Essentials of Statistics for Business & Economics

David R. Anderson; Dennis J. Sweeney; Thomas A. Williams; Jeffrey D. Camm; James J. Cochran

Chapter 7

Sampling and Sampling Distributions - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider a finite population with five elements labeled A, B, C, D, and E. Ten possible simple random samples of size 2 can be selected.
a. List the 10 samples beginning with $\mathrm{AB}, \mathrm{AC}$, and so on.
b. Using simple random sampling, what is the probability that each sample of size 2 is selected?
c. Assume random number 1 corresponds to A , random number 2 corresponds to B , and so on. List the simple random sample of size 2 that will be selected by using the random digits 8057532 .

Ryan Mcalister

Ryan Mcalister

Numerade Educator

Problem 2

Assume a finite population has 350 elements. Using the last three digits of each of the following five-digit random numbers (e.g., $601,022,448, \ldots$ ), determine the first four elements that will be selected for the simple random sample.
$$
\begin{array}{lllllllll}
98601 & 73022 & 83448 & 02147 & 34229 & 27553 & 84147 & 93289 & 14209
\end{array}
$$

Nick Johnson

Numerade Educator

Problem 2

Suppose a random sample of size 50 is selected from a population with $\sigma=10$. Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate).
a. The population size is infinite.
b. The population size is $N=50,000$.
c. The population size is $N=5000$.
d. The population size is $N=500$.

Sheryl Ezze

Numerade Educator

Problem 3

Industrial Stock Performance. Fortune publishes data on sales, profits, assets, stockholders' equity, market value, and earnings per share for the 500 largest U.S. industrial corporations every year. Assume that you want to select a simple random sample of 10 corporations from the Fortune 500 list. Use the last three digits in column 9 of Table 7.1 , beginning with 554 . Read down the column and identify the numbers of the 10 corporations that would be selected.

Nick Johnson

Numerade Educator

Problem 4

Investigating Trading Practices. The 10 most active stocks on the New York Stock Exchange for a given week, are shown here.
$$
\begin{array}{lllll}
\text { AT\&T } & \text { Alcatel Lucent } & \text { Exxon Mobile } & \text { Petrobras } & \text { Vale SA } \\
\text { Pfizer } & \text { Verizon } & \text { Gen. Elect. } & \text { Citigroup } & \text { Ford }
\end{array}
$$
Exchange authorities decided to investigate trading practices using a sample of three of these stocks.
a. Beginning with the first random digit in column 6 of Table 7.1, read down the column to select a simple random sample of three stocks for the exchange authorities.
b. Using the information in the third Note and Comment, determine how many different simple random samples of size 3 can be selected from the list of 10 stocks.

Nick Johnson

Numerade Educator

Problem 5

Pass-Fail Grading. A student government organization is interested in estimating the proportion of students who favor a mandatory "pass-fail" grading policy for elective courses. A list of names and addresses of the 645 students enrolled during the current quarter is available from the registrar's office. Using three-digit random numbers in row 10 of Table 7.1 and moving across the row from left to right, identify the first 10 students who would be selected using simple random sampling. The three-digit random numbers begin with 816,283 , and 610 .

Nick Johnson

Numerade Educator

Problem 6

Census Bureau County Data. The County and City Data Book, published by the Census Bureau, lists information on 3139 counties throughout the United States. Assume that a national study will collect data from 30 randomly selected counties. Use four-digit random numbers from the last column of Table 7.1 to identify the numbers corresponding to the first five counties selected for the sample. Ignore the first digits and begin with the four-digit random numbers $9945,8364,5702$, and so on.

Nick Johnson

Numerade Educator

Problem 7

Sampling Doctors. Assume that we want to identify a simple random sample of 12 of the 372 doctors practicing in a particular city. The doctors' names are available from a local medical organization. Use the eighth column of five-digit random numbers in Table 7.1 to identify the 12 doctors for the sample. Ignore the first two random digits in each five-digit grouping of the random numbers. This process begins with random number 108 and proceeds down the column of random numbers.

Nick Johnson

Numerade Educator

Problem 8

DJIA Stocks. The following stocks make up the Dow Jones Industrial Average.$$
\begin{array}{|c|c|c|}
\hline \text { 1. } 3 \mathrm{M} & \text { 11. Exxon Mobil } & \text { 21. Microsoft } \\
\hline \text { 2. American Express } & \text { 12. General Electric } & \text { 22. Nike } \\
\hline \text { 3. Apple } & \text { 13. Goldman Sachs } & \text { 23. Pfizer } \\
\hline \text { 4. Boeing } & \text { 14. Home Depot } & \text { 24. Procter \& Gamble } \\
\hline \text { 5. Caterpillar } & \text { 15. IBM } & \text { 25. Travelers Companies, Inc } \\
\hline \text { 6. Chevron } & \text { 16. Intel } & \text { 26. United Technologies } \\
\hline \text { 7. Cisco } & \text { 17. Johnson \& Johnson } & \text { 27. UnitedHealth } \\
\hline \text { 8. Coca-Cola } & \text { 18. JP Morgan Chase } & \text { 28. Verizon } \\
\hline \text { 9. Disney } & \text { 19. McDonald's } & \text { 29. Visa } \\
\hline \text { 0. DowDuPont, Inc. } & \text { 20. Merck } & \text { 30. Wal-Mart } \\
\hline
\end{array}
$$
Suppose you would like to select a sample of six of these companies to conduct an indepth study of management practices. Use the first two digits in each row of the ninth column of Table 7.1 to select a simple random sample of six companies.

Nick Johnson

Numerade Educator

Problem 9

Returns on Mutual Finds. The Wall Street Journal provides the net asset value, the year-to-date percent return, and the three-year percent return for 882 mutual funds at the end of 2017. Assume that a simple random sample of 12 of the 882 mutual funds will be selected for a follow-up study on the size and performance of mutual funds. Use the fourth column of the random numbers in Table 7.1, beginning with 51102, to select the simple random sample of 12 mutual funds. Begin with mutual fund 102 and use the last three digits in each row of the fourth column for your selection process. What are the numbers of the 12 mutual funds in the simple random sample?

Nick Johnson

Numerade Educator

Problem 10

Sampling from Infinite Populations. Indicate which of the following situations involve sampling from a finite population and which involve sampling from an infinite population. In cases where the sampled population is finite, describe how you would construct a frame.
a. Obtain a sample of licensed drivers in the state of New York.
b. Obtain a sample of boxes of cereal produced by the Breakfast Choice company.
c. Obtain a sample of cars crossing the Golden Gate Bridge on a typical weekday.
d. Obtain a sample of students in a statistics course at Indiana University.
e. Obtain a sample of the orders that are processed by a mail-order firm.

Sheryl Ezze

Numerade Educator

Problem 11

The following data are from a simple random sample.
$\begin{array}{llllll}5 & 8 & 10 & 7 & 10 & 14\end{array}$
a. What is the point estimate of the population mean?
b. What is the point estimate of the population standard deviation?

Andrew Kim

Numerade Educator

Problem 13

Monthly Sales Data. A sample of 5 months of sales data provided the following information:
a. Develop a point estimate of the population mean number of units sold per month.
b. Develop a point estimate of the population standard deviation.

MW

Michan Walsh

Numerade Educator

Problem 14

Morningstar Stock Data. Morningstar publishes ratings data on 1208 company stocks. A sample of 40 of these stocks is contained in the file Morningstar. Use the Morningstar data set to answer the following questions.
a. Develop a point estimate of the proportion of the stocks that receive Morningstar's highest rating of 5 Stars.
b. Develop a point estimate of the proportion of the Morningstar stocks that are rated Above Average with respect to business risk.
c. Develop a point estimate of the proportion of the Morningstar stocks that are rated 2 Stars or less.

Dominador Tan

Numerade Educator

Problem 15

Rating Wines. According to Wine-Searcher, wine critics generally use a wine-scoring scale to communicate their opinions on the relative quality of wines. Wine scores range from 0 to 100 , with a score of $95-100$ indicating a great wine, $90-94$ indicating an outstanding wine, $85-89$ indicating a very good wine, $80-84$ indicating a good wine, 75-79 indicating a mediocre wine, and below 75 indicating that the wine is not recommended. Random ratings of a pinot noir recently produced by a newly established vineyard in 2018 follow:
$$
\begin{array}{llllll}
87 & 91 & 86 & 82 & 72 & 91 \\
60 & 77 & 80 & 79 & 83 & 96
\end{array}
$$
a. Develop a point estimate of mean wine score for this pinot noir.
b. Develop a point estimate of the standard deviation for wine scores received by this pinot noir.

Sheryl Ezze

Numerade Educator

Problem 16

AARP Survey. In a sample of 426 U.S. adults age 50 and older, AARP asked how important a variety of issues were in choosing whom to vote for in the next presidential election.
a. What is the sampled population for this study?
b. Social Security and Medicare was cited as "very important" by 350 respondents. Estimate the proportion of the population of U.S. adults age 50 and over who believe this issue is very important.
c. Education was cited as "very important" by $74 \%$ of the respondents. Estimate the number of respondents who believe this issue is very important.
d. Job Growth was cited as "very important" by 354 respondents. Estimate the proportion of U.S. adults age 50 and over who believe job growth is very important.
e. What is the target population for the inferences being made in parts (b) and (d)? Is it the same as the sampled population you identified in part (a)? Suppose you later learn that the sample was restricted to members of the American Association of Retired People (AARP). Would you still feel the inferences being made in parts (b) and (d) are valid? Why or why not?

Sheryl Ezze

Numerade Educator

Problem 17

Attitudes Toward Automation. The Pew American Trends Survey includes a series of questions on attitudes toward automation. The May 2018 results showed that 2977 of 4135 respondents are worried about a future in which robots and computers can do many human jobs, 2770 are worried about the development of algorithms that can evaluate and hire job candidates, and 2233 are worried about the development of driverless vehicles.
a. Develop a point estimate of the proportion of respondents who are worried about a future in which robots and computers can do many human jobs.
b. Develop a point estimate of the proportion of respondents who are worried about the development of algorithms that can evaluate and hire job candidates.
c. Develop a point estimate of the proportion of respondents who are worried about the development of driverless vehicles.

Sheryl Ezze

Numerade Educator

Problem 18

A population has a mean of 200 and a standard deviation of 50 . A sample of size 100 will be taken and the sample mean $\bar{x}$ will be used to estimate the population mean.
a. What is the expected value of $\bar{x}$ ?
b. What is the standard deviation of $\bar{x}$ ?
c. Show the sampling distribution of $\bar{x}$.
d. What does the sampling distribution of $\bar{x}$ show?

Nick Johnson

Numerade Educator

Problem 19

A population has a mean of 200 and a standard deviation of 50 . Suppose a sample of size 100 is selected and $\bar{x}$ is used to estimate $\mu$.
a. What is the probability that the sample mean will be within $\pm 5$ of the population mean?
b. What is the probability that the sample mean will be within $\pm 10$ of the population mean?

Rashmi Sinha

Numerade Educator

Problem 20

Assume the population standard deviation is $\sigma=25$. Compute the standard error of the mean, $\sigma_{\bar{S}}$, for sample sizes of $50,100,150$, and 200 . What can you say about the size of the standard error of the mean as the sample size is increased?

Foster Wisusik

Numerade Educator

Problem 22

Sampling Distribution for Electronic Associates, Inc., Managers. Refer to the EAI sampling problem. Suppose a simple random sample of 60 managers is used.
a. Sketch the sampling distribution of $\bar{x}$ when simple random samples of size 60 are used.
b. What happens to the sampling distribution of $\bar{x}$ if simple random samples of size 120 are used?
c. What general statement can you make about what happens to the sampling distribution of $\bar{x}$ as the sample size is increased? Does this generalization seem logical? Explain.

Sheryl Ezze

Numerade Educator

Problem 23

Finding Probabilities for Electronic Associates, Inc., Managers. In the EAI sampling problem (see Figure 7.5), we showed that for $n=30$, there was . 5034 probability of obtaining a sample mean within $$\pm \$ 500$$ of the population mean.
a. What is the probability that $\bar{x}$ is within $$\$ 500$$ of the population mean if a sample of size 60 is used?
b. Answer part (a) for a sample of size 120 .

Hoan Nguyen

Numerade Educator

Problem 24

U.S. Unemployment. Barron's reported that the average number of weeks an individual is unemployed is 17.5 weeks. Assume that for the population of all unemployed individuals the population mean length of unemployment is 17.5 weeks and that the population standard deviation is 4 weeks. Suppose you would like to select a sample of 50 unemployed individuals for a follow-up study.
a. Show the sampling distribution of $\bar{x}$, the sample mean average for a sample of 50 unemployed individuals.
b. What is the probability that a simple random sample of 50 unemployed individuals will provide a sample mean within 1 week of the population mean?
c. What is the probability that a simple random sample of 50 unemployed individuals will provide a sample mean within $1 / 2$ week of the population mean?

Sheryl Ezze

Numerade Educator

Problem 25

SAT Scores. In May 2018, The College Board reported the following mean scores for two parts of the Scholastic Aptitude Test (SAT):
Evidence-Based Reading and Writing
533
Mathematics
527
Assume that the population standard deviation on each part of the test is $\sigma=100$.
a. What is the probability a sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 533 on the Evidence-Based Reading and Writing part of the test?
b. What is the probability a sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 527 on the Mathematics part of the test?
c. Comment on the differences between the values computed in parts (a) and (b).

Sheryl Ezze

Numerade Educator

Problem 26

Federal Income Tax Returns. The Wall Street Journal reports that 33\% of taxpayers with adjusted gross incomes between $$\$ 30,000$$ and $$\$ 60,000$$ itemized deductions on their federal income tax return. The mean amount of deductions for this population of taxpayers was $$\$ 16,642$$. Assume the standard deviation is $$\sigma=\$ 2400$$.
a. What is the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within $$\$ 200$$ of the population mean for each of the following sample sizes: $30,50,100$, and 400 ?
b. What is the advantage of a larger sample size when attempting to estimate the population mean?

Sheryl Ezze

Numerade Educator

Problem 27

College Graduate-Level Wages. The Economic Policy Institute periodically issues reports on worker's wages. The institute reported that mean wages for male college graduates were $$\$ 37.39$$ per hour and for female college graduates were $$\$ 27.83$$ per hour in 2017. Assume the standard deviation for male graduates is $$\$ 4.60$$, and for female graduates it is $$\$ 4.10$$.
a. What is the probability that a sample of 50 male graduates will provide a sample mean within $$\$ 1.00$$ of the population mean, $$\$ 37.39$$ ?
b. What is the probability that a sample of 50 female graduates will provide a sample mean within $$\$ 1.00$$ of the population mean, $$\$ 27.83$$ ?
c. In which of the preceding two cases, part (a) or part (b), do we have a higher probability of obtaining a sample estimate within $$\$ 1.00$$ of the population mean? Why?
d. What is the probability that a sample of 120 female graduates will provide a sample mean more than $$\$ .60$$ below the population mean, 27.83 ?

Sheryl Ezze

Numerade Educator

Problem 28

State Rainfalls. The state of California has a mean annual rainfall of 22 inches, whereas the state of New York has a mean annual rainfall of 42 inches. Assume that the standard deviation for both states is 4 inches. A sample of 30 years of rainfall for California and a sample of 45 years of rainfall for New York has been taken.
a. Show the probability distribution of the sample mean annual rainfall for California.
b. What is the probability that the sample mean is within 1 inch of the population mean for California?
c. What is the probability that the sample mean is within 1 inch of the population mean for New York?
d. In which case, part (b) or part (c), is the probability of obtaining a sample mean within 1 inch of the population mean greater? Why?

Sheryl Ezze

Numerade Educator

Problem 29

Income Tax Return Preparation Fees. The CPA Practice Advisor reports that the mean preparation fee for 2017 federal income tax returns was $$\$ 273$$. Use this price as the population mean and assume the population standard deviation of preparation fees is $$\$ 100$$.
a. What is the probability that the mean price for a sample of 30 federal income tax returns is within $$\$ 16$$ of the population mean?
b. What is the probability that the mean price for a sample of 50 federal income tax returns is within $$\$ 16$$ of the population mean?
c. What is the probability that the mean price for a sample of 100 federal income tax returns is within $$\$ 16$$ of the population mean?
d. Which, if any, of the sample sizes in parts (a), (b), and (c) would you recommend to ensure at least a .95 probability that the sample mean is within $$\$ 16$$ of the population mean?

Sheryl Ezze

Numerade Educator

Problem 30

Employee Ages. To estimate the mean age for a population of 4000 employees, a simple random sample of 40 employees is selected.
a. Would you use the finite population correction factor in calculating the standard error of the mean? Explain.
b. If the population standard deviation is $\sigma=8.2$ years, compute the standard error both with and without the finite population correction factor. What is the rationale for ignoring the finite population correction factor whenever $n / N \leq .05$ ?
c. What is the probability that the sample mean age of the employees will be within $\pm 2$ years of the population mean age?

Foster Wisusik

Numerade Educator

Problem 31

A sample of size 100 is selected from a population with $p=.40$.
a. What is the expected value of $\bar{p}$ ?
b. What is the standard error of $\bar{p}$ ?
c. Show the sampling distribution of $\bar{p}$.
d. What does the sampling distribution of $\bar{p}$ show?

Sheryl Ezze

Numerade Educator

Problem 32

A population proportion is .40 . A sample of size 200 will be taken and the sample proportion $\bar{p}$ will be used to estimate the population proportion.
a. What is the probability that the sample proportion will be within $\pm .03$ of the population proportion?
b. What is the probability that the sample proportion will be within $\pm .05$ of the population proportion?

Sheryl Ezze

Numerade Educator

Problem 33

Assume that the population proportion is .55 . Compute the standard error of the proportion, $\sigma_p$, for sample sizes of $100,200,500$, and 1000 . What can you say about the size of the standard error of the proportion as the sample size is increased?

Andrew Kim

Numerade Educator

Problem 34

The population proportion is .30 . What is the probability that a sample proportion will be within $\pm .04$ of the population proportion for each of the following sample sizes?
a. $n=100$
b. $n=200$
c. $n=500$
d. $n=1000$
e. What is the advantage of a larger sample size?

Sheryl Ezze

Numerade Educator

Problem 35

Orders from First-Time Customers. The president of Doerman Distributors, Inc., believes that $30 \%$ of the firm's orders come from first-time customers. A random sample of 100 orders will be used to estimate the proportion of first-time customers.
a. Assume that the president is correct and $p=.30$. What is the sampling distribution of $\bar{p}$ for this study?
b. What is the probability that the sample proportion $\bar{p}$ will be between .20 and .40 ?
c. What is the probability that the sample proportion will be between .25 and .35 ?

Andrew Kim

Numerade Educator

Problem 36

Ages of Entrepreneurs. The Wall Street Journal reported that the age at first startup for $55 \%$ of entrepreneurs was 29 years of age or less and the age at first startup for $45 \%$ of entrepreneurs was 30 years of age or more.
a. Suppose a sample of 200 entrepreneurs will be taken to learn about the most important qualities of entrepreneurs. Show the sampling distribution of $\bar{p}$ where $\bar{p}$ is the sample proportion of entrepreneurs whose first startup was at 29 years of age or less.
b. What is the probability that the sample proportion in part (a) will be within $\pm .05$ of its population proportion?
c. Suppose a sample of 200 entrepreneurs will be taken to learn about the most important qualities of entrepreneurs. Show the sampling distribution of $\bar{p}$ where $\bar{p}$ is now the sample proportion of entrepreneurs whose first startup was at 30 years of age or more.
d. What is the probability that the sample proportion in part (c) will be within $\pm .05$ of its population proportion?
e. Is the probability different in parts (b) and (d)? Why?
f. Answer part (b) for a sample of size 400 . Is the probability smaller? Why?

Sheryl Ezze

Numerade Educator

Problem 37

Food Waste. In 2017 , the Restaurant Hospitality website reported that only $10 \%$ of surplus food is being recovered in the food-service and restaurant sector, leaving approximately 1.5 billion meals per year uneaten. Assume this is the true population proportion and that you plan to take a sample survey of 525 companies in the in the food-service and restaurant sector to further investigate their behavior.
a. Show the sampling distribution of $\bar{p}$, the proportion of food recovered by your sample respondents.
b. What is the probability that your survey will provide a sample proportion within $\pm .03$ of the population proportion?
c. What is the probability that your survey will provide a sample proportion within $\pm .015$ of the population proportion?

Sheryl Ezze

Numerade Educator

Problem 38

Unnecessary Medical Care. According to Reader's Digest, $42 \%$ of primary care doctors think their patients receive unnecessary medical care.
a. Suppose a sample of 300 primary care doctors was taken. Show the sampling distribution of the proportion of the doctors who think their patients receive unnecessary medical care.
b. What is the probability that the sample proportion will be within $\pm .03$ of the population proportion?
c. What is the probability that the sample proportion will be within $\pm .05$ of the population proportion?
d. What would be the effect of taking a larger sample on the probabilities in parts (b) and (c)? Why?

Sheryl Ezze

Numerade Educator

Problem 39

Better Business Bureau Complaints. In 2016 the Better Business Bureau settled $80 \%$ of complaints they received in the United States. Suppose you have been hired by the Better Business Bureau to investigate the complaints they received this year involving new car dealers. You plan to select a sample of new car dealer complaints to estimate the proportion of complaints the Better Business Bureau is able to settle. Assume the population proportion of complaints settled for new car dealers is .80 , the same as the overall proportion of complaints settled in 2016.
a. Suppose you select a sample of 200 complaints involving new car dealers. Show the sampling distribution of $\bar{p}$.
b. Based upon a sample of 200 complaints, what is the probability that the sample proportion will be within .04 of the population proportion?
c. Suppose you select a sample of 450 complaints involving new car dealers. Show the sampling distribution of $\bar{p}$.
d. Based upon the smaller sample of only 450 complaints, what is the probability that the sample proportion will be within .04 of the population proportion?
e. As measured by the increase in probability, how much do you gain in precision by taking the larger sample in part (d)?

Dominador Tan

Numerade Educator

Problem 40

Product Labeling. The Grocery Manufacturers of America reported that $76 \%$ of consumers read the ingredients listed on a product's label. Assume the population proportion is $p=.76$ and a sample of 400 consumers is selected from the population.
a. Show the sampling distribution of the sample proportion $\bar{p}$ where $\bar{p}$ is the proportion of the sampled consumers who read the ingredients listed on a product's label.
b. What is the probability that the sample proportion will be within $\pm .03$ of the population proportion?
c. Answer part (b) for a sample of 750 consumers.

Foster Wisusik

Numerade Educator

Problem 41

Household Grocery Expenditures. The Food Marketing Institute shows that $17 \%$ of households spend more than $$\$ 100$$ per week on groceries. Assume the population proportion is $p=.17$ and a sample of 800 households will be selected from the population.
a. Show the sampling distribution of $\bar{p}$, the sample proportion of households spending more than $$\$ 100$$ per week on groceries.
b. What is the probability that the sample proportion will be within $\pm .02$ of the population proportion?
c. Answer part (b) for a sample of 1600 households.

Rashmi Sinha

Numerade Educator

Problem 42

A population has a mean of 400 and a standard deviation of 100 . A sample of size 100,000 will be taken, and the sample mean $\bar{x}$ will be used to estimate the population mean.
a. What is the expected value of $\bar{x}$ ?
b. What is the standard deviation of $\bar{x}$ ?
c. Show the sampling distribution of $\bar{x}$.
d. What does the sampling distribution of $\bar{x}$ show?

Sheryl Ezze

Numerade Educator

Problem 43

Assume the population standard deviation is $\sigma=25$. Compute the standard error of the mean, $\sigma_{\mathrm{S}}$, for sample sizes of 500,$000 ; 1,000,000 ; 5,000,000 ; 10,000,000$; and $100,000,000$. What can you say about the size of the standard error of the mean as the sample size is increased?

Sheryl Ezze

Numerade Educator

Problem 44

A sample of size 100,000 is selected from a population with $p=.75$.
a. What is the expected value of $\bar{p}$ ?
b. What is the standard error of $\bar{p}$ ?
c. Show the sampling distribution of $\bar{p}$.
d. What does the sampling distribution of $\bar{p}$ show?

Sheryl Ezze

Numerade Educator

Problem 45

Assume that the population proportion is .44 . Compute the standard error of the proportion, $\sigma_p$, for sample sizes of 500,$000 ; 1,000,000 ; 5,000,000 ; 10,000,000$; and $100,000,000$. What can you say about the size of the standard error of the sample proportion as the sample size is increased?

Sheryl Ezze

Numerade Educator

Problem 46

Vacation Hours Earned by Blue-Collar and Service Employees. The U.S. Burcau of Labor Statistics (BLS) reported that the mean annual number of hours of vacation time earned by blue-collar and service employees who work for small private establishments and have at least 10 years of service is 100 . Assume that for this population the standard deviation for the annual number of vacation hours earned is 48 . Suppose the BLS would like to select a sample of 15,000 individuals from this population for a follow-up study.
a. Show the sampling distribution of $\bar{x}$, the sample mean for a sample of 15,000 individuals from this population.
b. What is the probability that a simple random sample of 15,000 individuals from this population will provide a sample mean that is within one hour of the population mean?
c. Suppose the mean annual number of hours of vacation time earned for a sample of 15,000 blue-collar and service employees who work for small private establishments and have at least 10 years of service differs from the population mean $\mu$ by more than one hour. Considering your results for part (b), how would you interpret this result?

Sheryl Ezze

Numerade Educator

Problem 47

MPG for New Cars. The New York Times reported that 17.2 million new cars and light trucks were sold in the United States in 2017, and the U.S. Environmental Protection Agency projects the average efficiency for these vehicles to be 25.2 miles per gallon. Assume that that the population standard deviation in miles per gallon for these automobiles is $\sigma=6$.
a. What is the probability a sample of 70,000 new cars and light trucks sold in the United States in 2017 will provide a sample mean miles per gallon that is within .05 miles per gallon of the population mean of 25.2 ?
b. What is the probability a sample of 70,000 new cars and light trucks sold in the United States in 2017 will provide a sample mean miles per gallon that is within .01 miles per gallon of the population mean of 25.2 ? Compare this probability to the value computed in part (a).
c. What is the probability a sample of 90,000 new cars and light trucks sold in the United States in 2017 will provide a sample mean miles per gallon that is within .01 of the population mean of 25.2 ? Comment on the differences between this probability and the value computed in part (b).
d. Suppose the mean miles per gallon for a sample of 70,000 new cars and light trucks sold in the United States in 2017 differs from the population mean $\mu$ by more than one gallon. How would you interpret this result?

Sheryl Ezze

Numerade Educator

Problem 48

Repeat Purchases. The president of Colossus.com. Inc., believes that $42 \%$ of the firm's orders come from customers who have purchased from Colossus.com in the past. A random sample of 108,700 orders from the past six months will be used to estimate the proportion of orders placed by repeat customers.
a. Assume that Colossus.com's president is correct and the population proportion $p=-42$. What is the sampling distribution of $\bar{p}$ for this study?
b. What is the probability that the sample proportion $\bar{p}$ will be within $.1 \%$ of the population proportion?
c. What is the probability that the sample proportion $\bar{p}$ will be within $.25 \%$ of the population proportion? Comment on the difference between this probability and the value computed in part (b).
d. Suppose the proportion of orders placed by repeat customers for a sample of 108,700 orders from the past six months differs from the population proportion $p$ by more than $1 \%$. How would you interpret this result?

Sheryl Ezze

Numerade Educator

Problem 49

Landline Telephone Service. According to the U.S. Department of Health and Human Services, only $49.2 \%$ of homes in the United States used landline telephone service in 2017.
a. Suppose a sample of 207,000 U.S. homes will be taken to learn about home telephone usage. Show the sampling distribution of $\bar{p}$ where $\bar{p}$ is the sample proportion of homes that use landline phone service.
b. What is the probability that the sample proportion in part (a) will be within $\pm$. 002 of the population proportion?
c. Suppose a sample of 86,800 entrepreneurs will be taken to learn about home telephone usage. Show the sampling distribution of $\bar{p}$ where $\bar{p}$ is the sample proportion of homes that use landline phone service.
d. What is the probability that the sample proportion in part (c) will be within $\pm .002$ of the population proportion?
e. Are the probabilities different in parts (b) and (d)? Why or why not?

Sheryl Ezze

Numerade Educator

Problem 50

Shadow Stocks. Jack Lawler, a financial analyst, wants to prepare an article on the Shadow Stock portfolio developed by the American Association of Individual Investors (AAII). A list of the 30 companies in the Shadow Stock portfolio is contained in the file ShadowStocks. Jack would like to select a simple random sample of 5 of these companies for an interview concerning management practices.
a. In the file ShadowStock, companies are listed in column A of an Excel worksheet. In column B we have generated a random number for each of the companies. Use these random numbers to select a simple random sample of 5 of these companies for Jack.
b. Generate a new set of random numbers and use them to select a new simple random sample. Did you select the same companies?

Sheryl Ezze

Numerade Educator

Problem 51

Personal Health Expenditures. Data made available through the Petersen-Kaiser Health System Tracker in May 2018 showed health expenditures were $$\$ 10,348$$ per person in the United States. Use $$\$ 10,348$$ as the population mean and suppose a survey research firm will take a sample of 100 people to investigate the nature of their health expenditures. Assume the population standard deviation is $$\$ 2500$$.
a. Show the sampling distribution of the mean amount of health care expenditures for a sample of 100 people.
b. What is the probability the sample mean will be within $$\pm \$ 200$$ of the population mean?
c. What is the probability the sample mean will be greater than $$\$ 12,000$$ ? If the survey research firm reports a sample mean greater than $$\$ 12,000$$, would you question whether the firm followed correct sampling procedures? Why or why not?

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 52

Foot Locker Store Productivity. Foot Locker uses sales per square foot as a measure of store productivity. Sales are currently running at an annual rate of $$\$ 406$$ per square foot. You have been asked by management to conduct a study of a sample of 64 Foot Locker stores. Assume the standard deviation in annual sales per square foot for the population of all 3400 Foot Locker stores is $$\$ 80$$.
a. Show the sampling distribution of $\bar{x}$, the sample mean annual sales per square foot for a sample of 64 Foot Locker stores.
b. What is the probability that the sample mean will be within $$\$ 15$$ of the population mean?
c. Suppose you find a sample mean of $$\$ 380$$. What is the probability of finding a sample mean of $$\$ 380$$ or less? Would you consider such a sample to be an unusually low-performing group of stores?

Sheryl Ezze

Numerade Educator

Problem 53

Airline Fares. The mean airfare for flights departing from Buffalo Niagara International Airport during the first three months of 2017 was $$\$ 320.51$$. Assume the standard deviation for this population of fares is known to be $$\$ 80$$. Suppose a random sample of 60 flights departing from Buffalo Niagara International Airport during the first three months of 2018 is taken.
a. If the mean and standard deviation of the population of airfares for flights departing from Buffalo Niagara International Airport didn't changed between the first three months of 2017 and the first three months of 2018 , what is the probability the sample mean will be within $$\$ 20$$ of the population mean cost per flight?
b. What is the probability the sample mean will be within $$\$ 10$$ of the population mean cost per flight?

Sheryl Ezze

Numerade Educator

Problem 54

University Costs. After deducting grants based on need, the average cost to attend the University of Southern Califormia (USC) is $$\$ 27,175$$. Assume the population standard deviation is $$\$ 7400$$. Suppose that a random sample of 60 USC students will be taken from this population.
a. What is the value of the standard error of the mean?
b. What is the probability that the sample mean will be more than $$\$ 27,175$$ ?
c. What is the probability that the sample mean will be within $$\$ 1000$$ of the population mean?
d. How would the probability in part (c) change if the sample size were increased to 100 ?

Sheryl Ezze

Numerade Educator

Problem 55

Inventory Costs. Three firms carry inventories that differ in size. Firm A's inventory contains 2000 items, firm B's inventory contains 5000 items, and firm C's inventory contains 10,000 items. The population standard deviation for the cost of the items in each firm's inventory is $\sigma=144$. A statistical consultant recommends that each firm take a sample of 50 items from its inventory to provide statistically valid estimates of the average cost per item. Managers of the small firm state that because it has the smallest population, it should be able to make the estimate from a much smaller sample than that required by the larger firms. However, the consultant states that to obtain the same standard error and thus the same precision in the sample results, all firms should use the same sample size regardless of population size.
a. Using the finite population correction factor, compute the standard error for each of the three firms given a sample of size 50 .
b. What is the probability that for each firm the sample mean $\bar{x}$ will be within $\pm 25$ of the population mean $\mu$ ?

Sheryl Ezze

Numerade Educator

Problem 56

Survey Research Results. A researcher reports survey results by stating that the standard error of the mean is 20 . The population standard deviation is 500 .
a. How large was the sample used in this survey?
b. What is the probability that the point estimate was within $\pm 25$ of the population mean?

Foster Wisusik

Numerade Educator

Problem 57

Production Quality Control. A production process is checked periodically by a quality control inspector. The inspector selects simple random samples of 30 finished products and computes the sample mean product weights. $\vec{x}$. If test results over a long period of time show that $5 \%$ of the $\bar{x}$ values are over 2.1 pounds and $5 \%$ are under 1.9 pounds, what are the mean and the standard deviation for the population of products produced with this process?

Andrew Kim

Numerade Educator

Problem 58

Australians and Smoking. Reuters reports that 15 percent of Australians smoke. By introducing tough laws banning brand labels on cigarette packages, Australia hopes to ultimately reduce the percentage of people smoking to $10 \%$. Answer the following questions based on a sample of 240 Australians.
a. Show the sampling distribution of $\bar{p}$, the proportion of Australians who are smokers.
b. What is the probability the sample proportion will be within $\pm .04$ of the population proportion?
c. What is the probability the sample proportion will be within $\pm .02$ of the population proportion?

Sheryl Ezze

Numerade Educator

Problem 59

Marketing Research Telephone Surveys. A market research firm conducts telephone surveys with a $40 \%$ historical response rate. What is the probability that in a new sample of 400 telephone numbers, at least 150 individuals will cooperate and respond to the questions? In other words, what is the probability that the sample proportion will be at least $150 / 400=.375$ ?

Andrew Kim

Numerade Educator

Problem 60

Internet Advertising. Advertisers contract with Internet service providers and search engines to place ads on websites. They pay a fee based on the number of potential customers who click on their ad. Unfortunately, click fraud-the practice of someone clicking on an ad solely for the purpose of driving up advertising revenue-has
become a problem. Businessweek reports that 40 percent of advertisers claim they have been a victim of click fraud. Suppose a simple random sample of 380 advertisers will be taken to learn more about how they are affected by this practice.
a. What is the probability that the sample proportion will be within $\pm .04$ of the population proportion experiencing click fraud?
b. What is the probability that the sample proportion will be greater than .45 ?

Kaylee Mcclellan

Kaylee Mcclellan

Numerade Educator

Problem 61

Traffic Tickets. The proportion of individuals insured by the All-Driver Automobile Insurance Company who received at least one traffic ticket during a five-year period is . 15 .
a. Show the sampling distribution of $\bar{p}$ if a random sample of 150 insured individuals is used to estimate the proportion having received at least one ticket.
b. What is the probability that the sample proportion will be within $\pm .03$ of the population proportion?

Sheryl Ezze

Numerade Educator

Problem 62

Textbook Publishing. Lori Jeffrey is a successful sales representative for a major publisher of college textbooks. Historically, Lori obtains a book adoption on $25 \%$ of her sales calls. Viewing her sales calls for one month as a sample of all possible sales calls, assume that a statistical analysis of the data yields a standard error of the proportion of .0625 .
a. How large was the sample used in this analysis? That is, how many sales calls did Lori make during the month?
b. Let $\bar{p}$ indicate the sample proportion of book adoptions obtained during the month. Show the sampling distribution of $\bar{p}$.
c. Using the sampling distribution of $\bar{p}$, compute the probability that Lori will obtain book adoptions on $30 \%$ or more of her sales calls during a one-month period.

Foster Wisusik

Numerade Educator

Problem 63

Life of Compact Fluorescent Lights. In 2018, the Simple Dollar website reported that the mean life of 14 -watt compact fluorescent lights (CFLs) is 8000 hours. Assume that for this population the standard deviation for CFL life is 480 . Suppose the U.S. Department of Energy would like to select a random sample of 35,000 from the population of 14 -watt CFLs for a follow-up study.
a. Show the sampling distribution of $\bar{x}$, the sample mean for a sample of 35,000 individuals from this population.
b. What is the probability that a simple random sample of 35,000 individuals from this population will provide a sample mean that is within four hours of the population mean?
c. What is the probability that a simple random sample of 35,000 individuals from this population will provide a sample mean that is within one hour of the population mean?
d. Suppose the mean life of a sample of 35,00014 -watt CFLs differs from the population mean life by more than four hours. How would you interpret this result?

Sheryl Ezze

Numerade Educator

Problem 64

Typical Home Internet Usage. According to USC Annenberg, the mean time spent by Americans on the Internet in their home per week is 17.6 hours. Assume that the standard deviation for the time spent by Americans on the Internet in their home per week is 5.1 hours. Suppose the Florida Department of State plans to select a random sample of 85,020 of the state's residents for a study of Floridians' Internet usage.
a. Using the U.S. population figures provided in the problem (the population mean and standard deviation of time spent by Americans on the Internet in their home per week are 17.6 hours and 5.1 hours, respectively), what is the sampling distribution of the sample mean for the sample of 85,020 Floridians?
b. Using the sampling distribution from part (a), what is the probability that a random sample of 85,020 Floridians will provide a sample mean that is within three minutes of the population mean?
c. Suppose the mean time spent on the Intemet in their home per week by the sample of 85,020 Floridians differs from the U.S population mean by more than three minutes? How would you interpret this result?

Nick Johnson

Numerade Educator

Problem 65

Undeliverable Mail Pieces. Of the 155 billion mailpieces the U.S. Postal Service (USPS) processed and delivered in $2017,4.3 \%$ were undeliverable as addressed. Suppose that a brief questionnaire about USPS service is attached to each mailpiece in a random sample of 114,250 mailpieces.
a. What is the sampling distribution of the sample proportion of undeliverable mailpieces $\bar{p}$ for this study?
b. What is the probability that the sample proportion of undeliverable mailpieces $\bar{p}$ will be within $.1 \%$ of the population proportion of undeliverable mailpieces?
c. What is the probability that the sample proportion of undeliverable mailpieces $\bar{p}$ will be within $.05 \%$ of the population proportion of undeliverable mailpieces? Comment on the difference between this probability and the probability computed in part (b).

Sheryl Ezze

Numerade Educator

Problem 66

U.S. Drivers and Speeding. ABC News reports that $58 \%$ of U.S. drivers admit to speeding. Suppose that a new satellite technology can instantly measure the speed of any vehicle on a U.S. road and determine whether the vehicle is speeding, and this satellite technology was used to take a random sample of 20,000 vehicles at 6 P.M. EST on a recent Tuesday afternoon.
a. For this investigation, what is the sampling distribution for sample proportion of vehicles on U.S. roads that speed?
b. What is the probability that the sample proportion of speeders $\bar{p}$ will be within $1 \%$ of the population proportion of speeders?
c. Suppose the sample proportion of speeders $\bar{p}$ differs from the U.S population proportion of seeders by more than $1 \%$ ? How would you interpret this result?

Sheryl Ezze

Numerade Educator