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 percent 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 percent confidence interval for the population mean.

A sample of 10 observations is selected from a normal population for which the population standard deviation is known to be 5 . 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 percent confidence interval even though the sample is less than \(30 .\) c. Determine the 95 percent confidence interval for the population mean.

A research firm conducted a survey to determine the mean amount steady smokers spend on cigarettes during a week. They found the distribution of amounts spent per week followed the normal distribution with a standard deviation of \(\$ 5 .\) A sample of 49 steady smokers revealed that \(\bar{x}=\$ 20 .\) a. What is the point estimate of the population mean? Explain what it indicates. b. Using the 95 percent level of confidence, determine the confidence interval for \(\mu\). Explain what it indicates.

Bob Nale is the owner of Nale's Texaco GasTown. 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 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 percent 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 standard deviation of 2.44 words per essay. For her 10 a.m. section of 40 students, the mean number of misspelled words per essay was \(6.05 .\) Construct a 95 percent 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 laid per chicken. A sample of 20 chickens shows they laid 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 percent confidence interval, what is the value of \(t ?\) d. Develop the 95 percent 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 American Sugar Producers Association wants to estimate the mean yearly sugar consumption. A sample of 16 people reveals the mean yearly consumption to be 60 pounds with a standard deviation of 20 pounds. 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 percent confidence interval, what is the value of \(t ?\) d. Develop the 90 percent confidence interval for the population mean. e. Would it be reasonable to conclude that the population mean is 63 pounds?

14\. The Greater Pittsburgh Area 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{tabular}{|llllllll|} \hline 29 & 38 & 38 & 33 & 38 & 21 & 45 & 34 \\ 40 & 37 & 37 & 42 & 30 & 29 & 35 & \\ \hline \end{tabular} Develop a 98 percent confidence interval for the population mean. Interpret the result.

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

16\. Ms. Maria Wilson is considering running for mayor of the town 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 percent 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 familyoriented comedy show. Before a final decision is made, network executives commission a sample of 400 viewers. After viewing the comedy, 250 indicated they would watch the new show and suggested it replace the crime investigation show. a. Estimate the value of the population proportion. b. Develop a 99 percent confidence interval for the population proportion. c. Interpret your findings.

Schadek Silkscreen Printing, Inc., purchases plastic cups on which to print 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. He found 15 to be defective. a. What is the estimated proportion defective in the population? b. Develop a 95 percent confidence interval for the proportion defective. c. Zack has an agreement with his supplier that he is to return lots that are 10 percent or more defective. Should he return this lot? Explain your decision.

Thirty-six items are randomly selected from a population of 300 items. The sample mean is 35 and the sample standard deviation 5. Develop a 95 percent confidence interval for the population mean.

Forty-nine items are randomly selected from a population of 500 items. The sample mean is 40 and the sample standard deviation 9. Develop a 99 percent confidence interval for the population mean.

The attendance at the Savannah Colts minor league baseball game last night was \(400 .\) A random sample of 50 of those in attendance revealed that the mean number of soft drinks consumed per person was 1.86 with a standard deviation of \(0.50 .\) Develop a 99 percent confidence interval for the mean number of soft drinks consumed per person.

There are 300 welders employed at Maine Shipyards Corporation. A sample of 30 welders revealed that 18 graduated from a registered welding course. Construct the 95 percent confidence interval for the proportion of all welders who graduated from a registered welding course.

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

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

The estimate of the population proportion is to be within plus or minus .05, with a 95 percent level of confidence. The best estimate of the population proportion is .15. How large a sample is required?

A survey is being planned to determine the mean amount of time corporation executives watch television. A pilot survey indicated that the mean time per week is 12 hours, with a standard deviation of 3 hours. It is desired to estimate the mean viewing time within one quarter hour. The 95 percent 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. To test the weight of the boxes, a few were checked. The mean weight was 20.4 pounds, the standard deviation 0.5 pounds. How many boxes must the processor sample to be 95 percent confident that the sample mean does not differ from the population mean by more than 0.2 pound?

Suppose the U.S. president wants an estimate of the proportion of the population who support his current policy toward revisions in the Social Security system. The president wants the estimate to be within . 04 of the true proportion. Assume a 95 percent level of confidence. The president's political advisors estimated the proportion supporting the current policy to be \(.60 .\) 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 percent of tourists going to Las Vegas to gamble during a weekend spend more than \(\$ 1,000 .\) Management wants to update this percentage. a. The new study is to use the 90 percent confidence level. The estimate is to be within 1 percent of the population proportion. What is the necessary sample size? b. Management said that 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 the average, they spent 6.5 years on the job before being promoted. The standard deviation of the sample was 1.7 years. Construct a 95 percent 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 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 pounds. a. What is the estimated population mean? b. Determine a 95 percent confidence interval for the population mean.

A recent study of 50 self-service gasoline stations in the Greater Cincinnati- Northern Kentucky metropolitan area revealed that the mean price of unleaded gas was \(\$ 2.799\) per gallon. The sample standard deviation was \(\$ 0.03\) per gallon. a. Determine a 99 percent confidence interval for the population mean price. b. Would it be reasonable to conclude that the population mean was \(\$ 2.50 ?\) Why or why not?

A recent survey of 50 executives who were laid off from their previous position 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 percent 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 current 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 of those included in 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 percent 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 meals per week. Construct a 97 percent confidence interval for the population mean.

The National Collegiate Athletic Association (NCAA) reported that the mean number of hours spent per week on coaching and recruiting by college football assistant coaches during the season was \(70 .\) 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 percent confidence interval for the population mean. b. Does the 99 percent confidence interval include the value suggested by the NCAA? Interpret this result. c. Suppose you decided to switch from a 99 to a 95 percent 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 percent 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 student conducted a study and reported that the 95 percent confidence interval for the mean ranged from 46 to \(54 .\) He was sure that the mean of the sample was 50 that the standard deviation of the sample was \(16,\) and that the sample was at least \(30,\) but could not remember the exact number. Can you help him out?

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

A study of 25 graduates of four-year colleges by the American Banker's Association revealed the mean amount owed by a student in student loans was \(\$ 14,381 .\) The standard deviation of the sample was \(\$ 1,892 .\) Construct a 90 percent confidence interval for the population mean. Is it reasonable to conclude that the mean of the population is actually \(\$ 15,000 ?\) Tell why or why not.

An important factor in selling a residential property is the number of people who look through the home. A sample of 15 homes recently sold in the Buffalo, New York, area revealed the mean number looking through each home was 24 and the standard deviation of the sample was 5 people. Develop a 98 percent confidence interval for the population mean.

As a condition of employment, Fashion Industries applicants must pass a drug test. Of the last 220 applicants 14 failed the test. Develop a 99 percent confidence interval for the proportion of applicants that fail the test. Would it be reasonable to conclude that more than 10 percent of the applicants are now failing the test? In addition to the testing of applicants, Fashion Industries randomly tests its employees throughout the year. Last year in the 400 random tests conducted, 14 employees failed the test. Would it be reasonable to conclude that less than 5 percent of the employees are not able to pass the random drug test?

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 percent confidence interval for the proportion of voters in the county who plan to vote for Ms. Miller. From this sample information, can you confirm she will be re-elected?

. 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 percent 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 (more than half) of the population believes the president is doing a qood iob?

Police Chief Edward Wilkin of River City reports 500 traffic citations were issued last month. A sample of 35 of these citations showed the mean amount of the fine was \(\$ 54,\) with a standard deviation of \(\$ 4.50 .\) Construct a 95 percent confidence interval for the mean amount of a citation in River City.

The First National Bank of Wilson has 650 checking account customers. A recent sample of 50 of these customers showed 26 to have a Visa card with the bank. Construct the 99 percent confidence interval for the proportion of checking account customers who have a Visa card with the bank.

It is estimated that 60 percent 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 percent level of confidence, how large of a sample is required?

52\. You need to estimate the mean number of travel days per year for pharmaceutical sales representatives. The mean of a small pilot study was 150 days, with a standard deviation of 14 days. If you must estimate the population mean within 2 days, how many sales representative in the pharmaceutical industry should you sample? Use the 90 percent confidence level.

You are to conduct a sample survey to determine the mean family income in a rural area of central Florida. 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 percent 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 mean annual health insurance premium for a family with coverage through an employer was \(\$ 10,979 .\) The standard deviation of the sample was \(\$ 1,000 .\) a. Based on this sample information, develop a 90 percent confidence interval for the population mean yearly premium. b. How large a sample is needed to find the population mean within \(\$ 250\) at 99 percent confidence?

55\. 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 corresponding air pressure on 30 randomly chosen flights. The study revealed a mean equivalent pressure of 8,000 feet with a standard deviation of 300 feet. a. Develop a 99 percent confidence interval for the population mean equivalent pressure. b. How large a sample is needed to find the population mean within 25 feet at 95 percent confidence?

A random sample of 25 people employed by the Florida Department of Transportation 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 percent 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 percent confidence?

A film alliance used a random sample of \(50 \mathrm{U.S.}\) citizens to estimate that the typical American spent 78 hours watching videos and DVDs last year. The standard deviation of this sample was 9 hours. a. Develop a 95 percent confidence interval for the population mean number of hours spent watching videos and DVDs last year. b. How large a sample should be used to be 90 percent confident the sample mean is within 1.0 hour of the population mean?

You plan to conduct a survey to find what proportion of the workforce has two or more jobs. You decide on the 95 percent confidence level and state that the estimated proportion must be within 2 percent of the population proportion. 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?

The proportion of public accountants who have changed companies within the last three years is to be estimated within 3 percent. The 95 percent level of confidence is to be used. A study conducted several years ago revealed that the percent of public accountants changing companies within three years was 21 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?