Essentials of Statistics for Business and Economics

David R. Anderson, Dennis J. Sweeney, Thomas A. Williams

Chapter 7

Sampling and Sampling Distributions - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider a finite population with five elements labeled $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},$ and $\mathrm{E}$. Ten possible $\operatorname{sim}$ ple 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.
\[
98601 \quad 73022 \quad 83448 \quad 02147 \quad 34229 \quad 27553 \quad 84147 \quad 93289 \quad 14209
\]

Nick Johnson

Numerade Educator

Problem 3

Fortune publishes data on sales, profits, assets, stockholders' equity, market value, and earnings per share for the 500 largest U.S. industrial corporations (Fortune 500,2006 ). 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

The 10 most active stocks on the New York Stock Exchange on March $6,2006,$ are shown here (The Wall Street Journal, March 7, 2006).
$\begin{array}{llll}\text { AT\&T } & \text { Lucent } & \text { Nortel } & \text { Qwest } & \text { Bell South } \\ \text { Pfizer } & \text { Texas Instruments } & \text { Gen. Elect. } & \text { iShrMSJpn } & \text { LSI Logic }\end{array}$
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

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

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

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

The following stocks make up the Dow Jones Industrial Average (Barron s , March 23 , 2009)
$1.3 \mathrm{M}$
2. AT\&T
3. Alcoa
4. American Express
5. Bank of America
6. Boeing
7. Caterpillar
8. Chevron
9. Cisco Systems
10. Coca-Cola
11. Disney
12. DuPont
13. ExxonMobil
14. General Electric
15. Hewlett-Packard
16. Home Depot
$17 . \mathrm{IBM}$
$18 .$ Intel
19. Johnson \& Johnson
$20 .$ Kraft Foods
21. McDonald's
22. Merck
23. Microsoft
24. J.P. Morgan
25. Pfizer
26. Procter \& Gamble
27. Travelers
28. United Technologies
29. Verizon
30. Walmart
Suppose you would like to select a sample of six of these companies to conduct an in-depth 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.

Shu Naito

Numerade Educator

Problem 9

The Wall Street Journal provides the net asset value, the year-to-date percent return, and the three-year percent return for 555 mutual funds (The Wall Street Journal, April 25, 2003 ). Assume that a simple random sample of 12 of the 555 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

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. $\quad$ 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.
\[
5 \quad 8 \quad 10 \quad 7 \quad 10 \quad 14
\]
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 12

A survey question for a sample of 150 individuals yielded 75 Yes responses, 55 No responses, and 20 No Opinions.
a. What is the point estimate of the proportion in the population who respond Yes?
b. What is the point estimate of the proportion in the population who respond No?

Foster Wisusik

Numerade Educator

Problem 13

A simple random sample of 5 months of sales data provided the following information:
\[
\begin{array}{lrrrrr}
\text {Month:} & 1 & 2 & 3 & 4 & 5 \\
\text {Units Sold:} & 94 & 100 & 85 & 94 & 92
\end{array}
\]
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.

Sheryl Ezze

Numerade Educator

Problem 14

Business Week published information on 283 equity mutual funds (Business Week, January 26,2004 ). A sample of 40 of those funds is contained in the data set MutualFund. Use the data set to answer the following questions.
a. Develop a point estimate of the proportion of the Business Week equity funds that are load funds.
b. Develop a point estimate of the proportion of funds that are classified as high risk.
c. Develop a point estimate of the proportion of funds that have a below-average risk rating.

Foster Wisusik

Numerade Educator

Problem 15

Many drugs used to treat cancer are expensive. Business Week reported on the cost per treatment of Herceptin, a drug used to treat breast cancer (Business Week, January 30,2006 ). Typical treatment costs (in dollars) for Herceptin are provided by a simple random sample of 10 patients.
\[
\begin{array}{lllll}
4376 & 5578 & 2717 & 4920 & 4495 \\
4798 & 6446 & 4119 & 4237 & 3814
\end{array}
\]
a. Develop a point estimate of the mean cost per treatment with Herceptin.
b. Develop a point estimate of the standard deviation of the cost per treatment with Herceptin.

Nick Johnson

Numerade Educator

Problem 16

A sample of 50 Fortune 500 companies (Fortune, April 14,2003 ) showed 5 were based in New York, 6 in California, 2 in Minnesota, and 1 in Wisconsin.
a. Develop an estimate of the proportion of Fortune 500 companies based in New York.
b. Develop an estimate of the number of Fortune 500 companies based in Minnesota.
c. Develop an estimate of the proportion of Fortune 500 companies that are not based in these four states.

Nick Johnson

Numerade Educator

Problem 17

The American Association of Individual Investors (AAII) polls its subscribers on a weekly basis to determine the number who are bullish, bearish, or neutral on the short-term prospects for the stock market. Their findings for the week ending March $2,2006,$ are consistent with the following sample results (AAII website, March 7, 2006).
Bullish $409 \quad$ Neutral $299 \quad$ Bearish $291$
Develop a point estimate of the following population parameters.
a. The proportion of all AAII subscribers who are bullish on the stock market.
b. The proportion of all AAII subscribers who are neutral on the stock market.
c. The proportion of all AAII subscribers who are bearish on the stock market.

Nick Johnson

Numerade Educator

Problem 18

A population has a mean of 200 and a standard deviation of $50 .$ A simple random 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 simple random 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 ±5 of the population mean?
b. What is the probability that the sample mean will be within ±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, $\sigma_{\bar{x}},$ 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 21

Suppose a simple 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 $N=500$.

Andrew Kim

Numerade Educator

Problem 22

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.

Foster Wisusik

Numerade Educator

Problem 23

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

Nick Johnson

Numerade Educator

Problem 24

Barron 's reported that the average number of weeks an individual is unemployed is 17.5 weeks (Barron $s$, February 18,2008 ). 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 random 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?

Foster Wisusik

Numerade Educator

Problem 25

The College Board reported the following mean scores for the three parts of the Scholastic Aptitude Test (SAT) (The World Almanac, 2009):
$$\begin{array}{ll}
\text { Critical Reading } & 502 \\
\text { Mathematics } & 515 \\
\text { Writing } & 494
\end{array}$$
Assume that the population standard deviation on each part of the test is $\sigma=100$
a. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the Critical Reading part of the test?
b. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 515 on the Mathematics part of the test? Compare this probability to the value computed in part (a).
c. What is the probability that a random sample of 100 test takers will provide a sample mean test score within 10 of the population mean of 494 on the writing part of the test? Comment on the differences between this probability and the values computed in parts
(a) and (b).

Andrew Kim

Numerade Educator

Problem 26

The mean annual cost of automobile insurance is $\$ 939$ (CNBC, February 23,2006 ). Assume that the standard deviation is $\sigma=\$ 245$
a. What is the probability that a simple random sample of automobile insurance policies will have a sample mean within $\$ 25$ 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?

Foster Wisusik

Numerade Educator

Problem 27

BusinessWeek conducted a survey of graduates from 30 top MBA programs ( Business Week ,
September 22,2003 ). On the basis of the survey, assume that the mean annual salary for male and female graduates 10 years after graduation is $\$ 168,000$ and $\$ 117,000,$ respectively. Assume the standard deviation for the male graduates is $\$ 40,000,$ and for the female graduates it is $\$ 25,000$
a. What is the probability that a simple random sample of 40 male graduates will provide a sample mean within $\$ 10,000$ of the population mean, $\$ 168,000 ?$
b. What is the probability that a simple random sample of 40 female graduates will provide a sample mean within $\$ 10,000$ of the population mean, $\$ 117,000 ?$
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 $\$ 10,000$ of the population mean? Why?
d. What is the probability that a simple random sample of 100 male graduates will provide a sample mean more than $\$ 4000$ below the population mean?

Andrew Kim

Numerade Educator

Problem 28

The average score for male golfers is 95 and the average score for female golfers is 106 (Golf Digest, April 2006). Use these values as the population means for men and women and assume that the population standard deviation is $\sigma=14$ strokes for both. A simple random sample of 30 male golfers and another simple random sample of 45 female golfers will be taken.
a. Show the sampling distribution of $\bar{x}$ for male golfers.
b. What is the probability that the sample mean is within three strokes of the population mean for the sample of male golfers?
c. What is the probability that the sample mean is within three strokes of the population mean for the sample of female golfers?
d. In which case, part (b) or part (c), is the probability of obtaining a sample mean within three strokes of the population mean higher? Why?

Foster Wisusik

Numerade Educator

Problem 29

The average price of a gallon of unleaded regular gasoline was reported to be $\$ 2.34$ in northern Kentucky (The Cincinnati Enquirer, January 21,2006 ). Use this price as the population mean, and assume the population standard deviation is $\$ .20$.
a. What is the probability that the mean price for a sample of 30 service stations is within $\$ .03$ of the population mean?
b. What is the probability that the mean price for a sample of 50 service stations is within $\$ .03$ of the population mean?
c. What is the probability that the mean price for a sample of 100 service stations is within $\$ .03$ of the population mean?
d. Which, if any, of the sample sizes in parts (a), (b), and (c) would you recommend to have at least a .95 probability that the sample mean is within $\$ .03$ of the population mean?

Andrew Kim

Numerade Educator

Problem 30

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 ±2 years of the population mean age?

Foster Wisusik

Numerade Educator

Problem 31

A simple random 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?

Andrew Kim

Numerade Educator

Problem 32

A population proportion is $.40 .$ A simple random 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 ±.03 of the population proportion?
b. What is the probability that the sample proportion will be within ±.05 of the population proportion?

Foster Wisusik

Numerade Educator

Problem 33

Assume that the population proportion is $.55 .$ Compute the standard error of the proportion, $\sigma_{\bar{p}},$ 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 ±.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

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

The Cincinnati Enquirer reported that, in the United States, $66 \%$ of adults and $87 \%$ of youths ages 12 to 17 use the Internet (The Cincinnati Enquirer, February 7,2006 ). Use the reported numbers as the population proportions and assume that samples of 300 adults and 300 youths will be used to learn about attitudes toward Internet security.
a. Show the sampling distribution of $\bar{p}$, where $\bar{p}$ is the sample proportion of adults using the Internet.
b. What is the probability that the sample proportion of adults using the Internet will be within ±.04 of the population proportion?
c. What is the probability that the sample proportion of youths using the Internet will be within ±.04 of the population proportion?
d. Is the probability different in parts (b) and (c)? If so, why?
e. Answer part (b) for a sample of size 600. Is the probability smaller? Why?

Sheryl Ezze

Numerade Educator

Problem 37

People end up tossing $12 \%$ of what they buy at the grocery store (Reader 's Digest, March, 2009 ). Assume this is the true population proportion and that you plan to take a sample survey of 540 grocery shoppers to further investigate their behavior.
a. Show the sampling distribution of $\bar{p},$ the proportion of groceries thrown out by your sample respondents.
b. What is the probability that your survey will provide a sample proportion within ±.03 of the population proportion?
c. What is the probability that your survey will provide a sample proportion within ±.015 of the population proportion?

Andrew Kim

Numerade Educator

Problem 38

Roper ASW conducted a survey to learn about American adults' attitudes toward money and happiness (Money, October 2003). Fifty-six percent of the respondents said they balance their checkbook at least once a month.
a. Suppose a sample of 400 American adults were taken. Show the sampling distribution of the proportion of adults who balance their checkbook at least once a month.
b. What is the probability that the sample proportion will be within ±.02 of the population proportion?
c. What is the probability that the sample proportion will be within ±.04 of the population proportion?

Foster Wisusik

Numerade Educator

Problem 39

In 2008 the Better Business Bureau settled $75 \%$ of complaints it received (USA Today, March 2,2009 ). Suppose you have been hired by the Better Business Bureau to investigate the complaints it 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 $.75,$ the same as the overall proportion of complaints settled in 2008
a. Suppose you select a sample of 450 complaints involving new car dealers. Show the sampling distribution of $\bar{p}$
b. Based upon a sample of 450 complaints, what is the probability that the sample proportion will be within .04 of the population proportion?
c. Suppose you select a sample of 200 complaints involving new car dealers. Show the sampling distribution of $\bar{p}$
d. Based upon the smaller sample of only 200 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 (b)?

Dominador Tan

Numerade Educator

Problem 40

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 ±.03 of the population proportion?
c. Answer part (b) for a sample of 750 consumers.

Foster Wisusik

Numerade Educator

Problem 41

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 simple random 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 ±.02 of the population proportion?
c. Answer part (b) for a sample of 1600 households.

Andrew Kim

Numerade Educator

Problem 42

U.S. News \& World Report publishes comprehensive information on America's best colleges (America s Best Colleges, 2009 ed.). Among other things, it provides a listing of the 133 best national universities. You would like to take a sample of these universities for a follow-up study on their students. Begin at the bottom of the third column of random digits in Table 7.1. Ignoring the first two digits in each five-number group and using the three-digit random numbers beginning with $959,$ read $u p$ the column to identify the number (from 1 to 133 ) of the first seven universities to be included in a simple random sample. Continue by starting at the bottom of the fourth and fifth columns and reading up if necessary.

Kari Hasz

Numerade Educator

Problem 43

Americans have become increasingly concerned about the rising cost of Medicare. In 1990 the average annual Medicare spending per enrollee was $\$ 3267 ;$ in $2003,$ the average annual Medicare spending per enrollee was $\$ 6883$ (Money, Fall 2003 ). Suppose you hired a consulting firm to take a sample of fifty 2003 Medicare enrollees to further investigate the nature of expenditures. Assume the population standard deviation for 2003 was $\$ 2000$.
a. Show the sampling distribution of the mean amount of Medicare spending for a sample of fifty 2003 enrollees.
b. What is the probability that the sample mean will be within $\pm \$ 300$ of the population mean?
c. What is the probability that the sample mean will be greater than $\$ 7500 ?$ If the consulting firm tells you the sample mean for the Medicare enrollees it interviewed was $\$ 7500,$ would you question whether the firm followed correct simple random sampling procedures? Why or why not?

Paul A.

California State Polytechnic University, Pomona

Problem 44

BusinessWeek surveyed MBA alumni 10 years after graduation $(\text {Business Week} \text { , September } 22,$
2003 ). One finding was that alumni spend an average of $\$ 115.50$ per week eating out socially. You have been asked to conduct a follow-up study by taking a sample of 40 of these MBA alumni. Assume the population standard deviation is $\$ 35$
a. Show the sampling distribution of $\bar{x}$, the sample mean weekly expenditure for the $40 \mathrm{MBA}$ alumni.
b. What is the probability that the sample mean will be within $\$ 10$ of the population mean?
c. Suppose you find a sample mean of $\$ 100 .$ What is the probability of finding a sample mean of $\$ 100$ or less? Would you consider this sample to be an unusually low spending group of alumni? Why or why not?

Foster Wisusik

Numerade Educator

Problem 45

The mean television viewing time for Americans is 15 hours per week (Money, November 2003 ). Suppose a sample of 60 Americans is taken to further investigate viewing habits. Assume the population standard deviation for weekly viewing time is $\sigma=4$ hours.
a. What is the probability that the sample mean will be within 1 hour of the population mean?
b. What is the probability that the sample mean will be within 45 minutes of the population mean?

Andrew Kim

Numerade Educator

Problem 46

After deducting grants based on need, the average cost to attend the University of Southern California (USC) is $\$ 27,175$ (U.S. News \& World Report, America 's Best Colleges,
$2009 \text { ed. }) .$ 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 $100 ?$

Foster Wisusik

Numerade Educator

Problem 47

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 ±25 of the population mean $\mu ?$

Andrew Kim

Numerade Educator

Problem 48

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 ±25 of the population mean?

Foster Wisusik

Numerade Educator

Problem 49

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 $\bar{x}$. 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 50

About $28 \%$ of private companies are owned by women (The Cincinnati Enquirer, January 26,2006)$.$ Answer the following questions based on a sample of 240 private companies.
a. Show the sampling distribution of $\bar{p},$ the sample proportion of companies that are owned by women.
b. What is the probability that the sample proportion will be within ±.04 of the population proportion?
c. What is the probability that the sample proportion will be within ±.02 of the population proportion?

Foster Wisusik

Numerade Educator

Problem 51

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 52

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. Forty percent of advertisers claim they have been a victim of click fraud (Business Week, March 13,2006 ). 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 ±.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 53

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 ±.03 of the population proportion?

Sheryl Ezze

Numerade Educator

Problem 54

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.

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