Basic Statistics for Business & Economics

Douglas A. Lind, William G.Marchal, Samuel A. Wathen

Chapter 10

One-Sample Tests of Hypothesis - all with Video Answers

Educators

Chapter Questions

Problem 1

For Exercises $1-4,$ answer the questions: (a) Is this a one- or two-tailed test? (b) What is the decision rule? (c) What is the value of the test statistic? (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it.
A sample of 36 observations is selected from a normal population. The sample mean is $49,$ and the population standard deviation is $5 .$ Conduct the following test of hypothesis using the .05 significance level.
$$
\begin{array}{l}
H_{0}: \mu=50 \\
H_{1}: \mu \neq 50
\end{array}
$$

Jon Southam

Jon Southam

Numerade Educator

Problem 2

Answer the questions: (a) Is this a one- or two-tailed test? (b) What is the decision rule? (c) What is the value of the test statistic? (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it.
A sample of 36 observations is selected from a normal population. The sample mean is $12,$ and the population standard deviation is $3 .$ Conduct the following test of hypothesis using the .01 significance level.
$$
\begin{array}{l}
H_{0}: \mu \leq 10 \\
H_{1}: \mu>10
\end{array}
$$

Jon Southam

Numerade Educator

Problem 3

Answer the questions: (a) Is this a one- or two-tailed test? (b) What is the decision rule? (c) What is the value of the test statistic? (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it.
A sample of 36 observations is selected from a normal population. The sample mean is 21 , and the population standard deviation is 5 . Conduct the following test of hypothesis using the .05 significance level.
$$
\begin{array}{l}
H_{0}: \mu \leq 20 \\
H_{1}: \mu>20
\end{array}
$$

Jon Southam

Numerade Educator

Problem 4

Answer the questions: (a) Is this a one- or two-tailed test? (b) What is the decision rule? (c) What is the value of the test statistic? (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it.
A sample of 64 observations is selected from a normal population. The sample mean is $215,$ and the population standard deviation is $15 .$ Conduct the following test of hypothesis using the .025 significance level.
$$
\begin{array}{l}
H_{0}: \mu \geq 220 \\
H_{1}: \mu<220
\end{array}
$$

Jon Southam

Numerade Educator

Problem 5

For Exercises 5-8: (a) State the null hypothesis and the alternate hypothesis. (b) State the decision rule. (c) Compute the value of the test statistic. (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it.
The manufacturer of the $X-15$ steel-belted radial truck tire claims that the mean mileage the tire can be driven before the tread wears out is 60,000 miles. Assume the mileage wear follows the normal distribution and the standard deviation of the distribution is 5,000 miles. Crosset Truck Company bought 48 tires and found that the mean mileage for its trucks is 59,500 miles. Is Crosset's experience different from that claimed by the manufacturer at the .05 significance level?

Jon Southam

Numerade Educator

Problem 6

(a) State the null hypothesis and the alternate hypothesis. (b) State the decision rule. (c) Compute the value of the test statistic. (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it.
The waiting time for customers at MacBurger Restaurants follows a normal distribution with a population standard deviation of 1 minute. At the Warren Road MacBurger, the quality assurance department sampled 50 customers and found that the mean waiting time was 2.75 minutes. At the . 05 significance level, can we conclude that the mean waiting time is less than 3 minutes?

Jon Southam

Numerade Educator

Problem 7

(a) State the null hypothesis and the alternate hypothesis. (b) State the decision rule. (c) Compute the value of the test statistic. (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it.
A recent national survey found that high school students watched an average (mean) of 6.8 movies per month with a population standard deviation of 1.8 . The distribution of number of movies watched per month follows the normal distribution. A random sample of 36 college students revealed that the mean number of movies watched last month was 6.2 . At the .05 significance level, can we conclude that college students watch fewer movies a month than high school students?

Jon Southam

Numerade Educator

Problem 8

(a) State the null hypothesis and the alternate hypothesis. (b) State the decision rule. (c) Compute the value of the test statistic. (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it.
At the time she was hired as a server at the Grumney Family Restaurant, Beth Brigden was told, "You can average $\$ 80$ a day in tips." Assume the population of daily tips is normally distributed with a standard deviation of $\$ 9.95 .$ Over the first 35 days she was employed at the restaurant, the mean daily amount of her tips was \$84.85. At the .01 significance level, can Ms. Brigden conclude that her daily tips average more than $\$ 80 ?$

Jon Southam

Numerade Educator

Problem 10

Given the following hypotheses:
$$\begin{array}{l}H_{0}: \mu=400 \\H_{1}: \mu \neq 400\end{array}$$
A random sample of 12 observations is selected from a normal population. The sample mean was 407 and the sample standard deviation was 6 . Using the .01 significance level:
a. State the decision rule.
b. Compute the value of the test statistic.
c. What is your decision regarding the null hypothesis?

Jon Southam

Numerade Educator

Problem 11

The Rocky Mountain district sales manager of Rath Publishing Inc., a college textbook publishing company, claims that the sales representatives make an average of 40 sales calls per week on professors. Several reps say that this estimate is too low. To investigate, a random sample of 28 sales representatives reveals that the mean number of calls made last week was $42 .$ The standard deviation of the sample is 2.1 calls. Using the .05 significance level, can we conclude that the mean number of calls per salesperson per week is more than $40 ?$

Jon Southam

Numerade Educator

Problem 12

The management of White Industries is considering a new method of assembling its golf cart. The present method requires a mean time of 42.3 minutes to assemble a cart. The mean assembly time for a random sample of 24 carts, using the new method, was 40.6 minutes, and the standard deviation of the sample was 2.7 minutes. Using the .10 level of significance, can we conclude that the assembly time using the new method is faster?

Jon Southam

Numerade Educator

Problem 13

The mean income per person in the United States is $\$ 50,000,$ and the distribution of incomes follows a normal distribution. A random sample of 10 residents of Wilmington, Delaware, had a mean of $\$ 60,000$ with a standard deviation of $\$ 10,000 .$ At the .05 level of significance, is that enough evidence to conclude that residents of Wilmington, Delaware, have more income than the national average?

Jon Southam

Numerade Educator

Problem 14

Most air travelers now use e-tickets. Electronic ticketing allows passengers to not worry about a paper ticket, and it costs the airline companies less than paper ticketing. However, recently the airlines have received complaints from passengers regarding their e-tickets, particularly when connecting flights and a change of airlines were involved. To investigate the problem, an independent watchdog agency contacted a random sample of 20 airports and collected information on the number of complaints the airport had with e-tickets for the month of March. The information is reported below.
$$\begin{array}{|llllllllll|}\hline 14 & 14 & 16 & 12 & 12 & 14 & 13 & 16 & 15 & 14 \\12 & 15 & 15 & 14 & 13 & 13 & 12 & 13 & 10 &13 \\\hline\end{array}$$
At the .05 significance level, can the watchdog agency conclude the mean number of complaints per airport is less than 15 per month?
a. What assumption is necessary before conducting a test of hypothesis?
b. Plot the number of complaints per airport in a frequency distribution or a dot plot. Is it reasonable to conclude that the population follows a normal distribution?
c. Conduct a test of hypothesis and interpret the results.

Jon Southam

Numerade Educator

Problem 15

Given the following hypotheses:
$$\begin{array}{l}H_{0}: \mu \geq 20 \\H_{1}: \mu>20\end{array}$$
A random sample of five resulted in the following values: $18,15,12,19,$ and $21 .$ Assume a normal population. Using the . 01 significance level, can we conclude the population mean is less than $20 ?$
a. State the decision rule.
b. Compute the value of the test statistic.
c. What is your decision regarding the null hypothesis?
d. Estimate the $p$ -value.

Jon Southam

Numerade Educator

Problem 16

Given the following hypotheses:
$$\begin{array}{l}H_{0}: \mu=100 \\H_{1}: \mu \neq 100\end{array}$$
A random sample of six resulted in the following values: $118,105,112,119,105,$ and 111 . Assume a normal population. Using the .05 significance level, can we conclude the mean is different from $100 ?$
a. State the decision rule.
b. Compute the value of the test statistic.
c. What is your decision regarding the null hypothesis?
d. Estimate the $p$ -value.

Jon Southam

Numerade Educator

Problem 17

The amount of water consumed each day by a healthy adult follows a normal distribution with a mean of 1.4 liters. A health campaign promotes the consumption of at least 2.0 liters per day. A sample of 10 adults after the campaign shows the following consumption in liters:
$$\begin{array}{llllllll}1.5 & 1.6 & 1.5 & 1.4 & 1.9 & 1.4 & 1.3 & 1.9 & 1.8 & 1.7\end{array}$$
At the .01 significance level, can we conclude that water consumption has increased? Calculate and interpret the $p$ -value.

Jon Southam

Numerade Educator

Problem 18

The liquid chlorine added to swimming pools to combat algae has a relatively short shelf life before it loses its effectiveness. Records indicate that the mean shelf life of a 5 -gallon jug of chlorine is 2,160 hours $(90$ days $)$. As an experiment, Holdlonger was added to the chlorine to find whether it would increase the shelf life. A sample of nine jugs of chlorine had these shelf lives (in hours):
$\begin{array}{lllllll}2,159 & 2,170 & 2,180 & 2,179 & 2,160 & 2,167 & 2,171 & 2,181 & 2,185\end{array}$
At the .025 level, has Holdlonger increased the shelf life of the chlorine? Estimate the $p$ -value.

Jon Southam

Numerade Educator

Problem 19

A Washington, D.C., "think tank" announces the typical teenager sent 67 text messages per day in 2017 . To update that estimate, you phone a sample of 12 teenagers and ask them how many text messages they sent the previous day. Their responses were:
$\begin{array}{llllllllll}51 & 175 & 47 & 49 & 44 & 54 & 145 & 203 & 21 & 59 & 42 & 100\end{array}$
At the .05 level, can you conclude that the mean number is greater than $67 ?$ Estimate the $p$ -value and describe what it tells you.

Jon Southam

Numerade Educator

Problem 20

Hugger Polls contends that an agent conducts a mean of 53 in-depth home surveys every week. A streamlined survey form has been introduced, and Hugger wants to evaluate its effectiveness. The number of in-depth surveys conducted during a week by a random sample of 15 agents are:
$\begin{array}{lllllllllll}53 & 57 & 50 & 55 & 58 & 54 & 60 & 52 & 59 & 62 & 60 & 60 & 51 & 59 & 56\end{array}$
At the .05 level of significance, can we conclude that the mean number of interviews conducted by the agents is more than 53 per week? Estimate the $p$ -value.

Jon Southam

Numerade Educator

Problem 21

According to the local union president, the mean gross income of plumbers in the Salt Lake City area follows the normal probability distribution with a mean of $\$ 45,000$ and a standard deviation of $\$ 3,000 .$ A recent investigative reporter for KYAK TV found, for a sample of 120 plumbers, the mean gross income was $\$ 45,500$. At the .10 significance level, is it reasonable to conclude that the mean income is not equal to $\$ 45,000 ?$ Determine the $p$ -value.

Jon Southam

Numerade Educator

Problem 22

Rutter Nursery Company packages its pine bark mulch in 50 -pound bags. From a long history, the production department reports that the distribution of the bag weights follows the normal distribution, and the standard deviation of the packaging process is 3 pounds per bag. At the end of each day, Jeff Rutter, the production manager, weighs 10 bags and computes the mean weight of the sample. Below are the weights of 10 bags from today's production.
$\begin{array}{lllllllll}45.6 & 47.7 & 47.6 & 46.3 & 46.2 & 47.4 & 49.2 & 55.8 & 47.5 & 48.5\end{array}$
a. Can Mr. Rutter conclude that the mean weight of the bags is less than 50 pounds? Use the .01 significance level.
b. In a brief report, tell why Mr. Rutter can use the $z$ distribution as the test statistic.
c. Compute the $p$ -value.

Jon Southam

Numerade Educator

Problem 23

A new weight-watching company, Weight Reducers International, advertises that those who join will lose an average of 10 pounds after the first two weeks. The standard deviation is 2.8 pounds. A random sample of 50 people who joined the weight reduction program revealed a mean loss of 9 pounds. At the .05 level of significance, can we conclude that those joining Weight Reducers will lose less than 10 pounds? Determine the $p$ -value.

Jon Southam

Numerade Educator

Problem 24

Dole Pineapple Inc. is concerned that the 16 -ounce can of sliced pineapple is being overfilled. Assume the standard deviation of the process is .03 ounce. The quality control department took a random sample of 50 cans and found that the arithmetic mean weight was 16.05 ounces. At the $5 \%$ level of significance, can we conclude that the mean weight is greater than 16 ounces? Determine the $p$ -value.

Jon Southam

Numerade Educator

Problem 25

According to a recent survey, Americans get a mean of 7 hours of sleep per night. A random sample of 50 students at West Virginia University revealed the mean length of time slept last night was 6 hours and 48 minutes $(6.8$ hours $)$. The standard deviation of the sample was 0.9 hour. At the $5 \%$ level of significance, is it reasonable to conclude that students at West Virginia sleep less than the typical American? Compute the $p$ -value.

Jon Southam

Numerade Educator

Problem 26

A statewide real estate sales agency, Farm Associates, specializes in selling farm property in the state of Nebraska. Its records indicate that the mean selling time of farm property is 90 days. Because of recent drought conditions, the agency believes that the mean selling time is now greater than 90 days. A statewide survey of 100 recently sold farms revealed a mean selling time of 94 days, with a standard deviation of 22 days. At the . 10 significance level, has there been an increase in selling time?

Jon Southam

Numerade Educator

Problem 27

According to the Census Bureau, 3.13 people reside in the typical American household. A sample of 25 households in Arizona retirement communities showed the mean number of residents per household was 2.86 residents. The standard deviation of this sample was 1.20 residents. At the .05 significance level, is it reasonable to conclude the mean number of residents in the retirement community household is less than 3.13 persons?

Jon Southam

Numerade Educator

Problem 28

A recent article in Vitality magazine reported that the mean amount of leisure time per week for American men is 40.0 hours. You believe this figure is too large and decide to conduct your own test. In a random sample of 60 men, you find that the mean is 37.8 hours of leisure per week and that the standard deviation of the sample is 12.2 hours. Can you conclude that the information in the article is untrue? Use the .05 significance level. Determine the $p$ -value and explain its meaning.

Jon Southam

Numerade Educator

Problem 29

A recent survey by nerdwallet.com indicated Americans paid a mean of $\$ 6,658$ interest on credit card debt in 2017 . A sample of 12 households with children revealed the following amounts. At the .05 significance level, is it reasonable to conclude that these households paid more interest?
$\begin{array}{lllllllll}7,077 & 5,744 & 6,753 & 7,381 & 7,625 & 6,636 & 7,164 & 7,348 & 8,060 & 5,848 & 9,275 & 7,052\end{array}$

Jon Southam

Numerade Educator

Problem 30

A recent article in The Wall Street Journal reported that the home equity loan rate is now less than $4 \% .$ A sample of eight small banks in the Midwest revealed the following home equity loan rates (in percent):
$\begin{array}{lllllll}3.6 & 4.1 & 5.3 & 3.6 & 4.9 & 4.6 & 5.0 & 4.4\end{array}$
At the .01 significance level, can we conclude that the home equity loan rate for small banks is less than $4 \%$ ? Estimate the $p$ -value.

Jon Southam

Numerade Educator

Problem 31

A recent study revealed the typical American coffee drinker consumes an average of 3.1 cups per day. A sample of 12 senior citizens revealed they consumed the following amounts of coffee, reported in cups, yesterday.
$\begin{array}{lllllllllll}3.1 & 3.3 & 3.5 & 2.6 & 2.6 & 4.3 & 4.4 & 3.8 & 3.1 & 4.1 & 3.1 & 3.2\end{array}$
At the .05 significance level, do these sample data suggest there is a difference between the national average and the sample mean from senior citizens?

Jon Southam

Numerade Educator

Problem 32

The postanesthesia care area (recovery room) at St. Luke's Hospital in Maumee, Ohio, was recently enlarged. The hope was that the change would increase the mean number of patients served per day to more than $25 .$ A random sample of 15 days revealed the following numbers of patients.
$\begin{array}{llllllllll}25 & 27 & 25 & 26 & 25 & 28 & 28 & 27 & 24 & 26 & 25 & 29 & 25 & 27 & 24\end{array}$
At the .01 significance level, can we conclude that the mean number of patients per day is more than 25 ? Estimate the $p$ -value and interpret it.

Sheryl Ezze

Numerade Educator

Problem 33

www.golfsmith.com receives an average of 6.5 returns per day from online shoppers. For a sample of 12 days, it received the following numbers of returns.
$\begin{array}{lllllllllll}0 & 4 & 3 & 4 & 9 & 4 & 5 & 9 & 1 & 6 & 7 & 10\end{array}$
At the .01 significance level, can we conclude the mean number of returns is less than $6.5 ?$

Jon Southam

Numerade Educator

Problem 34

During recent seasons, Major League Baseball has been criticized for the length of the games. A report indicated that the average game lasts 3 hours and 30 minutes. A sample of 17 games revealed the following times to completion. (Note that the minutes have been changed to fractions of hours, so that a game that lasted 2 hours and 24 minutes is reported at 2.40 hours.)
$\begin{array}{lllllllll}2.98 & 2.40 & 2.70 & 2.25 & 3.23 & 3.17 & 2.93 & 3.18 & 2.80 \\ 2.38 & 3.75 & 3.20 & 3.27 & 2.52 & 2.58 & 4.45 & 2.45 & \end{array}$
Can we conclude that the mean time for a game is less than 3.50 hours? Use the .05 significance level.

Jon Southam

Numerade Educator

Problem 35

Watch Corporation of Switzerland claims that its watches on average will neither gain nor lose time during a week. A sample of 18 watches provided the following gains (t) or losses (-) in seconds per week.
$\begin{array}{llllllll}-0.38 & -0.20 & -0.38 & -0.32 & +0.32 & -0.23 & +0.30 & +0.25 & -0.10 \\ -0.37 & -0.61 & -0.48 & -0.47 & -0.64 & -0.04 & -0.20 & -0.68 & +0.05\end{array}$
Is it reasonable to conclude that the mean gain or loss in time for the watches is 0 ? Use the .05 significance level. Estimate the $p$ -value.

Jon Southam

Numerade Educator

Problem 36

Listed below is the annual rate of return (reported in percent) for a sample of 12 taxable mutual funds.
$\begin{array}{llllllllll}4.63 & 4.15 & 4.76 & 4.70 & 4.65 & 4.52 & 4.70 & 5.06 & 4.42 & 4.51 & 4.24 & 4.52\end{array}$
Using the .05 significance level, is it reasonable to conclude that the mean rate of return is more than $4.50 \% ?$

Jon Southam

Numerade Educator

Problem 37

Many grocery stores and large retailers such as Kroger and Walmart have installed self-checkout systems so shoppers can scan their own items and cash out themselves. How do customers like this service and how often do they use it? Listed below is the number of customers using the service for a sample of 15 days at a Walmart location.
$\begin{array}{rrrrrrrrr}120 & 108 & 120 & 114 & 118 & 91 & 118 & 92 & 104 & 104 \\ 112 & 97 & 118 & 108 & 117 & & & & & \end{array}$
Is it reasonable to conclude that the mean number of customers using the self-checkout system is more than 100 per day? Use the .05 significance level.

Jon Southam

Numerade Educator

Problem 38

For a recent year, the mean fare to fly from Charlotte, North Carolina, to Chicago, Illinois, on a discount ticket was $\$ 267 .$ A random sample of 13 round-trip discount fares on this route last month shows:
$\begin{array}{llllllllll}\$ 321 & \$ 286 & \$ 290 & \$ 330 & \$ 310 & \$ 250 & \$ 270 & \$ 280 & \$ 299 & \$ 265 & \$ 291 & \$ 275 & \$ 281\end{array}$
At the .01 significance level, can we conclude that the mean fare has increased? What is the $p$ -value?

Jon Southam

Numerade Educator

Problem 39

The publisher of Celebrity Living claims that the mean sales for personality magazines that feature people such as Megan Fox or Jennifer Lawrence are 1.5 million copies per week. A sample of 10 comparable titles shows a mean weekly sales last week of 1.3 million copies with a standard deviation of 0.9 million copies. Do these data contradict the publisher's claim? Use the .01 significance level.

Jon Southam

Numerade Educator

Problem 40

A United Nations report shows the mean family income for Mexican migrants to the United States is $\$ 27,000$ per year. A FLOC (Farm Labor Organizing Committee) evaluation of 25 Mexican family units reveals the mean to be $\$ 30,000$ with a sample standard deviation of $\$ 10,000 .$ Does this information disagree with the United Nations report? Apply the .01 significance level.

Jon Southam

Numerade Educator

Problem 41

The number of "destination weddings" has skyrocketed in recent years. For example, many couples are opting to have their weddings in the Caribbean. A Caribbean vacation resort recently advertised in Bride Magazine that the cost of a Caribbean wedding was less than $\$ 30,000$. Listed below is a total cost in $\$ 000$ for a sample of eight Caribbean weddings.
$\begin{array}{lllllll}29.7 & 29.4 & 31.7 & 29.0 & 29.1 & 30.5 & 29.1 & 29.8\end{array}$
At the .05 significance level, is it reasonable to conclude the mean wedding cost is less than $\$ 30,000$ as advertised?

Jon Southam

Numerade Educator

Problem 42

The American Water Works Association reports that the per capita water use in a single-family home is 69 gallons per day. Legacy Ranch is a relatively new housing development. The builders installed more efficient water fixtures, such as low-flush toilets, and subsequently conducted a survey of the residences. Thirty-six owners responded, and the sample mean water use per day was 64 gallons with a standard deviation of 8.8 gallons per day. At the .10 level of significance, is that enough evidence to conclude that residents of Leqacy Ranch use less water on average?

Jon Southam

Numerade Educator

Problem 43

A cola-dispensing machine is set to dispense 9.00 ounces of cola per cup, with a standard deviation of 1.00 ounce. The manufacturer of the machine would like to set the control limit in such a way that, for samples of $36,5 \%$ of the sample means will be greater than the upper control limit, and $5 \%$ of the sample means will be less than the lower control limit.
a. At what value should the control limit be set?
b. If the population mean shifts to $8.6,$ what is the probability of detecting the change?
c. If the population mean shifts to $9.6,$ what is the probability of detecting the change?

Jon Southam

Numerade Educator

Problem 44

The North Valley Real Estate data report information on the homes sold last year.
a. Adam Marty recently joined North Valley Real Estate and was assigned 20 homes to market and show. When he was hired, North Valley assured him that the 20 homes would be fairly assigned to him. When he reviewed the selling prices of his assigned homes, he thought that the prices were much below the average of $\$ 357,000$. Adam was able to find the data of how the other agents in the firm were assigned to the homes. Use statistical inference to analyze the "fairness" of assigning homes to agents.

Jon Southam

Numerade Educator

Problem 45

Refer to the Baseball 2016 data, which report information on the 30 Major League Baseball teams for the 2016 season.
a. Conduct a test of hypothesis to determine whether the mean salary of the teams was different from $\$ 100.0$ million. Use the .05 significance level.
b. Using a $5 \%$ significance level, conduct a test of hypothesis to determine whether the mean attendance was more than 2,000,000 per team.

Jon Southam

Numerade Educator

Problem 46

Refer to the Lincolnville School District bus data.
a. Select the variable for the number of miles traveled last month. Conduct a hypothesis test to determine whether the mean miles traveled last month equals $10,000 .$ Use the .01 significance level. Find the $p$ -value and explain what it means.
b. A study of school bus fleets reports that the average maintenance cost per bus is $\$ 4,000$ per year. Using the maintenance cost variable, conduct a hypothesis test to determine whether the mean maintenance cost for Lincolnville's bus fleet is more than $\$ 4,000$ at the .05 significance level. Determine the $p$ -value and report the results.

Jon Southam

Numerade Educator