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 Data were collected on the amount spent by 64 customers for lunch at a major Houston restaurant. These data are contained in the WEBfile named Houston. Based upon past studies the population standard deviation is known with σ = $6.
a. At 99% confidence, what is the margin of error?
b. Develop a 99% confidence interval estimate of the mean amount spent for lunch.
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8.6 In an attempt to assess total daily travel taxes in various cities, the Global Business Travel Association conducted a study of daily travel taxes on lodging, rental car, and meals (GBTA Foundation website, October 30, 2012). The data contained in the WEBfile named TravelTax are consistent with the findings of that study for business travel to Chicago. Assume the population standard deviation is known to be $8.50 and develop a 95% confidence interval of the population mean total daily travel taxes for Chicago.
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8.7
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8.8
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8.9
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8.10 Costs are rising for all kinds of medical care. The mean monthly rent at assisted-living facilities was reported to have increased 17% over the last five years to $3486 (The Wall Street Journal, October 27, 2012). Assume this cost estimate is based on a sample of 120 facilities and, from past studies, it can be assumed that the population standard deviation is σ = $650.
a. Develop a 90% confidence interval estimate of the population mean monthly rent.
b. Develop a 95% confidence interval estimate of the population mean monthly rent.
c. Develop a 99% confidence interval estimate of the population mean monthly rent.
d. What happens to the width of the confidence interval as the confidence level is increased? Does this 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 A sample containing years to maturity and yield for 40 corporate bonds is contained in the WEBfile named CorporateBonds (Barron’s, April 2, 2012).
a. What is the sample mean years to maturity for corporate bonds and what is the sample standard deviation?
b. Develop a 95% confidence interval for the population mean years to maturity.
c. What is the sample mean yield on corporate bonds and what is the sample standard deviation?
d. Develop a 95% confidence interval for the population mean yield on corporate bonds.
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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 Older people often have a hard time finding work. AARP reported on the number of weeks it takes a worker aged 55 plus to find a job. The data on number of weeks spent searching for a job contained in the WEBfile named JobSearch are consistent with the AARP findings (AARP Bulletin, April 2008).
a. Provide a point estimate of the population mean number of weeks it takes a worker aged 55 plus to find a job.
b. At 95% confidence, what is the margin of error?
c. What is the 95% confidence interval estimate of the mean?
d. Discuss the degree of skewness found in the sample data. What suggestion would you make for a repeat of this study?
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8.19
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8.20 The average annual premium for automobile insurance in the United States is $1503 (Insure.com website, March 6, 2014). The following annual premiums ($) are representative of the website’s findings for the state of Michigan....Assume the population is approximately normal.
a. Provide a point estimate of the mean annual automobile insurance premium in Michigan.b. Develop a 95% confidence interval for the mean annual automobile insurance premium in Michigan.c Does the 95% confidence interval for the annual automobile insurance premium in Michigan include the national average for the United States? What is your interpretation of the relationship between auto insurance premiums in Michigan and the national average?
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8.21 Health insurers are beginning to offer telemedicine services online that replace the common office visit. Wellpoint provides a video service that allows subscribers to connect with a physician online and receive prescribed treatments (Bloomberg Businessweek, March 4–9, 2014). Wellpoint claims that users of its LiveHealth Online service saved a significant amount of money on a typical visit. The data shown below ($), for a sample of 20 online doctor visits, are consistent with the savings per visit reported by Wellpoint....Assuming the population is roughly symmetric, construct a 95% confidence interval for the mean savings for a televisit to the doctor as opposed to an office visit.
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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 U.S. Energy Information Administration (US EIA) reported that the average price for a gallon of regular gasoline is $3.94 (US EIA website, April 6, 2012). The US EIA updates its estimates of average gas prices on a weekly basis. Assume the standard deviation is $.25 for the price of a gallon of regular gasoline and recommend the appropriate sample size for the US EIA to use if it wishes to report each of the following margins of error at 95% confidence.
a. The desired margin of error is $.10.
b. The desired margin of error is $.07.
c. The desired margin of error is $.05.
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8.27
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8.28 Many medical professionals believe that eating too much red meat increases the risk of heart disease and cancer (WebMD website, March 12, 2014). Suppose you would like to conduct a survey to determine the yearly consumption of beef by a typical American and want to use 3 pounds as the desired margin of error for a confidence interval estimate of the population mean amount of beef consumed annually. Use 25 pounds as a planning value for the population standard deviation and recommend a sample size for each of the following situations.
a. A 90% confidence interval is desired for the mean amount of beef consumed.
b. A 95% confidence interval is desired for the mean amount of beef consumed.
c. A 99% confidence interval is desired for the mean amount of beef consumed.
d. When the desired margin of error is set, what happens to the sample size as the confidence level is increased? Would you recommend using a 99% confidence interval in this case? Discuss.
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8.29 Customers arrive at a movie theater at the advertised movie time only to find that they have to sit through several previews and prepreview ads before the movie starts. Many complain that the time devoted to previews is too long (The Wall Street Journal, October 12, 2012). A preliminary sample conducted by The Wall Street Journal showed that the standard deviation of the amount of time devoted to previews was 4 minutes. Use that as a planning value for the standard deviation in answering the following questions.
a. If we want to estimate the population mean time for previews at movie theaters with a margin of error of 75 seconds, what sample size should be used? Assume 95% confidence.
b. If we want to estimate the population mean time for previews at movie theaters with a margin of error of 1 minute, what sample size should be used? Assume 95% confidence.
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8.30
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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 One of the questions on a survey of 1000 adults asked if today’s children will be better off than their parents (Rasmussen Reports website, October 26, 2012). Representative data are shown in the WEBfile named ChildOutlook. A response of Yes indicates that the adult surveyed did think today’s children will be better off than their parents. A response of No indicates that the adult surveyed did not think today’s children will be better off than their parents. A response of Not Sure was given by 23% of the adults surveyed.
a. What is the point estimate of the proportion of the population of adults who do think that today’s children will be better off than their parents?
b. At 95% confidence, what is the margin of error?
c. What is the 95% confidence interval for the proportion of adults who do think that today’s children will be better off than their parents?
d. What is the 95% confidence interval for the proportion of adults who do not think that today’s children will be better off than their parents?
e. Which of the confidence intervals in parts (c) and (d) has the smaller margin of error? Why?
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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 that 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
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8.40
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8.41 Fewer young people are driving. In 1983, 87% of 19-year-olds had a driver’s license. Twenty-five years later that percentage had dropped to 75% (University of Michigan Transportation Research Institute website, April 7, 2012). Suppose these results are based on a random sample of 1200 19-year-olds in 1983 and again in 2008.
a. At 95% confidence, what is the margin of error and the interval estimate of the number of 19-year-old drivers in 1983?
b. At 95% confidence, what is the margin of error and the interval estimate of the number of 19-year-old drivers in 2008?
c. Is the margin of error the same in parts (a) and (b)? Why or why not?
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8.42 A poll for the presidential campaign sampled 491 potential voters in June. A primary purpose of the poll was to obtain an estimate of the proportion of potential voters who favored each candidate. Assume a planning value of p* =.50 and a 95% confidence level.
a. For p* =.50, what was the planned margin of error for the June poll?...
b. Closer to the November election, better precision and smaller margins of error are desired. Assume the following margins of error are requested for surveys to be conducted during the presidential campaign. Compute the recommended sample size for each survey.
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8.43 The Pew Research Center Internet Project, conducted on the 25th anniversary of the Internet, involved a survey of 857 Internet users (Pew Research Center website, April 1, 2014). It provided a variety of statistics on Internet users. For instance, in 2014, 87% of American adults were Internet users. In 1995 only 14% of American adults used the Internet.
a. The sample survey showed that 90% of respondents said the Internet has been a good thing for them personally. Develop a 95% confidence interval for the proportion of respondents who say the Internet has been a good thing for them personally.
b. The sample survey showed that 67% of Internet users said the Internet has generally strengthened their relationship with family and friends. Develop a 95% confidence interval for the proportion of respondents who say the Internet has strengthened their relationship with family and friends.
c. Fifty-six percent of Internet users have seen an online group come together to help a person or community solve a problem, whereas only 25% have left an online group because of unpleasant interaction. Develop a 95% confidence interval for the proportion of Internet users who say online groups have helped solve a problem.
d. Compare the margin of error for the interval estimates in parts (a), (b), and (c). How is the margin of error related to the sample proportion?
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8.44
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8.45
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8.46
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8.47
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8.48 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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8.49 A survey conducted by the American Automobile Association (AAA) 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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8.50 The 92 million Americans of age 50 and over control 50% of all discretionary income (AARP Bulletin, March 2008). AARP estimated that the average annual expenditure on restaurants and carryout food was $1873 for individuals in this age group. Suppose this estimate is based on a sample of 80 persons and that the sample standard deviation is $550.
a. At 95% confidence, what is the margin of error?
b. What is the 95% confidence interval for the population mean amount spent on restaurants and carryout food?
c. What is your estimate of the total amount spent by Americans of age 50 and over on restaurants and carryout food?
d. If the amount spent on restaurants and carryout food is skewed to the right, would you expect the median amount spent to be greater or less than $1873?
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8.51 Russia has recently started a push for stronger smoking regulations much like those in Western countries concerning cigarette advertising, smoking in public places, and so on. The WEBfile named Russia contains sample data on smoking habits of Russians that are consistent with those reported by The Wall Street Journal (The Wall Street Journal, October 16, 2012). Analyze the data using Excel and answer the following questions.
a. Develop a point estimate and a 95% confidence interval for the proportion of Russians who smoke.
b. Develop a point estimate and a 95% confidence interval for the mean annual per capita consumption (number of cigarettes) of a Russian.
c. For those Russians who do smoke, estimate the number of cigarettes smoked per day.
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8.52 The Health Care Cost Institute tracks health care expenditures for beneficiaries under the age of 65 who are covered by employer-sponsored private health insurance (Health Care Cost Institute website, November 4, 2012). The data contained in the WEBfile named DrugCost are consistent with the institute’s findings concerning annual prescription costs per employee. Analyze the data using Excel and answer the following questions.
a. Develop a 90% confidence interval for the annual cost of prescription drugs.
b. Develop a 90% confidence interval for the amount of out-of-pocket expense per employee.
c. What is your point estimate of the proportion of employees who incurred no prescription drug costs?
d. Which, if either, of the confidence intervals in parts (a) and (b) has a larger margin of error. Why?
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8.53
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8.54 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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8.55 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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8.56 Annual salary plus bonus data for chief executive officers are presented in an 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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8.57 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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8.58 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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8.59 The Pew Research Center has conducted extensive research on the young adult population (Pew Research website, November 6, 2012). One finding was that 93% of adults aged 18 to 29 use the Internet. Another finding was that 21% of those aged 18 to 28 are married. Assume the sample size associated with both findings is 500.
a. Develop a 95% confidence interval for the proportion of adults aged 18 to 29 who use the Internet.
b. Develop a 99% confidence interval for the proportion of adults aged 18 to 28 who are married.
c. In which case, part (a) or part (b), is the margin of error larger? Explain why.
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8.60 A survey of 750 likely voters in Ohio was conducted by the Rasmussen Poll just prior to the general election (Rasmussen Reports website, November 4, 2012). The state of the economy was thought to be an important determinant of how people would vote. Among other things, the survey found that 165 of the respondents rated the economy as good or excellent and 315 rated the economy as poor.
a. Develop a point estimate of the proportion of likely voters in Ohio who rated the economy as good or excellent.
b. Construct a 95% confidence interval for the proportion of likely voters in Ohio who rated the economy as good or excellent.
c. Construct a 95% confidence interval for the proportion of likely voters in Ohio who rated the economy as poor.
d. Which of the confidence intervals in parts (b) and (c) is wider? Why?
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8.61 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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8.62 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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8.63
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8.64 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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