Chapter 9, Hypothesis Testing Video Solutions, Business statistics : communicating with numbers | Numerade (2024)

Explain why the following hypotheses are not constructed correctly.
a. $\mathrm{H}_0: \mu \leq 10 ; \mathrm{H}_A: \mu \geq 10$
b. $\mathrm{H}_0: \mu \neq 500 ; \mathrm{H}_A: \mu=500$
c. $\mathrm{H}_0: \mathrm{p} \leq 0.40 ; \mathrm{H}_A: \mathrm{p}>0.42$
d. $H_0: \bar{X} \leq 128 ; H_A: \bar{X}>128$

Which of the following statements are valid null and alternative hypotheses? If they are invalid hypotheses, explain why.
a. $H_0: \bar{X} \leq 210 ; H_A: \bar{X}>210$
b. $\mathrm{H}_0: \mu=120 ; \mathrm{H}_A: \mu \neq 120$
c. $\mathrm{H}_0: \mathrm{p} \leq 0.24 ; \mathrm{H}_A: \mathrm{p}>0.24$
d. $\mathrm{H}_0: \mu<252 ; \mathrm{H}_A: \mu>252$

Explain why the following statements are not correct.
a. "With my methodological approach, I can reduce the Type I error with the given sample information without changing the Type II error."
b. "I have already decided how much of the Type I error l am going to allow. A bigger sample will not change either the Type I or Type II error."
c. "I can reduce the Type II error by making it difficult to reject the null hypothesis."
d. "By making it easy to reject the null hypothesis, I am reducing the Type I error."

Which of the following statements are correct? Explain if incorrect.
a. "I accept the null hypothesis since sample evidence is not inconsistent with the null hypothesis."
b. "Since sample evidence cannot be supported by the null hypothesis, I reject the null hypothesis."
c. "I can establish a given claim if sample evidence is consistent with the null hypothesis."
d. "I cannot establish a given claim if the null hypothesis is not rejected."

Construct the null and the alternative hypotheses for the following tests:
a. Test if the mean weight of cereal in a cereal box differs from 18 ounces.
b. Test if the stock price increases on more than $60 \%$ of the trading days.
c. Test if Americans get an average of less than seven hours of sleep.

Define the consequences of Type I and Type II errors for each of the tests considered in the preceding question.

Construct the null and the alternative hypotheses for the following claims:
a. "I am going to get the majority of the votes to win this election."
b. "I suspect that your 10 -inch pizzas are, on average, less than 10 inches in size."
c. "I will have to fine the company since its tablets do not contain an average of $250 \mathrm{mg}$ of ibuprofen as advertised."

Discuss the consequences of Type I and Type II errors for each of the claims considered in the preceding question.

A polygraph (lie detector) is an instrument used to determine if an individual is telling the truth. These tests are considered to be $95 \%$ reliable. In other words, if an individual lies, there is a 0.95 probability that the test will detect a lie. Let there also be a 0.005 probability that the test erroneously detects a lie even when the individual is actually telling the truth. Consider the null hypothesis, "the individual is telling the truth," to answer the following questions.
a. What is the probability of a Type lerror?
b. What is the probability of a Type II error?
c. What are the consequences of Type I and Type II errors?
d. What is wrong with the statement, "I can prove that the individual is telling the truth on the basis of the polygraph result"?

The manager of a large manufacturing firm is considering switching to new and expensive software that promises to reduce its assembly costs. Before purchasing the software, the manager wants to conduct a hypothesis test to determine if the new software does reduce its assembly costs.
a. Would the manager of the manufacturing firm be more concerned about a Type lerror or a Type II error? Explain.
b. Would the software company be more concerned about a Type I error or a Type II error? Explain.

The screening process for detecting a rare disease is not perfect. Researchers have developed a blood test that is considered fairly reliable. It gives a positive reaction in $98 \%$ of the people who have that disease. However, it erroneously gives a positive reaction in $3 \%$ of the people who do not have the disease. Consider the null hypothesis "the individual does not have the disease" to answer the following questions.
a. What is the probability of a Type I error?
b. What is the probability of a Type II error?
c. What are the consequences of Type I and Type II errors?
d. What is wrong with the nurse's analysis, "The blood test result has proved that the individual is free of disease"?

A consumer group has accused a restaurant of using higher fat content than what is reported on its menu. The group has been asked to conduct a hypothesis test to substantiate its claims.
a. Is the manager of the restaurant more concemed about a Type I error or a Type II error? Explain.
b. Is the consumer group more concerned about a Type I error or a Type II error? Explain.

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: \mu \leq 12.6 \\
& H_A: \mu>12.6
\end{aligned}
$$
A sample of 25 observations yields a sample mean of 13.4. Assume that the sample is drawn from a normal population with a population standard deviation of 3.2 .
a. Calculate the $\mathrm{p}$-value. What is the conclusion if $\alpha=0.10$ ?
b. Calculate the $p$-value if the above sample mean was based on a sample of 100 observations. What is the conclusion if $\alpha=0.10$ ?

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: \mu=100 \\
& H_A: \mu \neq 100
\end{aligned}
$$
A sample of 16 observations yields a sample mean of 95 . Assume that the sample is drawn from a normal population with a population standard deviation of 10 .
a. Calculate the value of the test statistic.
b. Find the p-value.
c. At the $10 \%$ significance level, what is the conclusion?

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: \mu \geq 150 \\
& H_A: \mu<150
\end{aligned}
$$
A sample of 80 observations results in a sample mean of 144 . The population standard deviation is known to be 28 .
a. Calculate the value of the test statistic and the $p$-value.
b. Does the above sample evidence enable us to reject the null hypothesis at $\alpha=0.01$ ?
c. Does the above sample evidence enable us to reject the null hypothesis at $\alpha=0.05$ ?

A researcher wants to determine if the population mean is greater than 45 . A random sample of 36 observations yields a sample mean of 47 . Assume that the population standard deviation is 8 .
a. Specify the competing hypotheses to test the researcher's claim.
b. Calculate the value of the test statistic.
c. Find the $p$-value.
d. At the $5 \%$ significance level, what is the conclusion?

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: \mu=1,800 \\
& H_A: \mu \neq 1,800
\end{aligned}
$$
The population is normally distributed with a population standard deviation of 440 . Compute the value of the test statistic and the resulting $p$-value for each of the following sample results. For each sample, determine if you can reject the null hypothesis at the $10 \%$ significance level.
a. $\bar{X}=1,850 ; n=110$
b. $\bar{X}=1,850 ; n=280$
c. $\bar{X}=1,650 ; n=32$
d. $\bar{X}=1,700 ; n=32$

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu \leq-5 \\
& H_A: \mu>-5
\end{aligned}
$$
A random sample of 50 observations yields a sample mean of -3 . The population standard deviation is 10 . Calculate the $\mathrm{p}$-value. What is the conclusion to the test if $\alpha=0.05 ?$

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu \leq 75 \\
& H_A: \mu>75
\end{aligned}
$$
A random sample of 100 observations yields a sample mean of 80 . The population standard deviation is 30 . Calculate the $p$-value. What is the conclusion to the test if $a=0.10$ ?

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu=-100 \\
& H_{A^*} \cdot \mu \neq-100
\end{aligned}
$$
A random sample of 36 observations yields a sample mean of -125 . The population standard deviation is 42 . Conduct the test at $\alpha=0.01$.

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: \mu=120 \\
& H_A: \mu \neq 120
\end{aligned}
$$
The population is normally distributed with a population standard deviation of 46 .
a. If $\bar{X}=132$ and $n=50$, what is the conclusion at the $5 \%$ significance level?
b. If $\bar{X}=108$ and $n=50$, what is the conclusion at the $10 \%$ significance level?

It is advertised that the average braking distance for a small car traveling at 65 miles per hour equals 120 feet. A transportation researcher wants to determine if the statement made in the advertisem*nt is false. She randomly test drives 36 small cars at 65 miles per hour and records the braking distance. The sample average braking distance is computed as 114 feet. Assume that the population standard deviation is 22 feet.
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a. State the null and the alternative hypotheses for the test.
b. Calculate the value of the test statistic and the $p$-value.
c. Use $\alpha=0.01$ to determine if the average breaking distance differs from 120 feet.

Customers at Costco spend an average of $$\$ 130$$ per trip (The Wall Street Journal, October $6,2010)$. One of Costco's rivals would like to determine whether its customers spend more per trip. A survey of the receipts of 25 customers found that the sample mean was $$\$ 135.25$$. Assume that the population standard deviation is $$\$ 10.50$$ and that spending follows a normal
distribution.
a. Specify the null and alternative hypotheses to test whether average spending at the rival's store is more than $$\$ 130$$.
b. Calculate the value of the test statistic and the p-value.
c. At the $5 \%$ significance level, what is the conclusion to the test?

In May $2008, C N N$ reported that sports utility vehicles (SUVs) are plunging toward the "endangered" list. Due to the uncertainty of oil prices and environmental concerns, consumers are replacing gas-guzzling vehicles with fuel-efficient smaller cars. As a result, there has been a big drop in the demand for new as well as used SUVs. A sales manager of a used car dealership for SUVs believes that it takes more than 90 days, on average, to sell an SUV. In order to test his claim, he samples 40 recently sold SUVS and finds that it took an average of 95 days to sell an SUV. He believes that the population standard deviation is fairly stable at 20 days.
a. State the null and the alternative hypotheses for the test.
b. What is the p-value?
c. Is the sales manager's claim justified at $\alpha=0.01$ ?

According to the Centers for Disease Control and Prevention (February 18, 2016), 1 in 3 American adults don't get enough sleep. A researcher wants to determine if Americans are sleeping less than the recommended 7 hours of sleep on weekdays. He takes a random sample of 150 Americans and computes the average sleep time of 6.7 hours on weekdays. Assume that the population is normally distributed with a known standard deviation of 2.1 hours. Test the researcher's claim at $\alpha=0.01$.

A local bottler in Hawaii wishes to ensure that an average of 16 ounces of passion fruit juice is used to fill each bottle. In order to analyze the accuracy of the bottling process, he takes a random sample of 48 bottles. The mean weight of the passion fruit juice in the sample is 15.80 ounces. Assume that the population standard deviation is 0.8 ounce.
a. State the null and the alternative hypotheses to test if the bottling process is inaccurate.
b. What is the value of the test statistic and the $p$-value?
c. At $\alpha=0.05$, what is the conclusion to the hypothesis test? Make a recommendation to the bottler.

A realtor in Mission Viejo, California, believes that the average price of a house is more than $$\$ 500,000$$.
a. State the null and the altemative hypotheses for the test.
b. The data accompanying this exercise show house prices. (Data are in $$\$ 1,000$$ s.)
Assume the population standard deviation is $$\$ 100$$ (in $$\$ 1,000$$ s). What is the value of the test statistic and the $p$-value?
c. At $\alpha=0.05$, what is the conclusion to the test? is the realtor's claim supported by the data?

The data accompanying this exercise show the weekly stock price for Home Depot. Assume that stock prices are normally distributed with a population standard deviation of $$\$ 3$$.
a. State the null and the alternative hypotheses in order to test whether or not the average weekly stock price differs from $$\$ 30$$.
b. Find the value of the test statistic and the $p$-value.
c. At $\alpha=0.05$, can you conclude that the average weekly stock price does not equal $$\$ 30$$ ?

An economist wants to test if the average hourly wage is less than
$$\$ 22$$. Assume that the population standard deviation is $$\$ 6$$.
a. State the null and the alternative hypotheses for the test.
b. The data accompanying this exercise show hourly wages. Find the value of the test statistic and the $p$-value.
c. At $\alpha=0.05$, what is the conclusion to the test? is the average hourly wage less than $$\$ 22 ?$$

On average, a college student graduates with $$\$ 27,200$$ in debt (The Boston Globe, May 27, 2012). A researcher collects data on debt from 40 recent undergraduates from Connecticut. Assume that the population standard deviation is $$\$ 5,000$$.
a. The researcher believes that recent undergraduates from Connecticut have less debt than the national average. Specify the competing hypotheses to test this belief.
b. Find the value of the test statistic and the $p$-value.
c. Do the data support the researcher's claim, at $\alpha=0.10$ ?

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: \mu \leq 210 \\
& H_A: \mu>210
\end{aligned}
$$
Find the $p$-value for this test based on the following sample information.
a. $\bar{x}=216 ; s=26 ; n=40$
b. $\bar{x}=216 ; s=26 ; n=80$
c. $\bar{x}=216 ; s=16 ; n=40$
d. $\bar{X}=214 ; s=16 ; n=40$

Which of the sample information in the preceding question enables us to reject the null hypothesis at $\alpha=0.01$ and at $\alpha=0.10$ ?

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: \mu=12 \\
& H_A: \mu \neq 12
\end{aligned}
$$
Find the $p$-value for this test based on the following sample information.
a. $\bar{X}=11 ; s=3.2 ; n=36$
b. $\bar{x}=13 ; s=3.2 ; n=36$
c. $\bar{X}=11 ; s=2.8 ; n=36$
d. $\bar{X}=11 ; s=2.8 ; n=49$

Which of the sample information in the preceding question enables us to reject the null hypothesis at $\alpha=0.01$ and at $\alpha=0.10$ ?

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: \mu=50 \\
& H_A: \mu \neq 50
\end{aligned}
$$
A sample of 16 observations yields a sample mean of 46 . Assume that the sample is drawn from a normal population with a sample standard deviation of 10 .
a. Calculate the value of the test statistic.
b. At the $5 \%$ significance level, does the population mean differ from 50 ? Explain.

In order to test if the population mean differs from 16 , you draw a random sample of 32 observations and compute the sample mean and the sample standard deviation as 15.2 and 0.6 , respectively. Conduct the test at the $1 \%$ level of significance.

In order to conduct a hypothesis test for the population mean, a random sample of 24 observations is drawn from a normally distributed population. The resulting sample mean and sample standard deviation are calculated as 4.8 and 0.8 , respectively. Conduct the following tests at $\alpha=0.05$.
a. $\mathrm{H}_0: \mu \leq 4.5$ against $\mathrm{H}_A: \mu>4.5$
b. $\mathrm{H}_0: \mu=4.5$ against $\mathrm{H}_A: \mu \neq 4.5$
Page 343

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: \mu \geq-10 \\
& H_A: \mu<-10
\end{aligned}
$$
A sample of 25 observations yields a sample mean of -12 . Assume that the sample is drawn from a normal population with a sample standard deviation of 4 .
a. Calculate the value of the test statistic.
b. At the $5 \%$ significance level, is the population mean less than -10? Explain.

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: \mu=8 \\
& H_A: \mu \neq 8
\end{aligned}
$$
The population is normally distributed. A sample produces the following observations:
$$
\begin{array}{|l|l|l|l|l|l|l|}
\hline 6 & 9 & 8 & 7 & 7 & 11 & 10 \\
\hline
\end{array}
$$
Conduct the test at the $5 \%$ level of significance.

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: \mu \geq 100 \\
& H_A: \mu<100
\end{aligned}
$$
The population is normally distributed. A sample produces the following observations:
$$
\begin{array}{|l|l|l|l|l|l|}
\hline 95 & 99 & 85 & 80 & 98 & 97 \\
\hline
\end{array}
$$
Conduct the test at the $1 \%$ level of significance.

A machine that is programmed to package 1.20 pounds of cereal in each cereal box is being tested for its accuracy. In a sample of 36 cereal boxes, the mean and the standard deviation are calculated as 1.22 pounds and 0.06 pound, respectively.
a. Set up the null and the alternative hypotheses to determine if the machine is working improperly—-that is, it is either underfilling or overfilling the cereal boxes.
b. Calculate the value of the test statistic and the $p$-value.
c. At the $5 \%$ level of significance, can you conclude that the machine is working improperly? Explain.

The manager of a small convenience store does not want her customers standing in line for too long prior to a purchase. In particular, she is willing to hire an employee for another cash register if the average wait time of the customers is more than five minutes. She randomly observes the wait time (in minutes) of customers during the day as:
$$
\begin{array}{|l|l|l|l|l|l|l|}
\hline 3.5 & 5.8 & 7.2 & 1.9 & 6.8 & 8.1 & 5.4 \\
\hline
\end{array}
$$
a. Set up the null and the alternative hypotheses to determine if the manager needs to hire another employee.
b. Calculate the value of the test statistic and the $p$-value. What assumption regarding the population is necessary to implement this step?
c. Decide whether the manager needs to hire another employee at $\alpha=0.10$.

Small, energy-efficient, Internet-centric, new computers are increasingly gaining popularity (The New York Times, July 20, 2008). Some of the biggest companies are wary of the new breed of computers because their low price could threaten PC makers' already thin profit margins. An analyst comments that the larger companies have a cause for concern since the mean price of these small computers has fallen below $$\$ 350$$. She examines six popular brands of these small computers and records their retail prices as:
$$
\begin{array}{|l|l|l|l|l|l|}
\hline 322 & 269 & 373 & 412 & 299 & 389 \\
\hline
\end{array}
$$
a. What assumption regarding the distribution of the price of small computers is necessary to test the analyst's claim?
b. Specify the null and alternative hypotheses to test the analyst's claim.
c. Calculate the value of the test statistic and the $p$-value.
d. At the $5 \%$ significance level, what is the conclusion to the test? Should the larger computer companies be concerned?

A local brewery wishes to ensure that an average of 12 ounces of beer is used to fill each bottle. In order to analyze the accuracy of the bottling process, the bottler takes a random sample of 48 bottles. The sample mean weight and the sample standard deviation of the bottles are 11.80 ounces and 0.8 ounce, respectively.
a. State the null and the altemative hypotheses to test if the accuracy of the bottling process is compromised.
b. Do you need to make any assumption regarding the population for testing?
c. Calculate the value of the test statistic and the $p$-value.
d. At $\alpha=0.05$, what is the conclusion to the test? Make a recommendation to the bottler.
Based on the average predictions of 47 members of the National Association of Business

Economists (NABE), the U.S. gross domestic product (GDP) will expand by $3.2 \%$ in 2011 (The Wall Street Journal, May 23, 2010). Suppose the sample standard deviation of their predictions was $1 \%$. At the $5 \%$ significance level, test if the mean forecast GDP of all NABE members is greater than $3 \%$.

In September 2007, U.S. home prices fell at a record pace, and price declines in Los Angeles and Orange counties in California outpaced other major metropolitan areas (Los Angeles Times, November 28, 2007). The report was based on the Standard & Poor's/Case-Shiller index that measures the value of single-family homes based on their sales histories. According to this index, the prices in San Diego dropped by an average of $9.6 \%$ from a year earlier. Assume that the survey was based on recent sales of 34 houses in San Diego that also resulted in a standard deviation of $5.2 \%$. Can we conclude that the mean drop of all home prices in San Diego is greater than the $7\%$ drop in Los Page 344 Angeles? Use a $1 \%$ level of significance for the analysis.

A car manufacturer is trying to develop a new sports car. Engineers are hoping that the average amount of time that the car takes to go from 0 to 60 miles per hour is below 6 seconds. The manufacturer tested 12 of the cars and clocked their performance times. Three of the cars clocked in at 5.8 seconds, 5 cars at 5.9 seconds, 3 cars at 6.0 seconds, and 1 car at 6.1 seconds. At the $5 \%$ level of significance, test if the new sports car is meeting its goal to go from 0 to 60 miles per hour in less than 6 seconds. Assume a normal distribution for the analysis.

A mortgage specialist would like to analyze the average mortgage rates for Atlanta, Georgia. He collects data on the annual percentage rates (APR in %) for 30 -year fixed loans as shown in the following table. If he is willing to assume that these rates are randomly drawn from a normally distributed population, can he conclude that the mean mortgage rate for the population exceeds $4.2 \%$ ? Test the hypothesis at the $10 \%$ level of significance.
$$
\begin{array}{|l|l|}
\hline \text { Financial Institution } & \text { APR } \\
\hline \text { G Squared Financial } & 4.125 \\
\hline \text { Best Possible Mortgage } & 4.250 \\
\hline \text { Hersch Financial Group } & 4.250 \\
\hline \text { Total Mortgages Services } & 4.375 \\
\hline \text { Wells Fargo } & 4.375 \\
\hline \text { Quicken Loans } & 4.500 \\
\hline \text { Amerisave } & 4.750 \\
\hline
\end{array}
$$

One of the consequences of the Great Recession was a free fall of the stock market's average price/earnings ratio, or P/E ratio (The Wall Street Journal, August 30, 2010). Generally, a high P/E ratio suggests that investors are expecting higher earnings growth in the future compared to companies with a lower $P / E$ ratio. An analyst wants to determine if the $\mathrm{P} / \mathrm{E}$ ratio of firms in the footwear industry is different from the overall average of 14.9 .
The following table shows the P/E ratios for a sample of seven firms in the footwear industry.
$$
\begin{array}{|l|c|}
\hline \text { Firm } & \text { P/E Ratio } \\
\hline \text { Brown Shoe Co., Inc. } & 20.54 \\
\hline \text { Collective Brands, Inc. } & 9.33 \\
\hline \text { Crocs, Inc. } & 22.63 \\
\hline \text { DSW, Inc. } & 14.42 \\
\hline \text { Nike, Inc. } & 18.68 \\
\hline \text { Skechers USA, Inc. } & 9.35 \\
\hline \text { Timberland Co. } & 14.93 \\
\hline
\end{array}
$$
a. State the null and the altemative hypotheses in order to test whether the P/E ratio of firms in the footwear industry differs from the overall average of 14.9 .
b. What assumption regarding the population is necessary?
c. Calculate the value of the test statistic and the $p$-value.
d. At $a=0.10$, does the P/E ratio of firms in the footwear industry differ from the overall average of 14.9 ? Explain.

The data accompanying this exercise show miles per gallon (MPG).
a. State the null and the altemative hypotheses in order to test whether the average MPG differs from 95 .
b. Calculate the value of the test statistic and the $p$-value.
c. At $\alpha=0.05$, can you conclude that the average MPG differs from 95 ?

A study found that consumers are making average monthly debt payments of $$\$ 983$$ (Experian.com, November 11, 2010). The data accompanying this exercise show the average debt payments (Debt, in $$\$ $$) for 26 metropolitan areas, a portion of which is shown in the following table.
$$
\begin{array}{|l|c|}
\hline \text { City } & \text { Debt } \\
\hline \text { Washington, D.C. } & 1285 \\
\hline \text { Seattle } & 1135 \\
\hline \vdots & \vdots \\
\hline \text { Pittsburgh } & 763 \\
\hline
\end{array}
$$
a. State the null and the alternative hypotheses in order to test whether average monthly debt payments are greater than $$\$ 900$$.
b. What assumption regarding the population is necessary to implement this step?
c. Calculate the value of the test statistic and the $p$-value.
d. At $\alpha=0.05$, are average monthly debt payments greater than $$\$ 900$$ ? Explain.

A police officer is concerned about speeds on a certain section
of Interstate 95. The data accompanying this exercise show the speeds of 40 cars on a Saturday afternoon.
a. The speed limit on this portion of Interstate 95 is $65 \mathrm{mph}$. Specify the competing hypotheses in order to determine if the average speed is greater than the speed limit.
b. Calculate the value of the test statistic and the $p$-value.
c. At $\alpha=0.01$, are the officer's concerns warranted? Explain.

An article found that Massachusetts residents spent an average of $$\$ 860.70$$ on the lottery in 2010 , more than three times the U.S. average (www.businessweek.com, March 14, 2012). A researcher at a Boston think tank believes that Massachusetts residents spend less than this amount. He surveys 100 Massachusetts residents and asks them about their annual expenditures on the lottery.
a. Specify the competing hypotheses to test the researcher's claim.
b. Calculate the value of the test statistic and the $p$-value.
c. At the $10 \%$ significance level, do the data support the researcher's claim? Explain.

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: p \geq 0.38 \\
& H_A: p<0.38
\end{aligned}
$$
Calculate the $p$-value based on the following sample information.
a. $x=22 ; n=74$
b. $x=110 ; n=300$
c. $\bar{p}=0.34 ; \mathrm{n}=50$
d. $\bar{p}=0.34 ; n=400$

Which sample information in the preceding question enables us to reject the null hypothesis at $\alpha=0.01$ and at $\alpha=0.10$ ?

Consider the following hypotheses:
$$
\begin{aligned}
& H_0: p=0.32 \\
& H_A: p \neq 0.32
\end{aligned}
$$
Calculate the $p$-value based on the following sample information
a. $x=20 ; n=66$
b. $x=100 ; n=264$
c. $\bar{\rho}=0.40 ; n=40$
d. $\bar{\rho}=0.38 ; \mathrm{n}=180$

Which sample information in the preceding question enables us to reject the null hypothesis at $\alpha=0.05$ and at $\alpha=0.10$ ?

In order to test if the population proportion differs from 0.40 , you draw a random sample of 100 observations and obtain a sample proportion of 0.48 .
a. Specify the competing hypotheses.
b. Is the normality condition satisfied? Explain.
c. Calculate the value of the test statistic and the $p$-value.
d. At the $5 \%$ significance level, does the population proportion differ from 0.40 ? Explain.

In order to conduct a hypothesis test for the population proportion, you sample 320 observations that result in 128 successes. Conduct the following tests at $\alpha=0.05$.
a. $\mathrm{H}_0: \mathrm{p} \geq 0.45 ; \mathrm{H}_A: \mathrm{p}<0.45$
b. $\mathrm{H}_0: \mathrm{p}=0.45 ; \mathrm{H}_A: \mathrm{p} \neq 0.45$

In order to test if the population proportion is greater than 0.65 , you draw a random sample of 200 observations and obtain a sample proportion of 0.72 .
a. Specify the competing hypotheses.
b. Is the normality condition satisfied? Explain.
c. Calculate the value of the test statistic and the $p$-value.
d. At the $5 \%$ significance level, is the population proportion greater than 0.65 ? Explain.

You would like to determine if the population probability of success differs from 0.70 . You find 62 successes in 80 binomial trials. Implement the test at the $1 \%$ level of significance.

You would like to determine if more than $50 \%$ of the observations in a population are below 10. At $\alpha=0.05$, conduct the test on the basis of the following 20 sample observations:
$$
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline 8 & 12 & 5 & 9 & 14 & 11 & 9 & 3 & 7 & 8 \\
\hline 12 & 6 & 8 & 9 & 2 & 6 & 11 & 4 & 13 & 10 \\
\hline
\end{array}
$$

A study by Allstate Insurance Co. finds that $82 \%$ of teenagers have used cell phones while driving (The Wall Street Journal, May 5, 2010). In October 2010, Massachusetts enacted a
law that forbids cell phone use by drivers under the age of 18 . A policy analyst would like to determine whether the law has decreased the proportion of drivers under the age of 18 who use a cell phone.
a. State the null and the alternative hypotheses to test the policy analyst's objective.
b. Suppose a sample of 200 drivers under the age of 18 results in 150 who still use a cell phone while driving. What is the value of the test statistic? What is the p-value?
c. At $\alpha=0.05$, has the law been effective?

In order to endure financial hardships such as unemployment and medical emergencies, Americans have increasingly been raiding their already fragile retirement accounts (MSN Money, July 16,2008 ). It is reported that between 1998 and 2004 , about $12 \%$ of families with $401(\mathrm{k})$ plans borrowed from them. An economist is concerned that this percentage now exceeds $20 \%$. He randomly surveys 190 households with $401(\mathrm{k})$ plans and finds that 50 are borrowing against them.
a. Set up the null and the alternative hypotheses to test the economist's concern.
b. Calculate the value of the test statistic and the $p$-value.
c. Determine if the economist's concern is justifiable at $a=0.05$.

The margarita is one of the most common tequila-based co*cktails, made with tequila mixed with triple sec and lime or lemon juice, often served with salt on the glass rim. A common ratio for a margarita is $2: 1: 1$, which includes $50 \%$ tequila, $25 \%$ triple sec, and $25 \%$ fresh lime or lemon juice. A manager at a local bar is concerned that the bartender uses incorrect proportions in more than $50 \%$ of margaritas. He secretly observes the bartender and finds that he used the correct proportions in only 10 out of 30 margaritas. Test if the manager's suspicion is justified at $\alpha=0.05$.

Research shows that many banks are unwittingly training their online customers to take risks with their passwords and other sensitive account information, leaving them more vulnerable to fraud (Yahoo.com, July 23, 2008). Even web-sawy surfers could find themselves the victims of identity theft because they have been conditioned to ignore potential signs about whether the banking site they are visiting is real or a bogus site served up by hackers. Researchers at the University of Michigan found design flaws in $78 \%$ of the 214 U.S. financial institution websites they studied. Is the sample evidence sufficient to conclude that more than three out of four financial institutions that offer online banking facilities are prone to fraud? Use a $5 \%$ significance level for the test.

Research commissioned by Vodafone suggests that older workers are the happiest employees (BBC News, July 21, 2008). The report documents that $70 \%$ of older workers in England feel fulfilled, compared with just $50 \%$ of younger workers. A demographer believes that an identical pattern does not exist in Asia. A survey of 120 older workers in Asia finds that 75 feel fulfilled. A similar survey finds that $58 \%$ of 210 younger workers feel fulfilled.
a. At the $5 \%$ level of significance, test if older workers in Asia feel less fulfilled than their British counterparts.
b. At the $5 \%$ level of significance, test if younger workers in Asia feel more fulfilled than their British counterparts.

A politician claims that he is supported by a clear majority of voters. In a recent survey, 24 out
of 40 randomly selected voters indicated that they would vote for the politician. is the politician's claim justified at the $5 \%$ level of significance?

A movie production company is releasing a movie with the hopes of many viewers retuming to see the movie in the theater for a second time. Their target is to have 30 million viewers, and they want more than $30 \%$ of the viewers to want to see the movie again. They show the movie to a test audience of 200 people, and after the movie they asked them if they would see the movie in theaters again. Of the test audience, 68 people said they would see the movie again.
a. At the $5 \%$ level of significance, test if more than $30 \%$ of the viewers will retum to see the movie again.
b. Repeat the analysis at the $10 \%$ level of significance.
c. Interpret your results.

With increasing out-of-pocket healthcare costs, it is claimed that more than $60 \%$ of senior citizens are likely to make serious adjustments to their lifestyle. Test this claim at the $1 \%$ level of significance if in a survey of 140 senior citizens, 90 reported that they have made serious adjustments to their lifestyle.

According to a report on workforce diversity, about $60 \%$ of the employees in high-tech firms in Silicon Valley are white and about $20 \%$ are Asian (umw.moneycnn.com, November 9, 2011). Women, along with blacks and Latinos, are highly underrepresented. Just about $30 \%$ of all employees are women, with blacks and Latinos accounting for only about $15 \%$ of the workforce. Tara Jones is a recent college graduate, working for a large high-tech firm in Silicon Valley. She wants to determine if her firm faces the same diversity as in the report. She collects gender and ethnicity information on 50 employees in her firm. A portion of the data is shown in the accompanying table.
$$
\begin{array}{|l|l|}
\hline \text { Gender } & \text { Ethnicity } \\
\hline \text { Female } & \text { White } \\
\hline \text { Male } & \text { White } \\
\hline \vdots & \vdots \\
\hline \text { Male } & \text { Nonwhite } \\
\hline
\end{array}
$$
a. At the $5 \%$ level of significance, determine if the proportion of women in Tara's firm is different from 0.30 .
b. At the $5 \%$ level of significance, determine if the proportion of whites in Tara's firm is more than 0.50 .

A pharmaceutical company has developed a new drug for depression. There is a concern, however, that the drug also raises the blood pressure of its users. A researcher wants to conduct a test to validate this claim. Would the manager of the pharmaceutical company be more concerned about a Type I error or a Type II error? Explain.

A company has developed a new diet that it claims will lower one's weight by more than 10 pounds. Health officials decide to conduct a test to validate this claim.
a. Would the manager of the company be more concerned about a Type I error or a Type II error? Explain.
b. Would the consumers be more concerned about a Type I error or a Type II error? Explain.

An advertisem*nt for a popular weight loss clinic suggests that participants in its new diet program lose, on average, more than 10 pounds. A consumer activist decides to test the authenticity of the claim. She follows the progress of 18 women who recently joined the weight reduction program. She calculates the mean weight loss of these participants as 10.8 pounds with a standard deviation of 2.4 pounds.
a. Set up the competing hypotheses to test the advertisem*nt's claim.
b. Calculate the value of the test statistic and the $p$-value.
c. At the $5 \%$ significance level, what does the consumer activist conclude?

A phone manufacturer wants to compete in the touch screen phone market. He understands that the lead product has a battery life of just 5 hours. The manufacturer claims that while the new touch screen phone is more expensive, its battery life is more than twice as long as that of the leading product. In order to test the claim, a researcher samples 45 units of the new phone and finds that the sample battery life averages 10.5 hours with a sample standard deviation of 1.8 hours.
a. Set up the competing hypotheses to test the manufacturer's claim.
b. Calculate the value of the test statistic and the $p$-value.
c. Test the phone manufacturer's claim at $\alpha=0.05$.

A city council is deciding whether or not to spend additional money to reduce the amount of traffic. The council decides that it will increase the transportation budget if the amount of waiting time for drivers exceeds 20 minutes. A sample of 32 main roads results in a mean waiting time of 22.08 minutes with a standard deviation of 5.42 minutes. Conduct a hypothesis test at the $1 \%$ level of significance to determine whether or not the city should increase its transportation budget.

A city council is deciding whether or not to spend additional money to reduce the amount of traffic. The council decides that it will increase the transportation budget if the amount of waiting time for drivers exceeds 20 minutes. A sample of 32 main roads results in a mean waiting time of 22.08 minutes with a standard deviation of 5.42 minutes. Conduct a hypothesis test at the $1 \%$ level of significance to determine whether or not the city should increase its transportation budget.

Rates on 30-year fixed mortgages continue to be at historic lows (Chron Business News, September 23, 2010). According to Freddie Mac, the average rate for 30-year fixed loans for the week was $4.37 \%$. An economist wants to test if there is any change in the mortgage rates in the following week. She searches the Internet for 30 -year fixed loans in the following week and reports the rates offered by seven banks as $4.25 \%, 4.125 \%, 4.375 \%$, $4.50 \%, 4.75 \%, 4.375 \%$, and $4.875 \%$. Assume that rates are normally distributed.
a. State the hypotheses to test if the average mortgage rate differs from $4.37 \%$.
b. Calculate the value of the test statistic and the $p$-value.
c. At the 5\% significance level, does the average mortgage rate differ from $4.37 \%$ ? Explain.

The Great Recession cost America trillions of dollars in lost wealth and also levied a heavy toll on the national psyche (The Wall Street Journal, December 21, 2009). According to a poll, just $33 \%$ of those surveyed said America was headed in the right direction. Suppose this poll was based on a sample of 1,000 people. Does the sample evidence suggest that the proportion of Americans who feel that America is headed in the right direction is below $35 \%$ ? Use a $5 \%$ level of significance for the analysis. What if the sample size was 2,000 ?

A retailer is looking to evaluate its customer service. Management has determined that if the retailer wants to stay competitive, then it will have to have at least a $90 \%$ satisfaction rate among its customers. Management will take corrective actions if the satisfaction rate falls below $90 \%$. A survey of 1,200 customers showed that 1,068 were satisfied with their customer service.
a. State the hypotheses to test if the retailer needs to improve its services.
b. What is the value of the test statistic?
c. Find the $p$-value.
d. Interpret the results at $\alpha=0.05$.

A national survey found that $33 \%$ of high school students said they texted or e-mailed while driving (The Boston Globe, June 8, 2012). These findings came a day after a Massachusetts teenager was convicted for causing a fatal crash while texting. A researcher wonders whether texting or e-mailing while driving is more prevalent among Massachusetts teens. He surveys 100 teens and $42 \%$ of them admitted that they texted or e-mailed while behind the wheel. Can he conclude at the $1 \%$ significance level that Massachusetts teens engage in this behavior at a rate greater than the national rate?

A television network is deciding whether or not to give its newest television show a spot during prime viewing time at night. For this to happen, it will have to move one of Page 353 its most viewed shows to another slot. The network conducts a survey asking its viewers which show they would rather watch. The network will keep its current lineup of shows unless the majority of the customers want to watch the new show. The network receives 827 responses, of which 428 indicate that they would like to see the new show in the lineup.
a. Set up the hypotheses to test if the television network should give its newest television show a spot during prime viewing time at night.
b. Calculate the value of the test statistic and the $p$-value.
c. At $a=0.01$, what should the television network do?

A survey finds that $17 \%$ of Americans cannot part with their landlines (The Washington Post, February 27, 2014). A researcher in the rural South collects data from 200 households and finds that 45 of them still have landlines.
a. The researcher believes that the proportion of households with landlines in the rural South is not representative of the national proportion. Specify the competing hypotheses to test her claim.
b. Calculate the value of the test statistic and the $p$-value.
c. At the $5 \%$ significance level, do the sample data support the researcher's belief?

Using data from the past 25 years, an investor wants to test whether the average return of Vanguard's Precious Metals and Mining Fund is greater than $12 \%$.
Assume returns are normally distributed with a population standard deviation of $30 \%$.
a. State the null and the alternative hypotheses for the test.
b. Calculate the value of the test statistic and the $p$-value.
c. At $a=0.05$, what is the conclusion? Is the return on Vanguard's Precious Metals and Mining Fund greater than $12 \%$ ?

On average, Americans drive 13,500 miles per year (The Boston Globe, June 7, 2012). An economist gathers data on the driving habits of 50 residents in the Midwest.
a. The economist believes that the average number of miles driven annually by Midwesterners is different from the U.S. average. Specify the competing hypotheses to test the economist's claim.
b. Calculate the value of the test statistic and the $p$-value.
c. At the $10 \%$ significance level, do the data support the researcher's claim? Explain.

An entrepreneur examines monthly sales (in $$\$ 1,000$$ ) for 40 convenience stores in Rhode Island.
a. State the null and the alternative hypotheses in order to test whether average sales differ from $$\$ 130,000$$.
b. Calculate the value of the test statistic and the $p$-value.
c. At $\alpha=0.05$, what is your conclusion to the test? Do average sales differ from $$\$ 130,000$$ ?

The euro-zone crisis has wreaked havoc on U.S. stock markets (The Wall Street Journal, June 8, 2012). A portfolio analyst wonders if the average trading volume on the Dow Jones Industrial Average (DJIA) has decreased since the beginning of the year. She gathers data on daily trading volumes for 30 days.
a. The average trading volume in the beginning of the year was about 4,000 shares (in millions). Specify the competing hypotheses to test her claim.
b. Calculate the value of the test statistic and the $p$-value.
c. At the $5 \%$ significance level, does it appear that the trading volume has increased since the beginning of the year?

Residents of Hawaii have the longest life expectancies in the United States, averaging 81.48 years (www.worldlifeexpectancy.com; data retrieved June 4, 2012). A sociologist collects data on the age at death for 50 recently deceased Michigan residents.
a. The sociologist believes that the life expectancies of Michigan residents are less than those of Hawaii residents. Specify the competing hypotheses to test this belief.
b. Calculate the value of the test statistic and the $p$-value.
c. At the $1 \%$ significance level, do the data support the sociologist's belief?

Thirty-three percent of children and teens in the United States are obese or overweight (Health, October 2010). A health practitioner in the Midwest collects data on 200 children
and teens and finds that 84 of them are either obese or overweight.
a. The health practitioner believes that the proportion of obese and overweight children in the Midwest is not representative of the national proportion. Specify the competing hypotheses to test her claim.
b. Calculate the value of the test statistic and the $p$-value.
c. At the $1 \%$ significance level, do the sample data support the health practitioner's belief?