2003 Statistical Abstract of the United States reported the percentage of people 18 years of age and older who smoke. Suppose that a study designed to collect new data on smokers and nonsmokers uses a preliminary estimate of the proportion who smoke of .30.
a. How large a sample should be taken to estimate the proportion of smokers in the population with a margin of error of .02? Use 95% confidence.
b. Assume that the study uses your sample size recommendation in part (a) and finds 520 smokers. What is the point estimate of the proportion of smokers in the population?
c. What is the 95% confidence interval for the proportion of smokers in the population?
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8.58 A well-known bank credit card firm wishes to estimate the proportion of credit card holders who carry a nonzero balance at the end of the month and incur an interest charge. Assume that the desired margin of error is .03 at 98% confidence.
a. How large a sample should be selected if it is anticipated that roughly 70% of the firm’s card holders carry a nonzero balance at the end of the month?
b. How large a sample should be selected if no planning value for the proportion could be specified?
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8.59 In a survey, 200 people were asked to identify their major source of news information; 110 stated that their major source was television news.
a. Construct a 95% confidence interval for the proportion of people in the population who consider television their major source of news information.
b. How large a sample would be necessary to estimate the population proportion with a margin of error of .05 at 95% confidence?
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8.60 Although airline schedules and cost are important factors for business travelers when choosing an airline carrier, a USA Today survey found that business travelers list an airline’s frequent flyer program as the most important factor. From a sample of n = 1993 business travelers who responded to the survey, 618 listed a frequent flyer program as the most important factor.
a. What is the point estimate of the proportion of the population of business travelers who believe a frequent flyer program is the most important factor when choosing an airline carrier?
b. Develop a 95% confidence interval estimate of the population proportion.
c. How large a sample would be required to report the margin of error of .01 at 95% confidence? Would you recommend that USA Today attempt to provide this degree of precision? Why or why not?
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