Chapter 6, Statistical Inference Video Solutions, Business Analytics | Numerade (2024)

The American League consists of 15 baseball teams. Suppose a sample of 5 teams is to be selected to conduct player interviews. The following table lists the 15 teams and the random numbers assigned by Excel's RAND function. Use these random numbers to select a sample of size 5 .
$$
\begin{array}{lclc}
\text { Team } & \text { Random Number } & \text { Team } & \text { Random Number } \\
\text { New York } & 0.178624 & \text { Boston } & 0.290197 \\
\text { Baltimore } & 0.578370 & \text { Tampa Bay } & 0.867778 \\
\text { Toronto } & 0.965807 & \text { Minnesota } & 0.811810 \\
\text { Chicago } & 0.562178 & \text { Cleveland } & 0.960271 \\
\text { Detroit } & 0.253574 & \text { Kansas City } & 0.326836 \\
\text { Oakland } & 0.288287 & \text { Los Angeles } & 0.895267 \\
\text { Texas } & 0.500879 & \text { Seattle } & 0.839071 \\
\text { Houston } & 0.713682 & &
\end{array}
$$

The U.S. Golf Association is considering a ban on long and belly putters. This has caused a great deal of controversy among both amateur golfers and members of the Professional Golf Association (PGA). Shown below are the names of the top 10 finishers in the recent PGA Tour McGladrey Classic golf tournament.
1. Tommy Gainey
2. David Toms
3. Jim Furyk
4. Brendon de Jonge
5. D. J. Trahan
6. Davis Love III
7. Chad Campbell
8. Greg Owens
9. Charles Howell III
10. Arjun Atwal
Select a simple random sample of 3 of these players to assess their opinions on the use of long and belly putters.

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.

Morningstar publishes ratings data on 1,208 company stocks. A sample of 40 of these stocks is contained in the file named 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.

One of the questions in the Pew Internet \& American Life Project asked adults if they used the Internet at least occasionally. The results showed that 454 out of 478 adults aged 18-29 answered Yes; 741 out of 833 adults aged 30-49 answered Yes; and 1,058 out of 1,644 adults aged 50 and over answered Yes.
a. Develop a point estimate of the proportion of adults aged 18-29 who use the Internet.
b. Develop a point estimate of the proportion of adults aged $30-49$ who use the Internet.
c. Develop a point estimate of the proportion of adults aged 50 and over who use the Internet.
d. Comment on any apparent relationship between age and Internet use.
e. Suppose your target population of interest is that of all adults (18 years of age and over). Develop an estimate of the proportion of that population who use the Internet.

In this chapter we showed how a simple random sample of 30 EAI employees can be used to develop point estimates of the population mean annual salary, the population standard deviation for annual salary, and the population proportion having completed the management training program.
a. Use Excel to select a simple random sample of $50 \mathrm{EAI}$ employees.
b. Develop a point estimate of the mean annual salary.
c. Develop a point estimate of the population standard deviation for annual salary.
d. Develop a point estimate of the population proportion having completed the management training program.

The College Board reported the following mean scores for the three parts of the SAT: Assume that the population standard deviation on each part of the test is $\sigma=100$.
$$
\begin{array}{ll}
\text { Critical Reading } & 502 \\
\text { Mathematics } & 515 \\
\text { Writing } & 494
\end{array}
$$
a. For a random sample of 30 test takers, what is the sampling distribution of $\bar{x}$ for scores on the Critical Reading part of the test?
b. For a random sample of 60 test takers, what is the sampling distribution of $\bar{x}$ for scores on the Mathematics part of the test?
c. For a random sample of 90 test takers, what is the sampling distribution of $\bar{x}$ for scores on the Writing part of the test?

For the year $2010,33 \%$ 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 that the standard deviation is $$\sigma=\$ 2,400$$.
a. What are the sampling distributions of $\bar{x}$ for itemized deductions for this population of taxpayers 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?

The Economic Policy Institute periodically issues reports on wages of entry-level workers. The institute reported that entry-level wages for male college graduates were $$\$ 21.68$$ per hour and for female college graduates were $$\$ 18.80$$ per hour in 2011. Assume that the standard deviation for male graduates is $$\$ 2.30$$ and for female graduates it is $$\$ 2.05$$.
a. What is the sampling distribution of $\bar{x}$ for a random sample of 50 male college graduates?
b. What is the sampling distribution of $\bar{x}$ for a random sample of 50 female college graduates?
c. In which of the preceding two cases, part (a) or part (b), is the standard error of $\bar{x}$ smaller? Why?

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 sampling distribution of the sample mean annual rainfall for California.
b. Show the sampling distribution of the sample mean annual rainfall for New York.
c. In which of the preceding two cases, part (a) or part (b), is the standard error of $\bar{x}$ smaller? Why?

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. Assume that the president is correct and $p=0.30$. What is the sampling distribution of $\bar{p}$ for this study?

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. 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.
c. Are the standard errors of the sampling distributions different in parts (a) and (b)?

People end up tossing $12 \%$ of what they buy at the grocery store. 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. Show the sampling distribution of $\bar{p}$, the proportion of groceries thrown out by your sample respondents.

Forty-two percent of primary care doctors think their patients receive unnecessary medical care.
a. Suppose a sample of 300 primary care doctors was taken. Show the distribution of the sample proportion of doctors who think their patients receive unnecessary medical care.
b. Suppose a sample of 500 primary care doctors was taken. Show the distribution of the sample proportion of doctors who think their patients receive unnecessary medical care.
c. Suppose a sample of 1,000 primary care doctors was taken. Show the distribution of the sample proportion of doctors who think their patients receive unnecessary medical care.
d. In which of the preceding three cases, part (a) or part (b) or part (c), is the standard error of $\bar{p}$ smallest? Why?

The International Air Transport Association surveys business travelers to develop quality ratings for transatlantic gateway airports. The maximum possible rating is 10 . Suppose a simple random sample of 50 business travelers is selected and each traveler is asked to provide a rating for the Miami International Airport. The ratings obtained from the sample of 50 business travelers follow.
$$
\begin{array}{rrrrrrrrrrrrr}
6 & 4 & 6 & 8 & 7 & 7 & 6 & 3 & 3 & 8 & 10 & 4 & 8 \\
7 & 8 & 7 & 5 & 9 & 5 & 8 & 4 & 3 & 8 & 5 & 5 & 4 \\
4 & 4 & 8 & 4 & 5 & 6 & 2 & 5 & 9 & 9 & 8 & 4 & 8 \\
9 & 9 & 5 & 9 & 7 & 8 & 3 & 10 & 8 & 9 & 6 & &
\end{array}
$$
Develop a $95 \%$ confidence interval estimate of the population mean rating for Miami.

A sample containing years to maturity and yield for 40 corporate bonds is contained in the file named CorporateBonds.
a. What is the sample mean years to maturity for corporate bonds and what is the sample standard deviation?
b. Develop a $95 \%$ confidence interval for the population mean years to maturity.
c. What is the sample mean yield on corporate bonds and what is the sample standard deviation?
d. Develop a $95 \%$ confidence interval for the population mean yield on corporate bonds.

Health insurers are beginning to offer telemedicine services online that replace the common office visit. WellPoint provides a video service that allows subscribers to connect with a physician online and receive prescribed treatments. Wellpoint claims that users of its LiveHealth Online service saved a significant amount of money on a typical visit. The data shown below ($$\$ $$), for a sample of 20 online doctor visits, are consistent with the savings per visit reported by Wellpoint.
$$
\begin{array}{rrr}
92 & 34 & 40 \\
105 & 83 & 55 \\
56 & 49 & 40 \\
76 & 48 & 96 \\
93 & 74 & 73 \\
78 & 93 & 100 \\
53 & 82 &
\end{array}
$$
Assuming that the population is roughly symmetric, construct a $95 \%$ confidence interval for the mean savings for a televisit to the doctor as opposed to an office visit.

The average annual premium for automobile insurance in the United States is $$\$ 1,503$$. The following annual premiums ($$\$ $$) are representative of the web site's findings for the state of Michigan.
$$
\begin{array}{lll}
1,905 & 3,112 & 2,312 \\
2,725 & 2,545 & 2,981 \\
2,677 & 2,525 & 2,627 \\
2,600 & 2,370 & 2,857 \\
2,962 & 2,545 & 2,675 \\
2,184 & 2,529 & 2,115 \\
2,332 & 2,442 &
\end{array}
$$
Assume the population is approximately normal.
a. Provide a point estimate of the mean annual automobile insurance premium in Michigan.
b. Develop a $95 \%$ confidence interval for the mean annual automobile insurance premium in Michigan.
c. Does the $95 \%$ confidence interval for the annual automobile insurance premium in Michigan include the national average for the United States? What is your interpretation of the relationship between auto insurance premiums in Michigan and the national average?

One of the questions on a survey of 1,000 adults asked if today's children will be better off than their parents. Representative data are shown in the file named ChildOutlook. A response of Yes indicates that the adult surveyed did think today's children will be better off than their parents. A response of No indicates that the adult surveyed did not think today's children will be better off than their parents. A response of Not Sure was given by $23 \%$ of the adults surveyed.
a. What is the point estimate of the proportion of the population of adults who do think that today's children will be better off than their parents?
b. At $95 \%$ confidence, what is the margin of error?
c. What is the $95 \%$ confidence interval for the proportion of adults who do think that today's children will be better off than their parents?
d. What is the $95 \%$ confidence interval for the proportion of adults who do not think that today's children will be better off than their parents?
e. Which of the confidence intervals in parts (c) and (d) has the smaller margin of error? Why?

According to Thomson Financial, last year the majority of companies reporting profits had beaten estimates. A sample of 162 companies showed that 104 beat estimates, 29 matched estimates, and 29 fell short.
a. What is the point estimate of the proportion that fell short of estimates?
b. Determine the margin of error and provide a $95 \%$ confidence interval for the proportion that beat estimates.
c. How large a sample is needed if the desired margin of error is 0.05 ?

The Pew Research Center Internet Project conducted a survey of 857 Internet users. This survey provided a variety of statistics on them.
a. The sample survey showed that $90 \%$ of respondents said the Internet has been a good thing for them personally. Develop a $95 \%$ confidence interval for the proportion of respondents who say the Internet has been a good thing for them personally.
b. The sample survey showed that $67 \%$ of Internet users said the Internet has generally strengthened their relationship with family and friends. Develop a $95 \%$ confidence interval for the proportion of respondents who say the Internet has strengthened their relationship with family and friends.
c. Fifty-six percent of Internet users have seen an online group come together to help a person or community solve a problem, whereas only $25 \%$ have left an online group because of unpleasant interaction. Develop a $95 \%$ confidence interval for the proportion of Internet users who say online groups have helped solve a problem.
d. Compare the margin of error for the interval estimates in parts (a), (b), and (c). How is the margin of error related to the sample proportion?

For many years businesses have struggled with the rising cost of health care. But recently, the increases have slowed due to less inflation in health care prices and employees paying for a larger portion of health care benefits. A recent Mercer survey showed that $52 \%$ of U.S. employers were likely to require higher employee contributions for health care coverage. Suppose the survey was based on a sample of 800 companies. Compute the margin of error and a $95 \%$ confidence interval for the proportion of companies likely to require higher employee contributions for health care coverage.

The manager of the Danvers-Hilton Resort Hotel stated that the mean guest bill for a weekend is $$\$ 600$$ or less. A member of the hotel's accounting staff noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of future weekend guest bills to test the manager's claim.
a. Which form of the hypotheses should be used to test the manager's claim? Explain.
$$
\begin{array}{lll}
H_0: \mu \geq 600 & H_0: \mu \leq 600 & H_0: \mu=600 \\
H_{\mathrm{a}}: \mu<600 & H_{\mathrm{a}}: \mu>600 & H_{\mathrm{a}}: \mu \neq 600
\end{array}
$$
b. What conclusion is appropriate when $H_0$ cannot be rejected?
c. What conclusion is appropriate when $H_0$ can be rejected?

The manager of an automobile dealership is considering a new bonus plan designed to increase sales volume. Currently, the mean sales volume is 14 automobiles per month. The manager wants to conduct a research study to see whether the new bonus plan increases sales volume. To collect data on the plan, a sample of sales personnel will be allowed to sell under the new bonus plan for a one-month period.
a. Develop the null and alternative hypotheses most appropriate for this situation.
b. Comment on the conclusion when $H_0$ cannot be rejected.
c. Comment on the conclusion when $H_0$ can be rejected.

A production line operation is designed to fill cartons with laundry detergent to a mean weight of 32 ounces. A sample of cartons is periodically selected and weighed to determine whether underfilling or overfilling is occurring. If the sample data lead to a conclusion of underfilling or overfilling, the production line will be shut down and adjusted to obtain proper filling.
a. Formulate the null and alternative hypotheses that will help in deciding whether to shut down and adjust the production line.
b. Comment on the conclusion and the decision when $H_0$ cannot be rejected.
c. Comment on the conclusion and the decision when $H_0$ can be rejected.

Because of high production-changeover time and costs, a director of manufacturing must convince management that a proposed manufacturing method reduces costs before the new method can be implemented. The current production method operates with a mean cost of $$\$ 220$$ per hour. A research study will measure the cost of the new method over a sample production period.
a. Develop the null and alternative hypotheses most appropriate for this study.
b. Comment on the conclusion when $H_0$ cannot be rejected.
c. Comment on the conclusion when $H_0$ can be rejected.

Duke Energy reported that the cost of electricity for an efficient home in a particular neighborhood of Cincinnati, Ohio, was $$\$ 104$$ per month. A researcher believes that the cost of electricity for a comparable neighborhood in Chicago, Illinois, is higher. A sample of homes in this Chicago neighborhood will be taken and the sample mean monthly cost of electricity will be used to test the following null and alternative hypotheses.
$$
\begin{aligned}
& H_0: \mu \leq 104 \\
& H_{\mathrm{a}}: \mu>104
\end{aligned}
$$
a. Assume that the sample data lead to rejection of the null hypothesis. What would be your conclusion about the cost of electricity in the Chicago neighborhood?
b. What is the Type I error in this situation? What are the consequences of making this error?
c. What is the Type II error in this situation? What are the consequences of making this error?

The label on a 3-quart container of orange juice states that the orange juice contains an average of 1 gram of fat or less. Answer the following questions for a hypothesis test that could be used to test the claim on the label.
a. Develop the appropriate null and alternative hypotheses.
b. What is the Type I error in this situation? What are the consequences of making this error?
c. What is the Type II error in this situation? What are the consequences of making this error?

Carpetland salespersons average $$\$ 8,000$$ per week in sales. Steve Contois, the firm's vice president, proposes a compensation plan with new selling incentives. Steve hopes that the results of a trial selling period will enable him to conclude that the compensation plan increases the average sales per salesperson.
a. Develop the appropriate null and alternative hypotheses.
b. What is the Type I error in this situation? What are the consequences of making this error?
c. What is the Type II error in this situation? What are the consequences of making this error?

Suppose a new production method will be implemented if a hypothesis test supports the conclusion that the new method reduces the mean operating cost per hour.
a. State the appropriate null and alternative hypotheses if the mean cost for the current production method is $\$ 220$ per hour.
b. What is the Type I error in this situation? What are the consequences of making this error?
c. What is the Type II error in this situation? What are the consequences of making this error?

Which is cheaper: eating out or dining in? The mean cost of a flank steak, broccoli, and rice bought at the grocery store is $$\$ 13.04$$. A sample of 100 neighborhood restaurants showed a mean price of $$\$ 12.75$$ and a standard deviation of $$\$ 2$$ for a comparable restaurant meal.
a. Develop appropriate hypotheses for a test to determine whether the sample data support the conclusion that the mean cost of a restaurant meal is less than fixing a comparable meal at home.
b. Using the sample from the 100 restaurants, what is the $p$ value?
c. At $\alpha=0.05$, what is your conclusion?

A shareholders' group, in lodging a protest, claimed that the mean tenure for a chief executive officer (CEO) was at least nine years. A survey of companies reported in the Wall Street Journal found a sample mean tenure of $\bar{x}=7.27$ years for CEOs with a standard deviation of $s=6.38$ years.
a. Formulate hypotheses that can be used to challenge the validity of the claim made by the shareholders' group.
b. Assume that 85 companies were included in the sample. What is the $p$ value for your hypothesis test?
c. At $\alpha=0.01$, what is your conclusion?

The national mean annual salary for a school administrator is $$\$ 90,000$$ a year. A school official took a sample of 25 school administrators in the state of Ohio to learn about salaries in that state to see if they differed from the national average.
a. Formulate hypotheses that can be used to determine whether the population mean annual administrator salary in Ohio differs from the national mean of $$\$ 90,000$$.
b. The sample data for 25 Ohio administrators is contained in the file named Administrator. What is the $p$ value for your hypothesis test in part (a)?
c. At $\alpha=0.05$, can your null hypothesis be rejected? What is your conclusion?

The time married men with children spend on child care averages 6.4 hours per week. You belong to a professional group on family practices that would like to do its own study to determine if the time married men in your area spend on child care per week differs from the reported mean of 6.4 hours per week. A sample of 40 married couples will be used with the data collected showing the hours per week the husband spends on child care. The sample data are contained in the file named ChildCare.
a. What are the hypotheses if your group would like to determine if the population mean number of hours married men are spending on child care differs from the mean reported by Time in your area?
b. What is the sample mean and the $p$ value?
c. Select your own level of significance. What is your conclusion?

The Coca-Cola Company reported that the mean per capita annual sales of its beverages in the United States was 423 eight-ounce servings. Suppose you are curious whether the consumption of Coca-Cola beverages is higher in Atlanta, Georgia, the location of Coca-Cola's corporate headquarters. A sample of 36 individuals from the Atlanta area showed a sample mean annual consumption of 460.4 eight-ounce servings with a standard deviation of $s=101.9$ ounces. Using $\alpha=0.05$, do the sample results support the conclusion that mean annual consumption of Coca-Cola beverage products is higher in Atlanta?

According to the National Automobile Dealers Association, the mean price for used cars is $$\$ 10,192$$. A manager of a Kansas City used car dealership reviewed a sample of 50 recent used car sales at the dealership in an attempt to determine whether the population mean price for used cars at this particular dealership differed from the national mean. The prices for the sample of 50 cars are shown in the file named UsedCars.
a. Formulate the hypotheses that can be used to determine whether a difference exists in the mean price for used cars at the dealership.
b. What is the $p$ value?
c. At $\alpha=0.05$, what is your conclusion?

What percentage of the population live in their state of birth? According to the U.S. Census Bureau's American Community Survey, the figure ranges from $25 \%$ in Nevada to $78.7 \%$ in Louisiana. The average percentage across all states and the District of Columbia is $57.7 \%$. The data in the file Homestate are consistent with the findings in the American Community Survey. The data are for a random sample of 120 Arkansas residents and for a random sample of 180 Virginia residents.
a. Formulate hypotheses that can be used to determine whether the percentage of stayat-home residents in the two states differs from the overall average of $57.7 \%$.
b. Estimate the proportion of stay-at-home residents in Arkansas. Does this proportion differ significantly from the mean proportion for all states? Use $\alpha=0.05$.
c. Estimate the proportion of stay-at-home residents in Virginia. Does this proportion differ significantly from the mean proportion for all states? Use $\alpha=0.05$.
d. Would you expect the proportion of stay-at-home residents to be higher in Virginia than in Arkansas? Support your conclusion with the results obtained in parts (b) and (c).

Last year, $46 \%$ of business owners gave a holiday gift to their employees. A survey of business owners indicated that $35 \%$ plan to provide a holiday gift to their employees. Suppose the survey results are based on a sample of 60 business owners.
a. How many business owners in the survey plan to provide a holiday gift to their employees?
b. Suppose the business owners in the sample do as they plan. Compute the $p$ value for a hypothesis test that can be used to determine if the proportion of business owners providing holiday gifts has decreased from last year.
c. Using a 0.05 level of significance, would you conclude that the proportion of business owners providing gifts has decreased? What is the smallest level of significance for which you could draw such a conclusion?

Ten years ago $53 \%$ of American families owned stocks or stock funds. Sample data collected by the Investment Company Institute indicate that the percentage is now $46 \%$.
a. Develop appropriate hypotheses such that rejection of $H_0$ will support the conclusion that a smaller proportion of American families own stocks or stock funds this year than 10 years ago.
b. Assume that the Investment Company Institute sampled 300 American families to estimate that the percent owning stocks or stock funds is $46 \%$ this year. What is the $p$ value for your hypothesis test?
c. At $\alpha=0.01$, what is your conclusion?

According to the University of Nevada Center for Logistics Management, $6 \%$ of all merchandise sold in the United States gets returned. A Houston department store sampled 80 items sold in January and found that 12 of the items were returned.
a. Construct a point estimate of the proportion of items returned for the population of sales transactions at the Houston store.
b. Construct a $95 \%$ confidence interval for the porportion of returns at the Houston store.
c. Is the proportion of returns at the Houston store significantly different from the returns for the nation as a whole? Provide statistical support for your answer.

Eagle Outfitters is a chain of stores specializing in outdoor apparel and camping gear. It is considering a promotion that involves mailing discount coupons to all its credit card customers. This promotion will be considered a success if more than $10 \%$ of those receiving the coupons use them. Before going national with the promotion, coupons were sent to a sample of 100 credit card customers.
a. Develop hypotheses that can be used to test whether the population proportion of those who will use the coupons is sufficient to go national.
b. The file named Eagle contains the sample data. Develop a point estimate of the population proportion.
c. Use $\alpha=0.05$ to conduct your hypothesis test. Should Eagle go national with the promotion?

One of the reasons health care costs have been rising rapidly in recent years is the increasing cost of malpractice insurance for physicians. Also, fear of being sued causes doctors to run more precautionary tests (possibly unnecessary) just to make sure they are not guilty of missing something. These precautionary tests also add to health care costs. Data in the file named LawSuit are consistent with findings in a Reader's Digest article and can be used to estimate the proportion of physicians over the age of 55 who have been sued at least once.
a. Formulate hypotheses that can be used to see if these data can support a finding that more than half of physicians over the age of 55 have been sued at least once.
b. Use Excel and the file named LawSuit to compute the sample proportion of physicians over the age of 55 who have been sued at least once. What is the $p$ value for your hypothesis test?
c. At $\alpha=0.01$, what is your conclusion?

The Port Authority sells a wide variety of cables and adapters for electronic equipment online. Last year the mean value of orders placed with the Port Authority was $$\$ 47.28$$, and management wants to assess whether the mean value of orders placed to date this year is the same as last year. The values of a sample of 49,896 orders placed this year are collected and recorded in the file PortAuthority.
a. Formulate hypotheses that can be used to test whether the mean value of orders placed this year differs from the mean value of orders placed last year.
b. Use the data in the file PortAuthority to conduct your hypothesis test. What is the $p$ value for your hypothesis test? At $\alpha=0.01$, what is your conclusion?

The Port Authority also wants to determine if the gender profile of its customers has changed since last year, when $59.4 \%$ of its orders placed were placed by males. The genders for a sample of 49,896 orders placed this year are collected and recorded in the file PortAuthority.
a. Formulate hypotheses that can be used to test whether the proportion of orders placed by male customers this year differs from the proportion of orders placed by male customers placed last year.
b. Use the data in the file PortAuthority to conduct your hypothesis test. What is the $p$ value for your hypothesis test? At $\alpha=0.05$, what is your conclusion?

Suppose a sample of 10,001 erroneous Federal income tax returns from last year has been taken and is provided in the file FedTaxErrors. A positive value indicates the taxpayer underpaid and a negative value indicates that the taxpayer overpaid.
a. What is the sample mean error made on erroneous Federal income tax returns last year?
b. Using $95 \%$ confidence, what is the margin of error?
c. Using the results from parts (a) and (b), develop the $95 \%$ confidence interval estimate of the mean error made on erroneous Federal income tax returns last year.

According to the Census Bureau, $2,475,780$ people are employed by the federal government in the United States. Suppose that a random sample of 3,500 of these federal employees was selected and the number of sick hours each of these employees took last year was collected from an electronic personnel database. The data collected in this survey are provided in the file FedSickHours.
a. What is the sample mean number of sick hours taken by federal employees last year?
b. Using $99 \%$ confidence, what is the margin of error?
c. Using the results from parts (a) and (b), develop the $99 \%$ confidence interval estimate of the mean number of sick hours taken by federal employees last year.
d. If the mean sick hours federal employees took two years ago was 62.2 , what would the confidence interval in part (c) lead you to conclude about last year?

Internet users were recently asked online to rate their satisfaction with the web browser they use most frequently. Of 102,519 respondents, 65,120 indicated they were very satisfied with the web browser they use most frequently.
a. What is the sample proportion of Internet users who are very satisfied with the web browser they use most frequently?
b. Using $95 \%$ confidence, what is the margin of error?
c. Using the results from parts (a) and (b), develop the $95 \%$ confidence interval estimate of the proportion of Internet users who are very satisfied with the web browser they use most frequently.

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 sample of 20,000 vehicles at $6: 00$ p.m. EST on a recent Tuesday afternoon. Of these 20,000 vehicles, 9,252 were speeding.
a. What is the sample proportion of vehicles on U.S. roads that speed?
b. Using $99 \%$ confidence, what is the margin of error?
c. Using the results from parts (a) and (b), develop the $99 \%$ confidence interval estimate of the proportion of vehicles on U.S. roads that speed.
d. What does the confidence interval in part (c) lead you to conclude about the ABC News report?

The Federal Government wants to determine if the mean number of business e-mails sent and received per business day by its employees differs from the mean number of e-mails sent and received per day by corporate employees, which is 101.5 . Suppose the department electronically collects information on the number of business e-mails sent and received on a randomly selected business day over the past year from each of 10,163 randomly selected Federal employees. The results are provided in the file FedEmail. Test the Federal Government's hypothesis at $\alpha=0.01$. Discuss the practical significance of the results.

CEOs who belong to a popular business-oriented social networking service have an average of 930 connections. Do other members have fewer connections than CEOs? The number of connections for a random sample of 7,515 members who are not CEOs is provided in the file SocialNetwork. Using this sample, test the hypothesis that other members have fewer connections than $\mathrm{CEOs}$ at $\alpha=0.01$. Discuss the practical significance of the results.

The American Potato Growers Association (APGA) would like to test the claim that the proportion of fast-food orders this year that include French fries exceeds the proportion of fast-food orders that included French fries last year. Suppose that a random sample of 49,581 electronic receipts for fast-food orders placed this year shows that 31,038 included French fries. Assuming that the proportion of fast-food orders that included French fries last year is 0.62 , use this information to test APGA's claim at $\alpha=0.05$. Discuss the practical significance of the results.

According to CNN, $55 \%$ of all U.S. smartphone users have used their GPS capability to get directions. Suppose a major provider of wireless telephone service in Canada wants to know how GPS usage by its customers compares with U.S. smartphone users. The company collects usage records for this year for a random sample of 547,192 of its Canadian customers and determines that 302,050 of these customers have used their telephone's GPS capability this year. Use this data to test whether Canadian smartphone users' GPS usage differs from U.S. smartphone users' GPS usage at $\alpha=0.01$. Discuss the practical significance of the results.

A well-respected polling agency has conducted a poll for an upcoming Presidential election. The polling agency has taken measures so that its random sample consists of 50,000 people and is representative of the voting population. The file Pedro contains survey data for 50,000 respondents in both a pre-election survey and a post-election poll.
a. Based on the data in the "Support Pedro in Pre-Election Poll" column, compute the $99 \%$ confidence interval on the population proportion of voters who support Pedro Ringer in the upcoming election. If Pedro needs at least $50 \%$ of the vote to win in the two-party election, should he be optimistic about winning the election?
b. Now suppose the election occurs and Pedro wins $55 \%$ of the vote. Explain how this result could occur given the sample information in part (a).
c. In an attempt to explain the election results (Pedro winning $55 \%$ of the vote), the polling agency has followed up with each of the respondents in their pre-election survey. The data in the "Voted for Pedro?" column corresponds to whether or not the respondent actually voted for Pedro in the election. Compute the $99 \%$ confidence interval on the population proportion of voters who voted for Pedro Ringer. Is this result consistent with the election results?
d. Use a PivotTable to determine the percentage of survey respondents who voted for Pedro that did not admit to supporting him in a pre-election poll. Use this result to explain the discrepancy between the pre-election poll and the actual election results. What type of error is occurring here?