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Statistics for Business and Economics

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

Chapter 9

Hypothesis Tests - all with Video Answers

Educators

Chapter Questions

01:21

Problem 1

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 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?

Breanna Ollech
Breanna Ollech
Numerade Educator
02:04

Problem 2

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.

Srikar Katta
Srikar Katta
Numerade Educator
00:48

Problem 3

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.

Eleanor Archer
Eleanor Archer
Numerade Educator
02:33

Problem 4

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.

Srikar Katta
Srikar Katta
Numerade Educator
01:29

Problem 5

Nielsen reported that young men in the United States watch 56.2 minutes of prime-time TV daily (The Wall Street Joumal Europe, November 18, 2003). A researcher believes that young men in Germany spend more time watching prime-time TV. A sample of German young men will be selected by the researcher and the time they spend watching TV in one day will be recorded. The sample results will be used to test the following null and alternative hypotheses.
$$
\begin{aligned}
& H_0: \mu \leq 56.2 \\
& H_{\mathrm{a}}: \mu>56.2
\end{aligned}
$$
a. What is the Type I error in this situation? What are the consequences of making this error?
b. What is the Type II error in this situation? What are the consequences of making thiserror?

Nick Johnson
Nick Johnson
Numerade Educator
02:25

Problem 6

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?

Srikar Katta
Srikar Katta
Numerade Educator
02:29

Problem 7

Carpetland salespersons average $\$ 8000$ 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?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:52

Problem 8

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?

Nick Johnson
Nick Johnson
Numerade Educator
02:42

Problem 9

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu \geq 20 \\
& H_{\mathrm{a}}: \mu<20
\end{aligned}
$$
A sample of 50 provided a sample mean of 19.4 . The population standard deviation is 2 .
a. Compute the value of the test statistic.
b. Whar is the p-value?
c. Using $a=.05$, what is your conclusion?
d. Whar is the rejection rule using the critical value? What is your conclusion?

Nick Johnson
Nick Johnson
Numerade Educator
02:05

Problem 10

Consider the following hypothesis test:
$$
\begin{aligned}
& H_{\mathrm{o}}: \mu \leq 25 \\
& H_{\mathrm{a}}: \mu>25
\end{aligned}
$$
A sample of 40 provided a sample mesan of 26.4. The population standard deviation is 6 .
a. Compute the value of the test statistic.
b. What is the $p$-value?
c. At $\alpha=.01$, what is your conclusion?
d. What is the rejection rule using the critical value? What is your conclusion?

Nick Johnson
Nick Johnson
Numerade Educator
05:13

Problem 11

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu=15 \\
& H_{\mathrm{a}}: \mu \neq 15
\end{aligned}
$$
A sample of 50 provided a sample mean of 14.15 . The population standard deviation is 3 .
a. Compute the value of the test statistic.
b. What is the p-value?
c. At $\alpha=.05$, what is your conclusion?
d. What is the rejection rule using the critical value? What is your conclusion?

Eleanor Archer
Eleanor Archer
Numerade Educator
04:37

Problem 12

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu \geq 80 \\
& H_{\mathrm{a}}: \mu<80
\end{aligned}
$$
A sample of 100 is used and the population standard deviation is 12 . Compute the p-value and state your conclusion for each of the following sample results. Use $a=.01$.
a. $\bar{x}=78.5$
b. $\bar{x}=77$
c. $\vec{x}=75.5$
d. $\tilde{\boldsymbol{x}}=81$

Nick Johnson
Nick Johnson
Numerade Educator
02:59

Problem 13

Consider the following hypothesis test:
$$
\begin{aligned}
& H_a: \mu \leq 50 \\
& H_{\mathrm{a}}: \mu>50
\end{aligned}
$$
A sample of 60 is used and the population standard deviation is 8 . Use the critical value approach to state your conclusion for each of the following sample results. Use $a=.05$.
a. $\tilde{x}=52.5$
b. $\bar{x}=51$
c. $\overline{\boldsymbol{x}}=51.8$

Nick Johnson
Nick Johnson
Numerade Educator
04:26

Problem 14

Consider the following hypothesis test:
$$
\begin{aligned}
& H_{\mathrm{o}}: \mu=22 \\
& H_{\mathrm{a}}: \mu \neq 22
\end{aligned}
$$
A sample of 75 is used and the population standard deviation is 10. Compute the $p$-value and state your conclusion for each of the following sample results. Use $\alpha=.01$.
a. $\bar{x}=23$
b. $\bar{x}=25,1$
c. $\tilde{x}=20$

Nick Johnson
Nick Johnson
Numerade Educator
01:58

Problem 15

Individuals filing federal income tax returns prior to March 31 received an average refund of $$\$10$$S6. Consider the population of "last-minute" fllers who mail their tax return during the last five days of the income tax period (typically April 10 to April 15).
a. A researcher suggests that a reason individuals wait until the last five days is that on average these individuals receive lower refunds than do early filers. Develop appropriate hypotheses such that rejection of $H_0$ will support the researcher's contention.
b. For a sample of 400 individuals who filed a tax return between April 10 and 15 , the sample mean refund was $$\$ 910$$. Based on prior experience a population standard deviation of $$\sigma=\$ 1600$$ may be assumed. What is the $p$-value?
c. At $a=.05$, what is your conclusion?
d. Repear the preceding hypochesis test using the critical value approach.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:41

Problem 16

In a study entitled How Undergraduate Students Use Credit Cards, it was reported thar undergraduate students have a mean credit card balance of $$\$ 3173$$ (Sallie Mac, April 2009). This figure was an all-time high and had increased $44 \%$ over the previous five years. Assume that a current study is being conducted to determine if it can be concluded that the mean credit card balance for undergraduate students has continued to increase compared to the April 2009 report. Based on previous studies, use a population standard deviation $\sigma=$ $$\$ 1000$$.
a. State the null and alvernative hypotheses.
b. What is the $p$-value for a sample of 180 undergraduate students with a sample mean credit card balance of $$\$ 3325$$ ?
c. Using a 05 level of significance, what is your conclusion?

Adriano Chikande
Adriano Chikande
Numerade Educator
01:53

Problem 17

Wall Street securities firms paid out record year-end bonuses of $$\$ 125,500$$ per employee for 2005 (Fortune, February 6, 2006). Suppose we would like to take a sample of employees at the Jones \& Ryan securities firm to see whether the mean year-end bonus is different from the reported mean of $$\$ 125,500$$ for the population.
a. State the null and alternative hypotheses you would use to test whether the year-end bonuses paid by Jones \& Ryan were different from the population mean.
b. Suppose a sample of 40 Jones & Ryan employees showed a sample mean year-end bonus of $$\$ 118,000$$. Assume a population standard deviation of $$\sigma=\$ 30,000$$ and compute the $p$-value.
c. With $a-05$ as the level of significance, what is your conclusion?
d. Repeat the preceding bypohesis test using the critical value approach.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:01

Problem 18

The average anmual total return for U.S. Diversified Equity mutual funds from 1999 to 2003 was $4.1 \%$ (Business Week, January 26, 2004). A researcher would like to conduct a hypothesis test to see whether the returns for mid-cap growth funds over the same period are significantly different from the average for U.S. Diversified Equity funds.
a. Formulate the hypotheses that can be used to determine whether the mean annual return for mid-cap growth funds differ from the mean for U.S. Diversified Equity funds.
b. A sample of 40 mid-cap growth funds provides a mean return of $\tilde{x}=3.4 \%$. Assume the population standard deviation for mid-cap growth funds is known from previous studies to be $\sigma-2 \%$. Use the sample results to compute the test statistic and $p$-value for the hypothesis test.
c. At $a=.05$, what is your conclusion?

Nick Johnson
Nick Johnson
Numerade Educator
01:28

Problem 19

The U.S. Department of Labor reported the average hourly earnings for U.S. production workers to be $$\$ 14.32$$ per hour in 2001 (The World Almanuc, 2003). A sample of 75 production workers during 2003 showed a sample mean of $$\$ 14.68$$ per hour. Assuming the population standard deviation $$\sigma=\$ 1.45$$, can we conclude that an increase occurred in the mean hourly eamings since 2001 ? Use $\alpha=.05$.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:50

Problem 20

For the United States, the mean monthly Internet bill is $$\$ 32.79$$ per household (CNBC, January 18,2006 ). A sample of 50 households in a southern state showed a sample mean of $$\$ 30.63$$. Use a population standard deviation of $$\sigma=\$ 5.60$$.
a. Formulate hypocheses for a test to determine whether the sample data support the conclusion that the mean monthly Intemet bill in the southern state is less than the national mean of $$\$ 32.79$$.
b. What is the value of the test statistic?
c. What is the p-value?
d. At $\alpha=.01$, what is your conclusion?

Nick Johnson
Nick Johnson
Numerade Educator
01:23

Problem 21

Fowle Marketing Research, Inc., bases charges to a client on the assumption that telephone surveys can be completed in a mean time of 15 minutes or less. If a longer mean survey time is necessary, a premium rate is charged. A sample of 35 sunveys provided the survey times shown in the file named Fowle. Based upon past studies, the population standard deviation is assumed known with $\sigma-4$ minutes. Is the premium rate justified?
a. Formulate the null and alternative hypotheses for this application.
b. Compute the value of the test statistic.
c. What is the p-value?
d. At $a=.01$, what is your conclusion?

Adriano Chikande
Adriano Chikande
Numerade Educator
04:59

Problem 22

$\mathrm{CCN}$ and ActMedia provided a television channel targeted to individuals waiting in supermarket checkout lines. The channel showed news, short features, and advertisements. The length of the program was based on the assumption that the population mean time a shopper stands in a supermarket checkout line is 8 minutes. A sample of actual waiting times will be used to test this assumption and determine whether actual mean waiting time differs from this standard.
a. Formulate the hypotheses for this application.
b. A sample of 120 shoppers showed a sample mean waiting time of 8.5 minutes. Assume a population standard deviation of $\sigma-3.2$ minutes. What is the p-value?
c. At $a=.05$, what is your conclusion?
d. Compute a $95 \%$ confidence interval for the population mean. Does it support your conclusion?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:58

Problem 23

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu \leq 12 \\
& H_{\mathrm{a}}: \mu>12
\end{aligned}
$$
A sample of 25 provided a sample mean $\bar{x}-14$ and a sample standard deviation $s=4.32$.
a. Compute the value of the test statistic.
b. Use the $t$ distribution table (Table 2 in Appendix B) to compute a range for the $p$-value.
c. At $\alpha=.05$, what is your conclusion?
d. What is the rejection rule using the critical value? What is your conclusion?

Nick Johnson
Nick Johnson
Numerade Educator
06:21

Problem 24

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu=18 \\
& H_{\mathrm{a}}: \mu \neq 18
\end{aligned}
$$
A sample of 48 provided a sample mean $\bar{x}-17$ and a sample standard deviation $s=4.5$.
a. Compute the value of the test statistic.
b. Use the $t$ distribution table (Table 2 in Appendix B) to compute a range for the p-value.
c. At $\alpha=.05$, what is your conclusion?
d. What is the rejection rule using the critical value? What is your conclusion?

Srikar Katta
Srikar Katta
Numerade Educator
04:17

Problem 25

Consider the following hypothesis test:
$$
\begin{aligned}
& H_{\mathrm{o}}: \mu \geq 45 \\
& H_{\mathrm{a}}: \mu<45
\end{aligned}
$$
A sample of 36 is used. Identify the $p$-value and state your conclusion for each of the following sample results. Use $a=.01$.
a. $\vec{x}=44$ and $s=5.2$
b. $\bar{x}=43$ and $s=4.6$
c. $\vec{x}=46$ and $s=5.0$

Nick Johnson
Nick Johnson
Numerade Educator
04:22

Problem 26

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu=100 \\
& H_2: \mu+100
\end{aligned}
$$
A sample of 65 is used. Identify the p-value and state your conclusion for each of the following sample results. Use $a=$.0S.
a. $\bar{x}=103$ and $s=11.5$
b. $\bar{x}=96.5$ and $s=11.0$
c. $\bar{x}=102$ and $s=10.5$

Nick Johnson
Nick Johnson
Numerade Educator
06:13

Problem 27

The Employment and Training Administration reported that the U.S. mean unemployment insurance benefit was $$\$ 238$$ per week (The World Almanac, 2003). A researcher in the state of Virginia anticipated that sample data would show evidence thar the mean weekly unemployment insurance benefit in Virginia was below the national average.
a. Develop appropriate hypotheses such that rejection of $H_0$ will support the researcher's contention.
b. For a sample of 100 individuals, the sample mean weekly unemployment insurance benefit was $$\$ 231$$ with a sample standard deviation of $\$ 80$. What is the p-value?
c. At $a=.05$, what is your conclusion?
d. Repear the preceding hypothesis test using the critical value approach.

Robin Corrigan
Robin Corrigan
Numerade Educator
01:43

Problem 28

A shareholders' group, in lodging a protest, claimed that the mean tenure for a chief exective office (CEO) was ar least nine years. A survey of companies reported in The Wall Sirect Joumal found a sample mean tenure of $\bar{x}-7.27$ years for CEOs with a standard deviation of $s-6.38$ years (The Wall Sireet Journal, January 2, 2007).
a. Formulate hypotheses that can be used to challenge the validity of the claim made by the sharcholders' group.
b. Assume 85 companies were included in the sample. What is the p-value for your hypochesis test?
c. At $a=.01$, what is your conclusion?

Adriano Chikande
Adriano Chikande
Numerade Educator
03:36

Problem 29

The cost of a one-carat VS2 clarity, H color diamond from Diamond Source USA is $$\$ 5600$$ (Diamond Source website, March 2003). A midwestern jeweler makes calls to contacts in the diamond district of New York Ciry to see whether the mean price of diamonds there differs from $$\$ 5600$$.
a. Formulate hypocheses that can be used to determine whether the mean price in New York City differs from $$\$ 5600$$.
b. A sample of 25 New York City contacts provided the prices shown in the file named Diamonds. What is the p-value?
c. At $a=, 05$, can the null hypochesis be rejected? What is your conclusion?
d. Repear the preceding hypochesis test using the critical value approach.

Nick Johnson
Nick Johnson
Numerade Educator
02:38

Problem 30

AOL Time Warner Inc.'s CNN has been the longtime ratings leader of cable television news. Nielsen Media Research indicated that the mean $\mathrm{CNN}$ viewing audience was 600,000 viewers per day during 2002 (The Wall Street Joumal, March 10, 2003). Assume that for a sample of 40 days during the first half of 2003 , the daily audience was 612,000 viewers with a sample standard deviation of 65,000 viewers.
a. What are the hypotheses if CNN management would like information on any change in the $\mathrm{CNN}$ viewing audience?
b. What is the $p$-value?
c. Select your own level of significance. What is your conclusion?
d. What recommendation would you make to $\mathrm{CNN}$ management in this application?

Nick Johnson
Nick Johnson
Numerade Educator
01:33

Problem 31

The Coca-Cola Company reported that the mean per capita annual sales of its beverages in the United States was 423 eight-ounce servings (Coca-Cola Company website, February 3,
2009). Suppose you are curious whether the consumption of Coca-Cola beverages is higher in Adlanta, Georgia, the location of Coca-Cola's corporate headquarters. A sample of 36 individuals from the Allanta area showed a sample mean annual consumption of 460.4 eight-ounce servings with a standard devistion of $s=101.9$ ounces. Using $\alpha=.05$, do the sample results support the conclusion that mean annusal consumption of Coca-Cola beverage products is higher in Aulanta?

Adriano Chikande
Adriano Chikande
Numerade Educator
01:35

Problem 32

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 hypocheses that can be used to determine whether a difference exists in the mean price for used cars at the dealership.
b. Whar is the p-value?
c. At $a=.05$, what is your conclusion?

Adriano Chikande
Adriano Chikande
Numerade Educator
01:47

Problem 33

Annual per capita consumption of milk is 21.6 gallons (Statistical Abstract of the Unired States: 2006). Being from the Midwest, you believe milk consumption is higher there and wish to support your opinion. A sample of 16 individuals from the midwestern town of Webster City showed a sample mean annual consumption of 24.1 gallons with a standard deviation of $s=4.8$.
a. Develop a hypothesis test that can be used to determine whether the mean annual consumption in Webster City is higher than the national mean.
b. What is a point estimate of the difference between mean annual consumption in Webster City and the national mean?
c. At $\alpha=.05$, test for a significant difference. What is your conclusion?

Nick Johnson
Nick Johnson
Numerade Educator
01:51

Problem 34

Joan's Nursery specializes in custom-designed landscaping for residential areas. The estimated labor cost associated with a particular landscaping proposal is based on the number of plantings of trees, shrubs, and so on to be used for the project. For costestimating purposes, managers use two hours of labor time for the planting of a medium-sized tree. Actual times from a sample of 10 plantings during the past month follow (times in hours).
$$
1.7
$$
1.5
2.6
2.2
2.4
2.3
2.6
3.0
1.4
2.3
With a 05 level of significance, test to see whether the mean tree-planting time differs from two hours.
a. State the null and alternative hypotheses.
b. Compute the sample mean.
c. Compute the sample standard deviation.
d. What is the p-value?
e. What is your conclusion?

Adriano Chikande
Adriano Chikande
Numerade Educator
01:39

Problem 35

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: p=.20 \\
& H_{\mathrm{a}}=p+.20
\end{aligned}
$$
A sample of 400 provided a sample proportion $\tilde{p}=.175$.
a. Compute the value of the test statistic.
b. What is the $p$-value?
c. At $a=.05$, what is your conclusion?
d. What is the rejection rule using the critical value? What is your conclusion?

Adriano Chikande
Adriano Chikande
Numerade Educator
04:20

Problem 36

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0=p \geq .75 \\
& H_{\mathrm{a}}=p<.75
\end{aligned}
$$
A sample of 300 items was selected. Compute the $p$-value and state your conclusion for each of the following sample results. Use $\alpha=.05$.
a. $\bar{p}=.68$
c. $\vec{p}=.70$
b. $\bar{p}=.72$
d. $\tilde{p}=.77$

Nick Johnson
Nick Johnson
Numerade Educator
01:39

Problem 37

A study found that, in $2005,12.5 \%$ of U.S. workers belonged to unions (The Wall Street Journal, January 21, 2006). Suppose a sample of 400 U.S. workers is collected in 2006 to determine whether union efforts to organize have increased union membership.
a. Formulate the hypotheses that can be used to determine whether union membership increased in 2006.
b. If the sample results show that 52 of the workers belonged to unions, what is the p-value for your hypothesis test?
c. At $a=.05$, what is your conclusion?

Nick Johnson
Nick Johnson
Numerade Educator
04:51

Problem 38

A study by Conswumer Reports showed that $64 \%$ of supermarket shoppers believe supermarket brands to be as good as national name brands. To investigate whether this result applies to its own product, the manufacturer of a national name-brand ketchup asked a sample of shoppers whether they believed that supermarket ketchup was as good as the national brand ketchup.
a. Formulate the bypotheses that could be used to determine whether the percentage of supermarket shoppers who believe that the supermarket ketchup was as good as the national brand ketchup differed from $64 \%$.
b. If a sample of 100 shoppers showed 52 stating that the supermarket brand was as good as the national brand, what is the $p$-value?
c. At $a=05$, what is your conclusion?
d. Should the narional brand ketchup manufacturer be pleased with this conclusion? Explain.

Srikar Katta
Srikar Katta
Numerade Educator
02:36

Problem 39

Accroding to the Pew Internet \& American Life Project, 75\% of American adults use the Internet (Pew Internet website, April 19, 2008). The Pew project authors also reported on the percentage of Americans who use the Internet by age group. The data in the file AgeCiroup are consistent with their findings. These data were obtained froen a sample of 100 Intemet users in the 30-49 age group and 200 Internet users in the $50-64$ age group. A Yes indicates the survey repondent had used the Internet; a No indicates the survey repondent had not.
a. Formulate hypotheses that could be used to determine whether the percentage of Internet users in the two age groups differs from the overall average of $75 \%$
b. Estimate the proportion of Intemet users in the 30-49 age group. Does this proportion differ significantly from the overall proportion of .75 ? Use $a=.05$
c. Estimate the proportion of Intemer users in the $50-64$ age group. Does this proportion differ significantly from the overall proportion of .75? Use $a=.05$
d. Would you expect the proportion of users in the $18-29$ age group to be larger or smaller than the proportion for the $30-49$ age group? Support you conclusion with the results obtained in parts (b) and (c).

Adriano Chikande
Adriano Chikande
Numerade Educator
01:23

Problem 40

Before the 2003 Super Bowl, ABC predicted that 22\% of the Super Bowl audience would express an interest in seeing one of its forthcoming new television shows, including 8 Simple Rries, Are You Hor?, and Dragnet. ABC ran commercials for these television shows during the Super Bowl. The day after the Super Bowl, Intermediate Advertising Giroup of New York sampled 1532 viewers who saw the commercials and found that 414 said that they would watch one of the ABC advertised television shows (The Wall Street Joumal, January 30,2003 ).
a. What is the point estimate of the proportion of the audience that said they would watch the television shows after seeing the television commercials?
b. At $a=.05$, deternine whether the intent to watch the ABC television shows significantly increased after seeing the television commercials. Formulate the appropriate hypotheses, compute the $p$-value, and state your conclusion.
c. Why are such studies valuable to companies and advertising firms?

Adriano Chikande
Adriano Chikande
Numerade Educator
03:15

Problem 41

Speaking to a group of analysts in January 2006 , a brokerage firm executive claimed that at least $70 \%$ of investors are currently confident of meeting their investment objectives. A UBS Investor Optimism Survey, conducted over the period January 2 to January 15,
found that $67 \%$ of investors were confident of meeting their investment objectives (CNBC. January 20,2006$)$.
a. Formulate the hypotheses that can be used to test the validity of the brokerage firm executive's claim.
b. Assume the UBS Investor Optimism Survey collected information from 300 investors. What is the p-value for the hypothesis test?
c. At $a=.05$, should the executive's claim be rejected?

Robin Corrigan
Robin Corrigan
Numerade Educator
08:49

Problem 42

Aceording to the University of Nevada Center for Logistics Management, $6 \%$ of all merchandise sold in the United Stares gets returned (BusinesrWeek, January 15, 2007). A Houston department store sampled 80 items sold in Janusry 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 retums at the Houston store.
c. Is the proportion of retums al the Houston store significantly different from the retums for the nation as a whole? Provide statistical support for your answer.

Srikar Katta
Srikar Katta
Numerade Educator
02:23

Problem 43

Eagle Outfiters is a chain of stores specializing in outoor apparel and camping gear. They are considering a promotion that involves mailing discount coupons to all their 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 Eagle contains the sample data. Develop a point estimate of the population proportion.
c. Use $\alpha=.05$ to conduct your hypothesis test. Should Eagle go national with the promotion?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
View

Problem 44

In a cover story, Bursiness Week published information about sleep habits of Americans (BusinessWeek, January 26,2004 ). The article noted that sleep deprivation causes a number of problems, including highway deachs. Fifty-one percent of adult drivers admit to driving while drowsy. A researcher hypothesized that this issue was an even bigger problem for night shift workers.
a. Formulate the bypotheses that can be used to help determine whether more than 515 of the population of night shift workers admit to driving while drowsy.
b. A sample of 400 night shift workers identified those who admitted to driving while drowsy. See the Drowsy file. Whar is the sample proportion? What is the $p$-value?
c. At $a=.01$, what is your conclusion?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:21

Problem 45

Many investors and financial analysts believe the Dow Jones Industrial Average (DJLA) provides a good barometer of the overall stock market. On January 31, 2006, 9 of the 30 stocks making up the DJIA increased in price (The Wall Sireet Journal, February 1 , $2006)$. On the basis of this fact, a financial analyst claims we can assume that $30 \%$ of the stocks traded on the New York Stock Exchange (NYSE) went up the same day.
a. Formulate null and alternative hypotheses to test the anslyst's claim.
b. A sample of 50 stocks traded on the NYSE that day showed that 24 went up. What is your point estimate of the population proportion of stocks that went up?
c. Conduct your hypothesis test using $\alpha=.01$ as the level of significance. What is your conclusion?

Nick Johnson
Nick Johnson
Numerade Educator
04:45

Problem 46

Consider the following hypothesis test.
$$
\begin{aligned}
& H_0: \mu \geqslant 10 \\
& H_{\mathrm{a}}=\mu<10
\end{aligned}
$$
The sample size is 120 and the population standard deviation is assumed known with $a=5$. Use $\alpha=.05$.
a. If the population mean is 9 , what is the probability that the sample mean leads to the conclusion do not reject $\mathrm{H}_0$ ?
b. What rype of error would be made if the actual population mean is 9 and we conclude that $H_0 ; \mu \geq 10$ is true?
c. What is the probability of making a Type II error if the actual population mean is 8 ?

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 47

Consider the following hypothesis test.
$$
\begin{aligned}
& H_0: \mu=20 \\
& H_{\mathrm{a}}: \mu+20
\end{aligned}
$$
A sample of 200 items will be taken and the population standard deviation is $\sigma=10$. Use $a=.05$. Compute the probability of making a Type II emor if the popalation mean is:
a. $\mu=18.0$
b. $\mu=22.5$
c. $\mu=21.0$

Nick Johnson
Nick Johnson
Numerade Educator
06:24

Problem 48

Fowle Marketing Research, Inc., bases charges to a client on the assumption that telephone surveys can be completed within 15 minutes or less. If more time is required, a premium rate is charged. With a sample of 35 surveys, a population standard deviation of 4 minutes, and a level of significance of .01, the sample mean will be used to test the null bypothesis $H_0: \mu \subseteq 15$.
a. What is your interpeetation of the Type Il error for this problem? What is its impact on the firm?
b. What is the probability of making a Type II error when the actual mean time is $\mu=17$ minutes?
c. What is the probability of making a Type II error when the actual mean time is $\mu=18$ minutes?
d. Sketch the general shape of the power curve for this test.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:58

Problem 49

A consumer research group is interested in testing an automobile manufacturer's claim that a new economy model will travel at least 25 miles per gallon of gasoline $\left(H_0 ; \mu \geq 25\right)$.
a. With a .02 level of significance and a sample of 30 cars, what is the rejection rule based on the value of $\bar{x}$ for the test to determine whether the manufacturer's claim should be rejected? Assume that $\sigma$ is 3 miles per gallon.
b. What is the probability of committing a Type II error if the actual mileage is 23 miles per gallon?
c. What is the probability of committing a Type II error if the actual mileage is 24 miles per gallon?
d. What is the probability of committing a Type II error if the actual mileage is 25.5 miles per gallon?

Adriano Chikande
Adriano Chikande
Numerade Educator
03:20

Problem 50

Young Adrlt magazine states the following hypotheses about the mean age of its subscribers.
$$
\begin{aligned}
& H_0: \mu=28 \\
& H_{\mathrm{a}}: \mu+28
\end{aligned}
$$
a. What would it mean to make a Type II error in this situation?
b. The population standard deviation is assumed known at $a-6$ years and the sample size is 100 . With $a=.05$, what is the probability of accepting $H_0$ for $\mu$ equal to 26,27 , 29 , and 30 ?
c. What is the power at $\mu-26$ ? What does this result tell you?

Adriano Chikande
Adriano Chikande
Numerade Educator
02:42

Problem 51

A production line operation is tested for filling weight accuracy using the following hypotheses.
$\begin{array}{ll}\text { Hypothesis } & \text { Conclusion and Action } \\ H_0: \mu=16 & \text { Filling okay; keep running } \\ H_{\mathrm{a}}: \mu \neq 16 & \text { Filling off standard; stop and adjust machine }\end{array}$
The sample size is 30 and the population standard deviation is $\sigma=.8$. Use $\alpha=.05$.
a. What would a Type II error mean in this situation?
b. What is the probability of making a Type II error when the machine is overfilling by 5 ounces?
c. What is the power of the statistical test when the machine is overfilling by .5 ounces?
d. Show the power curve for this hypochesis test. What information does it contain for the production manager?

Adriano Chikande
Adriano Chikande
Numerade Educator
01:31

Problem 52

Refer to exercise 48. Assume the firm selects a sample of 50 surveys and repeat parts (b) and (c). What observation can you make about how increasing the sample size affects the probability of making a Type II error?

Adriano Chikande
Adriano Chikande
Numerade Educator
02:22

Problem 53

Sparr Investments, Inc., specializes in tax-deferred investment opportunities for its clients. Recently Sparr offered a payroll deduction investment program for the employees of a particular company. Sparr estimates that the employees are currently averaging $$\$ 100$$ or less per month in tax-deferred investments. A sample of 40 employees will be used to test Sparr's hypothesis about the current level of investment activity among the population of employees. Assume the employee monthly tax-deferred investment amounts have a standard deviation of $$\$ 75$$ and that a .05 level of significance will be used in the hypothesis test. a. What is the Type II error in this situation?
b. What is the probability of the Type II error if the actual mean employee monthly investment is $$\$ 120$$ ?
c. What is the probability of the Type II error if the actual mean employee monthly investment is $$\$ 130$$ ?
d. Assume a sample size of 80 employees is used and repeat parts (b) and (c).

Adriano Chikande
Adriano Chikande
Numerade Educator
01:05

Problem 54

Consider the following hypothesis test.
$$
\begin{aligned}
& H_0: \mu \geq 10 \\
& H_{\mathrm{a}}: \mu<10
\end{aligned}
$$
The sample size is 120 and the population standard deviation is 5 . Use $\alpha=.05$. If the actual population mean is 9 , the probability of a Type Il error is .2912. Suppose the researcher wants to reduce the probability of a Type II error to .10 when the actual population mean is 9 . What sample size is recommended?

Adriano Chikande
Adriano Chikande
Numerade Educator
01:17

Problem 55

Consider the following hypothesis test.
$$
\begin{aligned}
& H_0: \mu=20 \\
& H_{\mathrm{a}}: \mu \neq 20
\end{aligned}
$$
The population standard deviation is 10 . Use $\alpha=.05$. How large a sample should be taken if the researcher is willing to accept a 05 probability of making a Type II error when the actual population mean is 22 ?

Adriano Chikande
Adriano Chikande
Numerade Educator
01:44

Problem 56

Suppose the project director for the Hilltop Coffee study (see Section 9.3) asked for a .10 probability of claiming that Hilltop was not in violation when it really was underfilling by 1 ounce ( $\mu_{\mathrm{a}}-2.9375$ pounds). What sample size would have been recommended?

Adriano Chikande
Adriano Chikande
Numerade Educator
01:18

Problem 57

A special industrial battery must have a life of at least 400 hours. A hypothesis test is to be conducted with a .02 level of significance. If the batteries from a particular production run have an actual mean use life of 385 hours, the production manager wants a sampling procedure that only $10 \%$ of the time would show erroneously that the batch is acceptable. What sample size is recommended for the hypothesis test? Use 30 hours as an estimate of the population standard deviation.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:25

Problem 58

Young Adult magazine states the following hypotheses about the mean age of its subscribers.
$$
\begin{aligned}
& H_{\mathrm{a}}=\mu-28 \\
& H_{\mathrm{a}}=\mu+28
\end{aligned}
$$
If the manager conducting the test will permit a .15 probability of making a Type If error when the true mean age is 29 , what sample size should be selected? Assume $\sigma=6$ and a .05 level of significance.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:17

Problem 59

An automobile mileage study tested the following bypocheses.
$$
\begin{aligned}
& \text { Hypothesis } \\
& H_0: \mu \geq 25 \mathrm{mpg} \\
& H_{\mathrm{a}}: \mu<25 \mathrm{mpg}
\end{aligned}
$$
Conclusion
Manufacturer's claim supported
Manufacturer's claim rejected; average mileage per gallon less than stated
For $\sigma=3$ and a 02 level of significance, what sample size would be recommended if the researcher wants an $80 \%$ chance of detecting that $\mu$ is less than 25 miles per gallon when it is actually 24 ?

Adriano Chikande
Adriano Chikande
Numerade Educator
03:47

Problem 60

A production line operates with a mean filling weight of 16 ounces per container. Overfilling or underfilling presents a serious problem and when detected requires the operator to shut down the production line to readjust the filling mechanism. From past data, a population standard deviation $\sigma=.8$ ounces is assumed. A quality control inspector selects a sample of 30 items every hour and at that time makes the decision of whether to shut down the line for readjustment. The level of significance is $a=.05$.
a. State the hypothesis test for this quality control application.
b. If a sample mean of $\bar{x}=16.32$ ounces were found, what is the $p$-value? What action would you recommend?
c. If a sample mean of $\bar{x}=15.82$ ounces were found, what is the p-value? What action would you recommend?
d. Use the critical value approach. What is the rejection rule for the preceding bypothesis vesting procedure? Repeat parts (b) and (c). Do you reach the same conclusion?

Kari Hasz
Kari Hasz
Numerade Educator
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Problem 61

At Western University the historical mean of scholarship examination scores for freshman applications is 900 . A historical population standard deviation $\sigma=180$ is assumed known. Each year, the assistant dean uses a sample of applications to determine whether the mean examination score for the new freshman applications has changed.
a. State the hypotheses.
b. What is the $95 \%$ confidence interval estimate of the population mean examination score if a sample of 200 applications provided a sample mean $\bar{x}=935$ ?
c. Use the confidence interval to conduct a hypochesis test. Using $\alpha-.05$, what is your conclusion?
d. What is the $p$-value?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:43

Problem 62

Playbill is a magazine distributed around the country to people attending musicals and other theatrical productions. The mean annual household income for the population of Playbill readers is $$\$ 119,155$$ (Playbill, January 2006). Assume the standard deviation is $$\sigma=\$ 20,700$$. A San Francisco civic group has asserted that the mean for theatergoers in the Bay Area is higher. A sample of 60 theater attendees in the Bay Area showed a sample mean bousehold income of $$\$ 126,100$$.
a. Develop hypotheses that can be used to determine whether the sample data support the conclusion that thearer attendees in the Bay Area have a higher mean bousebold income than that for all Playbill readers.
b. What is the $p$-value based on the sample of 60 theater attendees in the Bay Area?
c. Use $\alpha-.01$ as the level of significance. What is your conclusion?

Adriano Chikande
Adriano Chikande
Numerade Educator
03:24

Problem 63

On Friday, Wall Street traders were anxiously awaiting the federal govemment's release of numbers on the January increase in nonfarm payrolls. The esrly consensus estimate among economists was for a growth of 250,000 new jobs (CNBC. February 3, 2006). However, a sample of 20 economists taken Thursday afternoon provided a sample mean of 266,000 with a sample standard deviation of 24,000 . Financial analysts often call such a sample mean, based on late-breaking news, the whisper number. Treat the "consensus estimate" as the population mean. Conduct a hypothesis test to determine whether the whisper number justifies a conclusion of a statistically significant increase in the consensus estimate of economists. Use $a=.01$ as the level of significance.

Robin Corrigan
Robin Corrigan
Numerade Educator
04:37

Problem 64

Data released by the National Center for Health Statistics showed that the mean age at which women had their first child was 25.0 in 2006 (The Wall Street Joumal, February 4 , 2009 ). The reporter, Sue Shellenbarger, noted that this was the first decrease in the average age at which women had their first child in several years. A recent sample of 42 women provided the data in the website file named FirstBirth concerning the age at which these women had their first child. Do the data indicate a change from 2006 in the mean age at which women had their first child? Use $a=.05$.

Srikar Katta
Srikar Katta
Numerade Educator
02:00

Problem 65

An extensive study of the cost of health care in the United States presented data showing that the mean spending per Medicare enrollee in 2003 was $$\$ 6883$$ (Money, Fall 2003). To investigate differences across the country, a researcher took a sample of 40 Medicare enrollees in Indianapolis. For the Indianapolis sample, the mean 2003 Medicare spending was $$\$ 5980$$ and the standard deviation was $$\$ 2518$$.
a. State the hypotheses that should be used if we would like to determine whether the mean annual Medicare spending in Indianapolis is lower than the national mean.
b. Use the preceding sample results to compute the test statistic and the $p$-value.
c. Use $a=.05$. What is your conclusion?
d. Repeat the hypothesis test using the critical value approach.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:54

Problem 66

The chamber of commerce of a Florida Gulf Coast community advertises that area residenthal property is available at a mean cost of $$\$ 125,000$$ or less per lot. Suppose a sample of 32 properties provided a sample mean of $$\$ 130,000$$ per lot and a sample standard deviation of $$\$ 12,500$$. Use a .05 level of significance to test the validity of the advertising claim.

Kari Hasz
Kari Hasz
Numerade Educator
View

Problem 67

The U.S. Energy Administration reported that the mean price for a gallon of regular gasoline in the United States was $$\$2.357$$ (U.S. Energy Administration, January 30, 2006). Data for a sample of regular gasoline prices at 50 service stations in the Lower Atlantic states are contained in the data file named Gasoline. Conduct a bypothesis test to determine whether the mean price for a gallon of gasoline in the Lower Adlantic states is different from the national mean. Use $\alpha=.05$ for the level of significance, and state your conclusion.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:57

Problem 68

A study by the Centers for Disease Control (CDC) found that $23.3 \%$ of adults are smokers and that roughly $70 \%$ of those who do smoke indicate that they want to quit (Associated Press, July 26, 2002). CDC reported that, of people who smoked at some point in their lives, $50 \%$ have been able to kick the habit. Part of the study suggested that the success rate for quitting rose by education level. Assume that a sample of 100 college graduates who smoked at some point in their lives showed that 64 had been able to successfully stop smoking.
a. State the hypotheses that can be used to determine whether the population of college graduates has a success rate higher than the overall population when it comes to breaking the smoking habit.
b. Given the sample data, what is the proportion of college graduates who, having smoked at some point in their lives, were able to stop smoking?
c. What is the p-value? At $a=.01$, what is your hypothesis testing conclusion?

Adriano Chikande
Adriano Chikande
Numerade Educator
01:41

Problem 69

An airline promotion to business travelers is based on the assumption that two-thirds of business travelers use a laptop computer on overnight business trips.
a. State the hypotheses that can be used to test the assumption.
b. What is the sample proportion from an American Express sponsored survey that found 355 of 546 business travelers use a laptop computer on ovemight business trips?
c. What is the p-value?
d. Use $a=.05$. What is your conclusion?

Kari Hasz
Kari Hasz
Numerade Educator
02:08

Problem 70

Virtual call centers are staffed by individuals working out of their homes. Most home agents earn $$\$ 10$$ to $$\$ 15$$ per hour without benefits versus $$\$ 7$$ to $$\$ 9$$ per hour with benefits at a traditional call center (BursinessWeek, January 23, 2006). Regional Airways is considering employing home agents, but only if a level of customer satisfaction greater than $80 \%$ can be maintained. A test was conducted with home service agents. In a sample of 300 customers, 252 reported that they were satisfied with service.
a. Develop hypotheses for a test to determine whether the sample data support the conclusion that customer service with home agents meets the Regional Airways criterion.
b. What is your point estimate of the percentage of satisfied customers?
c. What is the p-value provided by the sample data?
d. What is your hypothesis testing conclusion? Use $\alpha-.05$ as the level of significance.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:03

Problem 71

During the 2004 election year, new polling results were reported daily. In an IBDVTIPP poll of 910 adults, 503 respondents reported that they were optimistic about the national outlook, and President Bush's leadership index jumped 4.7 points to 55.3 (Invesfor s Bussiness Daily, January 14, 2004).
a. What is the sample proportion of respondents who are optimistic about the national outlook?
b. A campaign manager wants to claim that this poll indicates that the majority of adults are optimistic about the national outlook. Construct a hypothesis test so that rejection of the null hypothesis will permit the conclusion that the proportion optimistic is greater than $50 \%$.
c. Use the polling data to compute the $p$-value for the hypothesis test in part (b). Explain to the manager what this $p$-value means about the level of significance of the results.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:50

Problem 72

A radio station in Myrtle Beach announced that at least $90 \%$ of the hotels and motels would be full for the Memorial Day weekend. The station advised listeners to make reservations in advance if they planned to be in the resort over the weekend. On Saturday night a sample of 58 hotels and motels showed 49 with a no-vacancy sign and 9 with vacancies. What is your reaction to the radio station's claim after seeing the sample evidence? Use $a=.05$ in making the statistical test. What is the $p$-value?

Adriano Chikande
Adriano Chikande
Numerade Educator
02:15

Problem 73

According to the federal government, $24 \%$ of workers covered by their company's health care plan were not required to contribute to the premium (Statistical Abstract of the United States: 2006). A recent study found that 81 out of 400 workers sampled were not required to contribute to their company's health care plan.
a. Develop hypotheses that can be used to test whether the percent of workers not required to contribute to their company's bealth care plan has declined.
b. What is a point estimate of the proportion receiving free company-sponsored health care insurance?
c. Has a statistically significant decline occurred in the proportion of workers receiving free company-sponsored health care insurance? Use $a-.05$.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:50

Problem 74

Shorney Construction Company bids on projects assuming that the mean idle time per worker is 72 or fewer minutes per day. A sample of 30 construction workers will be used to test this assumption. Assume that the population standard deviation is 20 minutes.
a. State the hypotheses to be tested.
b. What is the probability of making a Type II error when the population mean idle time is 80 minutes?
c. What is the probability of making a Type II error when the population mean idle time is 75 minutes?
d. What is the probability of making a Type II error when the population mean idle time is 70 minutes?
e. Sketch the power curve for this problem.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:14

Problem 75

A federal funding program is available to low-income neighborhoods. To qualify for the funding, a neighborhood must have a mean household income of less than $$\$ 15,000$$ per year. Neighborhoods with mean annual household income of $$\$ 15,000$$ or more do not qualify. Funding decisions are based on a sample of residents in the neighborhood. A hypochesis test with a .02 level of significance is conducted. If the funding guidelines call for a maximum probability of .05 of not funding a neighborhood with a mean annual housebold income of $$\$ 14,000$$, what sample size should be used in the funding decision study? Use $$\sigma-\$ 4000$$ as a planning value.

Adriano Chikande
Adriano Chikande
Numerade Educator
03:27

Problem 76

$H_0: \mu=120$ and $H_2: \mu+120$ are used to test whether a bath sosp production process is meeting the standard output of 120 bars per batch. Use a .05 level of significance for the test and a planning value of 5 for the standard deviation.
a. If the mean output drops to 117 bars per batch, the firm wants to have a $98 \%$ chance of concluding that the standard production output is not being met. How large a sample should be selected?
b. With your sample size from part (a), what is the probability of concluding that the process is operating satisfactorily for each of the following actual mean outputs: $117,118,119,121,122$, and 123 bars per batch? That is, what is the probability of a Type II error in each case?

Adriano Chikande
Adriano Chikande
Numerade Educator