Chapter 9

A sample of 49 observations is taken from a normal population with a standard deviation of \(10 .\) The sample mean is \(55 .\) Determine the \(99 \%\) confidence interval for the population mean.

A sample of 81 observations is taken from a normal population with a standard deviation of \(5 .\) The sample mean is \(40 .\) Determine the \(95 \%\) confidence interval for the population mean.

A sample of 250 observations is selected from a normal population with a population standard deviation of \(25 .\) The sample mean is 20 . a. Determine the standard error of the mean. b. Explain why we can use formula \((9-1)\) to determine the \(95 \%\) confidence interval. c. Determine the \(95 \%\) confidence interval for the population mean.

A research firm conducted a survey of 49 randomly selected Americans to determine the mean amount spent on coffee during a week. The sample mean was 20 per week. The population distribution is normal with a standard deviation of 5 a. What is the point estimate of the population mean? Explain what it indicates. b. Using the 95% level of confidence, determine the confidence interval for mu . Explain what it indicates.

Bob Nale is the owner of Nale's Quick Fill. Bob would like to estimate the mean number of gallons of gasoline sold to his customers. Assume the number of gallons sold follows the normal distribution with a population standard deviation of 2.30 gallons. From his records, he selects a random sample of 60 sales and finds the mean number of gallons sold is \(8.60 .\) a. What is the point estimate of the population mean? b. Develop a \(99 \%\) confidence interval for the population mean. c. Interpret the meaning of part (b).

Dr. Patton is a professor of English. Recently she counted the number of misspelled words in a group of student essays. She noted the distribution of misspelled words per essay followed the normal distribution with a population standard deviation of 2.44 words per essay. For her 10 a.m. section of 40 students, the mean number of misspelled words was \(6.05 .\) Construct a \(95 \%\) confidence interval for the mean number of misspelled words in the population of student essays.

The owner of Britten's Egg Farm wants to estimate the mean number of eggs produced per chicken. A sample of 20 chickens shows they produced an average of 20 eggs per month with a standard deviation of 2 eggs per month. a. What is the value of the population mean? What is the best estimate of this value? b. Explain why we need to use the \(t\) distribution. What assumption do you need to make? c. For a \(95 \%\) confidence interval, what is the value of \(t ?\) d. Develop the \(95 \%\) confidence interval for the population mean. e. Would it be reasonable to conclude that the population mean is 21 eggs? What about 25 eggs?

The U.S. Dairy Industry wants to estimate the mean yearly milk consumption. A sample of 16 people reveals the mean yearly consumption to be 45 gallons with a standard deviation of 20 gallons. Assume the population distribution is normal. a. What is the value of the population mean? What is the best estimate of this value? b. Explain why we need to use the \(t\) distribution. What assumption do you need to make? c. For a 90\% confidence interval, what is the value of \(t ?\) d. Develop the \(90 \%\) confidence interval for the population mean. e. Would it be reasonable to conclude that the population mean is 48 gallons?

Merrill Lynch Securities and Health Care Retirement Inc. are two larqe employers in downtown Toledo, Ohio. They are considering jointly offering child care for their employees. As a part of the feasibility study, they wish to estimate the mean weekly child care cost of their employees. A sample of 10 employees who use child care reveals the following amounts spent last week. $$\left.\begin{array}{llllllll}\$ 107 & \$ 92 & \$ 97 & \$ 95 & \$ 105 & \$ 101 & \$ 91 & \$ 99 & \$ 95 & \$ 104\end{array}\right)$$ Develop a \(90 \%\) confidence interval for the population mean. Interpret the result.

The Buffalo. New York, Chamber of Commerce wants to estimate the mean time workers who are employed in the downtown area spend getting to work. A sample of 15 workers reveals the following number of minutes spent traveling. $$\begin{array}{llllllll}14 & 24 & 24 & 19 & 24 & 7 & 31 & 20 \\ 26 & 23 & 23 & 28 & 16 & 15 & 21 &\end{array}$$ Develop a \(98 \%\) confidence interval for the population mean. Interpret the result.

The owner of the West End Kwick Fill Gas Station wishes to determine the propor tion of customers who pay at the pump using a credit card or debit card. He surveys 100 customers and finds that 80 paid at the pump. a. Estimate the value of the population proportion. b. Develop a \(95 \%\) confidence interval for the population proportion. c. Interpret your findings.

Ms. Maria Wilson is considering running for mayor of Bear Gulch, Montana. Before completing the petitions, she decides to conduct a survey of voters in Bear Gulch. A sample of 400 voters reveals that 300 would support her in the November election. a. Estimate the value of the population proportion. b. Develop a \(99 \%\) confidence interval for the population proportion. c. Interpret your findings.

The Fox TV network is considering replacing one of its prime-time crime investigation shows with a new family-oriented comedy show. Before a final decision is made, network executives designed an experiment to estimate the proportion of their viewers who would prefer the comedy show over the crime investigation show. A random sample of 400 viewers was selected and asked to watch the new comedy show and the crime investigation show. After viewing the shows, 250 indicated they would watch the new comedy show and suggested it replace the crime investigation show. a. Estimate the value of the population proportion of people who would prefer the comedy show. b. Develop a \(99 \%\) confidence interval for the population proportion of people who would prefer the comedy show. c. Interpret your findings.

Schadek Silkscreen Printing Inc. purchases plastic cups and imprints them with logos for sporting events, proms, birthdays, and other special occasions. Zack Schadek, the owner, received a large shipment this morning. To ensure the quality of the shipment, he selected a random sample of 300 cups and inspected them for defects. He found 15 to be defective a. What is the estimated proportion defective in the population? b. Develop a \(95 \%\) confidence interval for the proportion defective. c. Zack has an agreement with his supplier that if \(10 \%\) or more of the cups are defective, he can return the order. Should he return this lot? Explain your decision.

A population's standard deviation is \(10 .\) We want to estimate the population mean within \(2,\) with a \(95 \%\) level of confidence. How large a sample is required?

We want to estimate the population mean within \(5,\) with a \(99 \%\) level of confidence. The population standard deviation is estimated to be \(15 .\) How large a sample is required?

The estimate of the population proportion should be within plus or minus. \(05,\) with a \(95 \%\) level of confidence. The best estimate of the population proportion is .15. How large a sample is required?

A large on-demand video streaming company is designing a large-scale survey to determine the mean amount of time corporate executives watch on-demand television. A small pilot survey of 10 executives indicated that the mean time per week is 12 hours, with a standard deviation of 3 hours. The estimate of the mean viewing time should be within one-quarter hour. The \(95 \%\) level of confidence is to be used. How many executives should be surveyed?

A processor of carrots cuts the green top off each carrot, washes the carrots, and inserts six to a package. Twenty packages are inserted in a box for shipment. Each box of carrots should weigh 20.4 pounds. The processor knows that the standard deviation of box weight is 0.5 pound. The processor wants to know if the current packing process meets the 20.4 weight standard. How many boxes must the processor sample to be \(95 \%\) confident that the estimate of the population mean is within 0.2 pound?

Suppose the U.S. president wants to estimate the proportion of the population that supports his current policy toward revisions in the health care system. The president wants the estimate to be within . 04 of the true proportion. Assume a \(95 \%\) level of confidence. The president's political advisors found a similar survey from two years ago that reported that \(60 \%\) of people supported health care revisions. a. How large of a sample is required? b. How large of a sample would be necessary if no estimate were available for the proportion supporting current policy?

Past surveys reveal that 30% of tourists going to Las Vegas to gamble spend more than \(\$ 1,000 .\) The Visitor's Bureau of Las Vegas wants to update this percentage. a. How many tourists should be randomly selected to estimate the population proportion with a 90% confidence level and a 1% margin of error? b. The Bureau feels the sample size determined above is too large. What can be done to reduce the sample? Based on your suggestion, recalculate the sample size.

A random sample of 85 group leaders, supervisors, and similar personnel at General Motors revealed that, on average, they spent 6.5 years in a particular job before being promoted. The standard deviation of the sample was 1.7 years. Construct a \(95 \%\) confidence interval.

A state meat inspector in lowa has been given the assignment of estimating the mean net weight of packages of ground chuck labeled "3 pounds." Of course, he realizes that the weights cannot always be precisely 3 pounds. A sample of 36 packages reveals the mean weight to be 3.01 pounds, with a standard deviation of 0.03 pound. a. What is the estimated population mean? b. Determine a \(95 \%\) confidence interval for the population mean.

As part of their business promotional package, the Milwaukee Chamber of Commerce would like an estimate of the mean cost per month to lease a one- bedroom apartment. The mean cost per month for a random sample of 40 apartments currently available for lease was 884. The standard deviation of the sample was 50. a. Develop a 98% confidence interval for the population mean. b. Would it be reasonable to conclude that the population mean is 950 per month?

A recent survey of 50 executives who were laid off during a recent recession revealed it took a mean of 26 weeks for them to find another position. The standard deviation of the sample was 6.2 weeks. Construct a \(95 \%\) confidence interval for the population mean. Is it reasonable that the population mean is 28 weeks? Justify your answer.

Marty Rowatti recently assumed the position of director of the YMCA of South Jersey. He would like some data on how long current members of the YMCA have been members. To investigate, suppose he selects a random sample of 40 current members. The mean length of membership for the sample is 8.32 years and the standard deviation is 3.07 years. a. What is the mean of the population? b. Develop a \(90 \%\) confidence interval for the population mean. c. The previous director, in the summary report she prepared as she retired, indicated the mean length of membership was now "almost 10 years." Does the sample information substantiate this claim? Cite evidence.

The American Restaurant Association collected information on the number of meals eaten outside the home per week by young married couples. A survey of 60 couples showed the sample mean number of meals eaten outside the home was 2.76 meals per week, with a standard deviation of 0.75 meal per week. Construct a \(99 \%\) confidence interval for the population mean.

The National Collegiate Athletic Association (NCAA) reported that college football assistant coaches spend a mean of 70 hours per week on coaching and recruiting during the season. A random sample of 50 assistant coaches showed the sample mean to be 68.6 hours, with a standard deviation of 8.2 hours. a. Using the sample data, construct a \(99 \%\) confidence interval for the population mean. b. Does the \(99 \%\) confidence interval include the value suggested by the NCAA? Interpret this result. c. Suppose you decided to switch from a \(99 \%\) to a \(95 \%\) confidence interval. Without performing any calculations, will the interval increase, decrease, or stay the same? Which of the values in the formula will change?

The human relations department of Electronics Inc. would like to include a dental plan as part of the benefits package. The question is: How much does a typical employee and his or her family spend per year on dental expenses? A sample of 45 employees reveals the mean amount spent last year was 1,820, with a standard deviation of 660. a. Construct a 95%$confidence interval for the population mean. b. The information from part (a) was given to the president of Electronics Inc. He indicated he could afford 1,700 of dental expenses per employee. Is it possible that the population mean could be 1,700 ? Justify your answer.

A recent study by the American Automobile Dealers Association surveyed a random sample of 20 dealers. The data revealed a mean amount of profit per car sold was 290, with a standard deviation of 125 . Develop a \(95 \%\) confidence interval for the population mean of profit per car.

A study of 25 graduates of four-year public colleges revealed the mean amount owed by a student in student loans was 55,051. The standard deviation of the sample was 7,568. Construct a 90 % confidence interval for the population mean. Is it reasonable to conclude that the mean of the population is actually 55,000 ? Explain why or why not.

An important factor in selling a residential property is the number of times real estate agents show a home. A sample of 15 homes recently sold in the Buffalo, New York, area revealed the mean number of times a home was shown was 24 and the standard deviation of the sample was 5 people. Develop a \(98 \%\) confidence interval for the population mean.

In 2003 , the Accreditation Council for Graduate Medical Education (ACGME) implemented new rules limiting work hours for all residents. A key component of these rules is that residents should work no more than 80 hours per week. The following is the number of weekly hours worked in 2017 by a sample of residents at the Tidelands Medical Center. $$ \left.\begin{array}{lllllllllll}84 & 86 & 84 & 86 & 79 & 82 & 87 & 81 & 84 & 78 & 74 & 86\end{array}\right)$$ a. What is the point estimate of the population mean for the number of weekly hours worked at the Tidelands Medical Center? b. Develop a \(90 \%\) confidence interval for the population mean. c. Is the Tidelands Medical Center within the ACGME quideline? Why?

Dr. Susan Benner is an industrial psychologist. She is currently studying stress among executives of Internet companies. She has developed a questionnaire that she believes measures stress. A score above 80 indicates stress at a dangerous level. A random sample of 15 executives revealed the following stress level scores. $$\begin{array}{llllllllllllll}94 & 78 & 83 & 90 & 78 & 99 & 97 & 90 & 97 & 90 & 93 & 94 & 100 & 75 & 84\end{array}$$ a. Find the mean stress level for this sample. What is the point estimate of the population mean? b. Construct a \(95 \%\) confidence level for the population mean. c. According to Dr. Benner's test, is it reasonable to conclude that the mean stress level of Internet executives is \(80 ?\) Explain.

Pharmaceutical companies promote their prescription drugs using television advertising. In a survey of 80 randomly sampled television viewers, 10 indicated that they asked their physician about using a prescription drug they saw advertised on TV. Develop a \(95 \%\) confidence interval for the proportion of viewers who discussed a drug seen on TV with their physician. Is it reasonable to conclude that \(25 \%\) of the viewers discuss an advertised drug with their physician?

HighTech Inc. randomly tests its employees about company policies. Last year, in the 400 random tests conducted, 14 employees failed the test. Develop a \(99 \%\) confidence interval for the proportion of applicants that fail the test. Would it be reasonable to conclude that \(5 \%\) of the employees cannot pass the company policy test? Explain.

During a national debate on changes to health care, a cable news service performs an opinion poll of 500 small-business owners. It shows that \(65 \%\) of small-business owners do not approve of the changes. Develop a \(95 \%\) confidence interval for the proportion opposing health care changes. Comment on the result.

There are 20,000 eligible voters in York County, South Carolina. A random sample of 500 York County voters revealed 350 plan to vote to return Louella Miller to the state senate. Construct a \(99 \%\) confidence interval for the proportion of voters in the county who plan to vote for Ms. Miller. From this sample information, is it reasonable to conclude that Ms. Miller will receive a majority of the votes?

In a poll to estimate presidential popularity, each person in a random sample of 1,000 voters was asked to agree with one of the following statements: 1\. The president is doing a good job. 2\. The president is doing a poor job. 3\. I have no opinion. A total of 560 respondents selected the first statement, indicating they thought the president was doing a good job. a. Construct a \(95 \%\) confidence interval for the proportion of respondents who feel the president is doing a good job. b. Based on your interval in part (a), is it reasonable to conclude that a majority of the population believes the president is doing a good job?

It is estimated that \(60 \%\) of U.S. households subscribe to cable TV. You would like to verify this statement for your class in mass communications. If you want your estimate to be within 5 percentage points, with a \(95 \%\) level of confidence, how many households should you sample?

You wish to estimate the mean number of travel days per year for salespeople. The mean of a small pilot study was 150 days, with a standard deviation of 14 days. If you want to estimate the population mean within 2 days, how many salespeople should you sample? Use the \(90 \%\) confidence level.

You want to estimate the mean family income in a rural area of central Indiana. The question is, how many families should be sampled? In a pilot sample of 10 families, the standard deviation of the sample was 500 . The sponsor of the survey wants you to use the \(95 \%\) confidence level. The estimate is to be within 100 How many families should be interviewed?

Families USA, a monthly magazine that discusses issues related to health and health costs, surveyed 20 of its subscribers. It found that the annual health insurance premiums for a family with coverage through an employer averaged 10,979 . The standard deviation of the sample was 1,000 . a. Based on this sample information, develop a \(90 \%\) confidence interval for the population mean yearly premium. b. How large a sample is needed to find the population mean within 250$ at 99% confidence?

Passenger comfort is influenced by the amount of pressurization in an airline cabin. Higher pressurization permits a closer-to-normal environment and a more relaxed flight. A study by an airline user group recorded the equivalent air pressure on 30 randomly chosen flights. The study revealed a mean equivalent air pressure of 8,000 feet with a standard deviation of 300 feet. a. Develop a \(99 \%\) confidence interval for the population mean equivalent air pressure. b. How large a sample is needed to find the population mean within 25 feet at \(95 \%\) confidence?

A survey of 25 randomly sampled judges employed by the state of Florida found that they earned an average wage (including benefits) of 65.00 per hour. The sample standard deviation was 6.25 per hour. a. What is the population mean? What is the best estimate of the population mean? b. Develop a 99 % confidence interval for the population mean wage (including benefits) for these employees. c. How large a sample is needed to assess the population mean with an allowable error of 1.00 at 95%confidence?

Based on a sample of 50 U.S. citizens, the American Film Institute found that a typical American spent 78 hours watching movies last year. The standard deviation of this sample was 9 hours. a. Develop a \(95 \%\) confidence interval for the population mean number of hours spent watching movies last year. b. How large a sample should be used to be \(90 \%\) confident the sample mean is within 1\. O hour of the population mean?

Dylan Jones kept careful records of the fuel efficiency of his new car. After the first nine times he filled up the tank, he found the mean was 23.4 miles per gallon (mpg) with a sample standard deviation of 0.9 mpg. a. Compute the \(95 \%\) confidence interval for his mpg. b. How many times should he fill his gas tank to obtain a margin of error below 0.1 mpg?

A survey of 36 randomly selected iPhone owners showed that the purchase price has a mean of \(\$ 650\) with a sample standard deviation of 24 . a. Compute the standard error of the sample mean. b. Compute the 95% confidence interval for the mean. c. How large a sample is needed to estimate the population mean within 10 ?

You plan to conduct a survey to find what proportion of the workforce has two or more jobs. You decide on the \(95 \%\) confidence level and a margin of error of \(2 \% .\) A pilot survey reveals that 5 of the 50 sampled hold two or more jobs. How many in the workforce should be interviewed to meet your requirements?

A study conducted several years ago reported that 21 percent of public accountants changed companies within 3 years. The American Institute of CPAs would like to update the study. They would like to estimate the population proportion of public accountants who changed companies within 3 years with a margin of error of \(3 \%\) and a \(95 \%\) level of confidence. a. To update this study, the files of how many public accountants should be studied? b. How many public accountants should be contacted if no previous estimates of the population proportion are available?