Chapter 8

Listed below are the 27 Nationwide Insurance agents in the Toledo, Ohio, metropolitan area. We would like to estimate the mean number of years employed with Nationwide. a. We want to select a random sample of four agents. The random numbers are: 02,59,51,25,14,29,77 \(69,\) and \(18 .\) Which dealers would be included in the sample? b. Use the table of random numbers to select your own sample of four agents. c. A sample is to consist of every seventh dealer. The number 04 is selected as the starting point. Which agents will be included in the sample?

A population consists of the following five values: 2,2 , \(4,4,\) and 8 a. List all samples of size \(2,\) and compute the mean of each sample. b. Compute the mean of the distribution of sample means and the population mean. Compare the two values. c. Compare the dispersion in the population with that of the sample means.

A population consists of the following five values: 12 , \(12,14,15,\) and 20 a. List all samples of size \(3,\) and compute the mean of each sample. b. Compute the mean of the distribution of sample means and the population mean. Compare the two values. c. Compare the dispersion in the population with that of the sample means.

A population consists of the following five values: 0,0 , 1,3,6 a. List all samples of size \(3,\) and compute the mean of each sample. b. Compute the mean of the distribution of sample means and the population mean. Compare the two values. c. Compare the dispersion in the population with that of the sample means.

Scrapper Elevator Company has 20 sales representatives who sell its product throughout the United States and Canada. The number of units sold last month by each representative is listed below. Assume these sales figures to be the population values. a. Draw a graph showing the population distribution. b. Compute the mean of the population. c. Select five random samples of 5 each. Compute the mean of each sample. Use the methods described in this chapter and \(\underline{\text { Appendix }} \mathrm{B} .6\) to determine the items to be included in the sample. d. Compare the mean of the sampling distribution of the sample means to the population mean. Would you expect the two values to be about the same? e. Draw a histogram of the sample means. Do you notice a difference in the shape of the distribution of sample means compared to the shape of the population distribution?

Consider all of the coins (pennies, nickels, quarters, etc.) in your pocket or purse as a population. Make a frequency table beginning with the current year and counting backward to record the ages (in years) of the coins. For example, if the current year is \(2007,\) then a coin with 2004 stamped on it is 3 years old. a. Draw a histogram or other graph showing the population distribution. b. Randomly select five coins and record the mean age of the sampled coins. Repeat this sampling process 20 times. Now draw a histogram or other graph showing the distribution of the sample means. c. Compare the shapes of the two histograms.

Consider the digits in the phone numbers on a randomly selected page of your local phone book a population. Make a frequency table of the final digit of 30 randomly selected phone numbers. For example, if a phone number is \(555-9704,\) record a 4 a. Draw a histogram or other graph of this population distribution. Using the uniform distribution, compute the population mean and the population standard deviation. b. Also record the sample mean of the final four digits (9704 would lead to a mean of 5). Now draw a histogram or other graph showing the distribution of the sample means. c. Compare the shapes of the two histograms.

A normal population has a mean of 60 and a standard deviation of \(12 .\) You select a random sample of \(9 .\) Compute the probability the sample mean is: a. Greater than \(63 .\) b. Less than 56 . c. Between 56 and \(63 .\)

A normal population has a mean of 75 and a standard deviation of \(5 .\) You select a sample of \(40 .\) Compute the probability the sample mean is: a. Less than 74 . b. Between 74 and 76 . c. Between 76 and 77 . d. Greater than 77 .

The rent for a one-bedroom apartment in Southern California follows the normal distribution with a mean of \(\$ 2,200\) per month and a standard deviation of \(\$ 250\) per month. The distribution of the monthly costs does not follow the normal distribution. In fact, it is positively skewed. What is the probability of selecting a sample of 50 one-bedroom apartments and finding the mean to be at least \(\$ 1,950\) per month?

According to an IRS study, it takes a mean of 330 minutes for taxpayers to prepare, copy, and electronically file a 1040 tax form. This distribution of times follows the normal distribution and the standard deviation is 80 minutes. A consumer watchdog agency selects a random sample of 40 taxpayers. a. What is the standard error of the mean in this example? b. What is the likelihood the sample mean is greater than 320 minutes? c. What is the likelihood the sample mean is between 320 and 350 minutes? d. What is the likelihood the sample mean is greater than 350 minutes?

What is sampling error? Could the value of the sampling error be zero? If it were zero, what would this mean?

The manufacturer of eMachines, an economy-priced computer, recently completed the design for a new laptop model. eMachine's top management would like some assistance in pricing the new laptop. Two market research firms were contacted and asked to prepare a pricing strategy. Marketing-Gets-Results tested the new eMachines laptop with 50 randomly selected consumers, who indicated they plan to purchase a laptop within the next year. The second marketing research firm, called Marketing-Reaps-Profits, \(\quad\) test-marketed the new eMachines laptop with 200 current laptop owners. Which of the marketing research companies' test results will be more useful? Discuss why.

Answer the following questions in one or two wellconstructed sentences. a. What happens to the standard error of the mean if the sample size is increased? b. What happens to the distribution of the sample means if the sample size in increased? c. When using the distribution of sample means to estimate the population mean, what is the benefit of using larger sample sizes?

Suppose your statistics instructor gave six examinations during the semester. You received the following grades (percent correct): 79,64,84,82,92 and 77 . Instead of averaging the six scores, the instructor indicated he would randomly select two grades and compute the final percent correct based on the two percents. a. How many different samples of two test grades are possible? b. List all possible samples of size two and compute the mean of each. c. Compute the mean of the sample means and compare it to the population mean. d. If you were a student, would you like this arrangement? Would the result be different from dropping the lowest score? Write a brief report.

At the downtown office of First National Bank there are five tellers. Last week the tellers made the following number of errors each: \(2,3,5,3,\) and \(5 .\) a. How many different samples of 2 tellers are possible? b. List all possible samples of size 2 and compute the mean of each. c. Compute the mean of the sample means and compare it to the population mean.

Mattel Corporation produces a remote-controlled car that requires three AA batteries. The mean life of these batteries in this product is 35.0 hours. The distribution of the battery lives closely follows the normal probability distribution with a standard deviation of 5.5 hours. As a part of its testing program Sony tests samples of 25 batteries. a. What can you say about the shape of the distribution of the sample mean? b. What is the standard error of the distribution of the sample mean? c. What proportion of the samples will have a mean useful life of more than 36 hours? d. What proportion of the sample will have a mean useful life greater than 34.5 hours? e. What proportion of the sample will have a mean useful life between 34.5 and 36.0 hours?

CRA CDs, Inc., wants the mean lengths of the "cuts" on a CD to be 135 seconds (2 minutes and 15 seconds). This will allow the disk jockeys to have plenty of time for commercials within each 10 -minute segment. Assume the distribution of the length of the cuts follows the normal distribution with a population standard deviation of 8 seconds. Suppose we select a sample of 16 cuts from various CDs sold by CRA CDs, Inc. a. What can we say about the shape of the distribution of the sample mean? b. What is the standard error of the mean? c. What percent of the sample means will be greater than 140 seconds? d. What percent of the sample means will be greater than 128 seconds? e. What percent of the sample means will be greater than 128 but less than 140 seconds?

Recent studies indicate that the typical 50 -year-old woman spends \(\$ 350\) per year for personal-care products. The distribution of the amounts spent follows a normal distribution with a standard deviation of \(\$ 45\) per year. We select a random sample of 40 women. The mean amount spent for those sampled is \(\$ 335 .\) What is the likelihood of finding a sample mean this large or larger from the specified population?

Information from the American Institute of Insurance indicates the mean amount of life insurance per household in the United States is \(\$ 110,000 .\) This distribution follows the normal distribution with a standard deviation of \(\$ 40,000 .\) a. If we select a random sample of 50 households, what is the standard error of the mean? b. What is the expected shape of the distribution of the sample mean? c. What is the likelihood of selecting a sample with a mean of at least \(\$ 112,000 ?\) d. What is the likelihood of selecting a sample with a mean of more than \(\$ 100,000 ?\) e. Find the likelihood of selecting a sample with a mean of more than \(\$ 100,000\) but less than \(\$ 112,000\).

The mean age at which men in the United States marry for the first time follows the normal distribution with a mean of 24.8 years. The standard deviation of the distribution is 2.5 years. For a random sample of 60 men, what is the likelihood that the age at which they were married for the first time is less than 25.1 years?

A recent study by the Greater Los Angeles Taxi Drivers Association showed that the mean fare charged for service from Hermosa Beach to Los Angeles International Airport is \(\$ 18.00\) and the standard deviation is \(\$ 3.50 .\) We select a sample of 15 fares. a. What is the likelihood that the sample mean is between \(\$ 17.00\) and \(\$ 20.00 ?\) b. What must you assume to make the above calculation?

Crossett Trucking Company claims that the mean weight of its delivery trucks when they are fully loaded is 6,000 pounds and the standard deviation is 150 pounds. Assume that the population follows the normal distribution. Forty trucks are randomly selected and weighed. Within what limits will 95 percent of the sample means occur?

The mean amount purchased by a typical customer at Churchill's Grocery Store is \(\$ 23.50\) with a standard deviation of \(\$ 5.00 .\) Assume the distribution of amounts purchased follows the normal distribution. For a sample of 50 customers, answer the following questions. a. What is the likelihood the sample mean is at least \(\$ 25.00 ?\) b. What is the likelihood the sample mean is greater than \(\$ 22.50\) but less than \(\$ 25.00 ?\) c. Within what limits will 90 percent of the sample means occur?

The mean SAT score for Division I student-athletes is 947 with a standard deviation of \(205 .\) If you select a random sample of 60 of these students, what is the probability the mean is below \(900 ?\)

Suppose we roll a fair die two times. a. How many different samples are there? b. List each of the possible samples and compute the mean. c. On a chart similar to Chart \(8-1\), compare the distribution of sample means with the distribution of the population. d. Compute the mean and the standard deviation of each distribution and compare them.

Human Resource Consulting (HRC) is surveying a sample of 60 firms in order to study health care costs for a client. One of the items being tracked is the annual deductible that employees must pay. The state Bureau of Labor reports the mean of this distribution is \(\$ 502\) with a standard deviation of \(\$ 100 .\) a. Compute the standard error of the sample mean for HRC. b. What is the chance HRC finds a sample mean between \(\$ 477\) and \(\$ 527 ?\) c. Calculate the likelihood that the sample mean is between \(\$ 492\) and \(\$ 512\). d. What is the probability the sample mean is greater that \(\$ 550 ?\)

The Oil Price Information Center reports the mean price per gallon of regular gasoline is \(\$ 3.26\) with a population standard deviation of \(\$ 0.18 .\) Assume a random sample of 40 gasoline stations is selected and their mean cost for regular gasoline is computed. a. What is the standard error of the mean in this experiment? b. What is the probability that the sample mean is between \(\$ 3.24\) and \(\$ 3.28 ?\) c. What is the probability that the difference between the sample mean and the population mean is less than \(0.01 ?\) d. What is the likelihood the sample mean is greater than \(\$ 3.34 ?\)

Nike's annual report says that the average American buys 6.5 pairs of sports shoes per year. Suppose the population standard deviation is 2.1 and that a sample of 81 customers will be examined next year. a. What is the standard error of the mean in this experiment? b. What is the probability that the sample mean is between 6 and 7 pairs of sports shoes? c. What is the probability that the difference between the sample mean and the population mean is less than 0.25 pairs? d. What is the likelihood the sample mean is greater than 7 pairs?

You need to find the "typical" or mean annual dividend per share for large banks. You decide to sample six banks listed on the New York Stock Exchange. These banks and their trading symbols follow. a. After numbering the banks from 01 to \(24,\) which banks would be included in a sample if the random numbers were \(14,08,24,25,05,44,02,\) and \(22 ?\) Go to the following website: \(\quad\) http://bigcharts.marketwatch.com. Enter the trading symbol for each of the sampled banks and record the price earnings ratio (P/E ratio). Determine the mean annual dividend per share for the sample of banks. b. Which banks are selected if you use a systematic sample of every fourth bank starting with the random number \(03 ?\)