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Statistics for Business and Economics: Global Edition

Newbold P., Carlson W.L., Thorne B.M.

Chapter 9

Hypothesis Tests of a Single Population - all with Video Answers

Educators

Chapter Questions

00:40

Problem 1

Mary Arnold wants to use the results of a random sample market survey to seek strong evidence that her brand of breakfast cereal has more than $20 \%$ of the total market. Formulate the null and alternative hypotheses, using $P$ as the population proportion.

Nick Johnson
Nick Johnson
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00:30

Problem 2

The Federal Reserve Board is meeting to decide if it should reduce interest rates in order to stimulate economic growth. State the null and alternative hypotheses regarding economic growth that the board would formulate to guide its decision.

Nick Johnson
Nick Johnson
Numerade Educator
00:58

Problem 3

John Stull, senior vice president of manufacturing, is seeking strong evidence to support his hope that new operating procedures have reduced the percentage of underfilled cereal packages from the Ames production line. State his null and alternative hypotheses and indicate the results that would provide strong evidence.

Nick Johnson
Nick Johnson
Numerade Educator
00:56

Problem 4

In the UK, some motorist groups want the current speed limit on motorways increased; they argue this would not be dangerous and would enable motorists to reach their destinations more quickly. However, some road-safety groups say speed can be a factor in accidents and believe it would be dangerous to increase the existing speed limit.
a. State the null and alternative hypotheses from the perspective of the motorist groups.
b. State the null and alternative hypotheses from the perspective of road-safety groups.

Nick Johnson
Nick Johnson
Numerade Educator
00:01

Problem 5

The branch manager of an international bank in Kuala Lumpur, Malaysia, has received a memorandum from senior executives at the head office of the bank instructing the manager to ensure that the average queuing time for customers waiting to see a cashier is no more than 5 minutes. Since receiving this directive, the manager has been informally checking queuing times and is very confident that the average time customers spend waiting to see a cashier is currently 5 minutes or less. You have now been brought in to undertake an audit of queuing times to check that they are in accordance with the senior executives' directive. State the null and alternative hypotheses you will be using in this instance.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:23

Problem 6

The 2000 presidential election in the United States was very close, and the decision came down to the results of the presidential voting in the state of Florida. The election was finally decided in favor of George W. Bush over Al Gore by a U.S. Supreme Court decision that stated that it was not appropriate to hand count ballots that had been rejected by the voting machines in various counties. At that time Bush had a small lead based on the ballots that had been counted. Imagine that you were a lawyer for $\mathrm{Al}$ Gore. State your null and alternative hypotheses concerning the population vote totals for each candidate. Given your hypotheses, what would you argue about the results of the proposed recount-if it had actually occurred?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:08

Problem 7

A random sample is obtained from a population with variance $\sigma^2=625$, and the sample mean is computed. Test the null hypothesis $H_0: \mu=100$ versus the alternative hypothesis $H_1: \mu>100$ with $\alpha=0.05$. Compute the critical value $\bar{x}_c$ and state your decision rule for the following options.
a. Sample size $n=25$
b. Sample size $n=16$
c. Sample size $n=44$
d. Sample size $n=32$

Nick Johnson
Nick Johnson
Numerade Educator
04:02

Problem 8

A random sample of $n=25$ is obtained from a population with variance $\sigma^2$, and the sample mean is computed. Test the null hypothesis $H_0: \mu=100$ versus the alternative hypothesis $H_1: \mu>100$ with $\alpha=0.05$.
Compute the critical value $\bar{x}_c$ and state your decision rule for the following options.
a. The population variance is $\sigma^2=225$.
b. The population variance is $\sigma^2=900$.
c. The population variance is $\sigma^2=400$.
d. The population variance is $\sigma^2=600$.

Nick Johnson
Nick Johnson
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02:16

Problem 9

A random sample is obtained from a population with a variance of $\sigma^2=400$, and the sample mean is computed to be $\bar{x}_c=70$. Consider the null hypothesis $H_0: \mu=80$ versus the alternative hypothesis $H_1: \mu<80$. Compute the $p$-value for the following options.
a. Sample size $n=25$
b. Sample size $n=16$
c. Sample size $n=44$
d. Sample size $n=32$

Nick Johnson
Nick Johnson
Numerade Educator
01:57

Problem 10

A random sample of $n=25$ is obtained from a population with variance $\sigma^2$, and the sample mean is computed to be $\bar{x}=70$. Consider the null hypothesis $H_0: \mu=80$ versus the alternative hypothesis $H_1: \mu<80$. Compute the $p$-value for the following options.
a. The population variance is $\sigma^2=225$.
b. The population variance is $\sigma^2=900$.
c. The population variance is $\sigma^2=400$.
d. The population variance is $\sigma^2=600$.

Nick Johnson
Nick Johnson
Numerade Educator
07:01

Problem 11

A manufacturer of detergent claims that the contents of boxes sold weigh on average at least 16 ounces. The distribution of weight is known to be normal, with a standard deviation of 0.4 ounce. A random sample of 16 boxes yielded a sample mean weight of 15.84 ounces. Test at the $10 \%$ significance level the null hypothesis that the population mean weight is at least 16 ounces.

Willis James
Willis James
Numerade Educator
01:44

Problem 12

A company that receives shipments of batteries tests a random sample of nine of them before agreeing to take a shipment. The company is concerned that the true mean lifetime for all batteries in the shipment should be at least 50 hours. From past experience it is safe to conclude that the population distribution of lifetimes is normal with a standard deviation of 3 hours. For one particular shipment the mean lifetime for a sample of nine batteries was 48.2 hours. Test at the $10 \%$ level the null hypothesis that the population mean lifetime is at least 50 hours.

Nick Johnson
Nick Johnson
Numerade Educator
04:08

Problem 13

A pharmaceutical manufacturer is concerned that the impurity concentration in pills should not exceed $3 \%$. It is known that from a particular production run impurity concentrations follow a normal distribution with a standard deviation of $0.4 \%$. A random sample of 64 pills from a production run was checked, and the sample mean impurity concentration was found to be $3.07 \%$.
a. Test at the $5 \%$ level the null hypothesis that the population mean impurity concentration is $3 \%$ against the alternative that it is more than $3 \%$.
b. Find the $p$-value for this test.
c. Suppose that the alternative hypothesis had been two-sided, rather than one-sided, with the null hypothesis $H_0: \mu=3$. State, without doing the calculations, whether the $p$-value of the test would be higher than, lower than, or the same as that found in part (b). Sketch a graph to illustrate your reasoning.
d. In the context of this problem, explain why a onesided alternative hypothesis is more appropriate than a two-sided alternative.

Nick Johnson
Nick Johnson
Numerade Educator
03:54

Problem 14

Test the hypotheses
$$
\begin{aligned}
& H_0: \mu \leq 100 \\
& H_1: \mu>100
\end{aligned}
$$
using a random sample of $n=25$, a probability of Type I error equal to 0.05 , and the following sample statistics.
a. $\bar{x}=106 ; s=15$
b. $\bar{x}=104 ; s=10$
c. $\bar{x}=95 ; s=10$
d. $\bar{x}=92 ; s=18$

Nick Johnson
Nick Johnson
Numerade Educator
04:46

Problem 15

Test the hypotheses
$$
\begin{aligned}
& H_0: \mu=100 \\
& H_1: \mu<100
\end{aligned}
$$
using a random sample of $n=36$, a probability of Type I error equal to 0.05 , and the following sample statistics.
a. $\bar{x}=106 ; s=15$
b. $\bar{x}=104 ; s=10$
c. $\bar{x}=95 ; s=10$
d. $\bar{x}=92 ; s=18$

Nick Johnson
Nick Johnson
Numerade Educator
01:15

Problem 16

An engineering research center claims that through the use of a new computer control system, automobiles should achieve, on average, an additional 3 miles per gallon of gas. A random sample of 100 automobiles was used to evaluate this product. The sample mean increase in miles per gallon achieved was 2.4, and the sample standard deviation was 1.8 miles per gallon. Test the hypothesis that the population mean is at least 3 miles per gallon. Find the $p$-value of this test, and interpret your findings.

Nick Johnson
Nick Johnson
Numerade Educator
01:15

Problem 17

A random sample of 1,562 undergraduates enrolled in management ethics courses was asked to respond on a scale from 1 (strongly disagree) to 7 (strongly agree) to this proposition: Senior corporate executives are interested in social justice. The sample mean response was 4.27 , and the sample standard deviation was 1.32 . Test at the $1 \%$ level, against a two-sided alternative, the null hypothesis that the population mean is 4 .

Nick Johnson
Nick Johnson
Numerade Educator
00:55

Problem 18

You have been asked to evaluate single-employer plans after the establishment of the Health Benefit Guarantee Corporation. A random sample of 76 percentage changes in promised health benefits was observed. The sample mean percentage change was 0.078 , and the sample standard deviation was 0.201 . Find and interpret the $p$-value of a test of the null hypothesis that the population mean percentage change is 0 against a two-sided alternative.

Nick Johnson
Nick Johnson
Numerade Educator
01:50

Problem 19

A random sample of 172 marketing students was asked to rate, on a scale from 1 (not important) to 5 (extremely important), health benefits as a job characteristic. The sample mean rating was 3.31 , and the sample standard deviation was 0.70 . Test at the $1 \%$ significance level the null hypothesis that the population mean rating is at most 3.0 against the alternative that it is larger than 3.0.

Nick Johnson
Nick Johnson
Numerade Educator
01:14

Problem 20

A random sample of 170 people was provided with a forecasting problem. Each sample member was given, in two ways, the task of forecasting the next value of a retail sales variable. The previous 20 values were presented both as numbers and as points on a graph. Subjects were asked to predict the next value. The absolute forecasting errors were measured. The sample then consisted of 170 differences in absolute forecast errors (numerical minus graphical). The sample mean of these differences was -2.91 , and the sample standard deviation was 11.33 . Find and interpret the $p$-value of a test of the null hypothesis that the population mean difference is 0 against the alternative that it is negative. (The alternative can be viewed as the hypothesis that, in the aggregate, people make better forecasts when they use graphs of past history compared to using numerical values from past history.)

Nick Johnson
Nick Johnson
Numerade Educator
01:23

Problem 21

The accounts of a corporation show that, on average, accounts payable are $$\$ 125.32$$. An auditor checked a random sample of 16 of these accounts. The sample mean was $$\$ 131.78$$ and the sample standard deviation was $$\$ 25.41$$. Assume that the population distribution is normal. Test at the 5\% significance level against a twosided alternative the null hypothesis that the population mean is $$\$ 125.32$$.

Nick Johnson
Nick Johnson
Numerade Educator
01:35

Problem 22

On the basis of a random sample the null hypothesis
$$
H_0: \mu=\mu_0
$$
is tested against the alternative
$$
H_1: \mu>\mu_0
$$
and the null hypothesis is not rejected at the $5 \%$ significance level.
a. Does this necessarily imply that $\mu_0$ is contained in the $95 \%$ confidence interval for $\mu$ ?
b. Does this necessarily imply that $\mu_0$ is contained in the $90 \%$ confidence interval for $\mu$ if the observed sample mean is larger than $\mu_0$ ?

Nick Johnson
Nick Johnson
Numerade Educator
00:01

Problem 23

A company selling licenses for new e-commerce computer software advertises that firms using this software obtain, on average during the first year, a yield of $10 \%$ on their initial investments. A random sample of 10 of these franchises produced the following yields for the first year of operation:
$\begin{array}{llllllllll}6.1 & 9.2 & 11.5 & 8.6 & 12.1 & 3.9 & 8.4 & 10.1 & 9.4 & 8.9\end{array}$
Assuming that population yields are normally distributed, test the company's claim.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:03

Problem 24

A process that produces bottles of shampoo, when operating correctly, produces bottles whose contents weigh, on average, 20 ounces. A random sample of nine bottles from a single production run yielded the following content weights (in ounces):
$\begin{array}{lllllllll}21.4 & 19.7 & 19.7 & 20.6 & 20.8 & 20.1 & 19.7 & 20.3 & 20.9\end{array}$
Assuming that the population distribution is normal, test at the $5 \%$ level against a two-sided alternative the null hypothesis that the process is operating correctly.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:07

Problem 25

A statistics instructor is interested in the ability of students to assess the difficulty of a test they have taken. This test was taken by a large group of students, and the average score was 78.5. A random sample of eight students was asked to predict this average score. Their predictions were as follows:
$\begin{array}{llllllll}72 & 83 & 78 & 65 & 69 & 77 & 81 & 71\end{array}$
Assuming a normal distribution, test the null hypothesis that the population mean prediction would be 78.5. Use a two-sided alternative and a $10 \%$ significance level.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:03

Problem 26

An IT consultancy in Singapore that offers telephony solutions to small businesses claims that its new callhandling software will enable clients to increase successful inbound calls by an average of 75 calls per week. For a random sample of 25 small-business users of this software, the average increase in successful inbound calls was 70.2 and the sample standard deviation was 8.4 calls. Test, at the $5 \%$ level, the null hypothesis that the population mean increase is at least 75 calls. Assume a normal distribution.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:00

Problem 27

In contract negotiations a company claims that a new incentive scheme has resulted in average weekly earnings of at least $$\$ 400$$ for all customer service workers. A union representative takes a random sample of 15 workers and finds that their weekly earnings have an average of $$\$ 381.35$$ and a standard deviation of $$\$ 48.60$$. Assume a normal distribution.
a. Test the company's claim.
b. If the same sample results had been obtained from a random sample of 50 employees, could the company's claim be rejected at a lower significance level than that used in part a?

Nick Johnson
Nick Johnson
Numerade Educator
03:14

Problem 28

A random sample of women is obtained, and each person in the sample is asked if she would purchase a new shoe model. The new shoe model would be successful in meeting corporate profit objective if more than $25 \%$ of the women in the population would purchase this shoe model. The following hypothesis test can be performed at a level of $\alpha=0.03$ using $\hat{p}$ as the sample proportion of women who said yes.
$$
\begin{aligned}
& H_0: P \leq 0.25 \\
& H_1: P>0.25
\end{aligned}
$$
What value of the sample proportion, $\hat{p}$, is required to reject the null hypothesis, given the following sample sizes?
a. $n=400$
c. $n=625$
b. $n=225$
d. $n=900$

Nick Johnson
Nick Johnson
Numerade Educator
02:11

Problem 29

A company is attempting to determine if it should retain a previously popular shoe model. A random sample of women is obtained, and each person in the sample is asked if she would purchase this existing shoe model. To determine if the old shoe model should be retained, the following hypothesis test is performed at a level of $\alpha=0.05$ using $\hat{p}$ as the sample proportion of women who said yes.
$$
\begin{aligned}
& H_0: P \geq 0.25 \\
& H_1: P<0.25
\end{aligned}
$$
What value of the sample proportion, $\hat{p}$, is required to reject the null hypothesis, given the following sample sizes?
a. $n=400$
c. $n=625$
b. $n=225$
d. $n=900$

Nick Johnson
Nick Johnson
Numerade Educator
01:57

Problem 30

In a random sample of 361 owners of small businesses that had gone into bankruptcy, 105 reported conducting no marketing studies prior to opening the business. Test the hypothesis that at most $25 \%$ of all members of this population conducted no marketing studies before opening their businesses. Use $\alpha=0.05$.

Nick Johnson
Nick Johnson
Numerade Educator
01:50

Problem 31

In a random sample of 360 export managers in the UK, 69 of the sample members indicated some measure of disagreement with this statement: The most important export market for UK manufacturers in 10 years' time will be the continent of Asia. Test, at the $5 \%$ level, the hypothesis that at least $25 \%$ of all members of this population would disagree with this statement.

Nick Johnson
Nick Johnson
Numerade Educator
01:38

Problem 32

In a random sample of 160 business school students, 72 sample members indicated some measure of agreement with this statement: Scores on a standardized entrance exam are less important for a student's chance to succeed academically than is the student's high school GPA. Test the null hypothesis that one-half of all business school graduates would agree with this statement against a two-sided alternative. Find and interpret the $p$-value of the test.

Nick Johnson
Nick Johnson
Numerade Educator
01:56

Problem 33

Of a random sample of 199 auditors, 104 indicated some measure of agreement with this statement: Cash flow is an important indication of profitability. Test at the $10 \%$ significance level against a twosided alternative the null hypothesis that one-half of the members of this population would agree with this statement. Also find and interpret the $p$-value of this test.

Nick Johnson
Nick Johnson
Numerade Educator
01:45

Problem 34

A random sample of 50 university admissions officers was asked about expectations in application interviews. Of these sample members, 28 agreed that the interviewer usually expects the interviewee to have volunteer experience doing community projects. Test the null hypothesis that one-half of all interviewers have this expectation against the alternative that the population proportion is larger than one-half. Use $\alpha=0.05$.

Nick Johnson
Nick Johnson
Numerade Educator
01:45

Problem 35

Of a random sample of 172 elementary school educators, 118 said that parental support was the most important source of a child's success. Test the hypothesis that parental support is the most important source of a child's success for at least $75 \%$ of elementary school educators against the alternative that the population percentage is less than $75 \%$. Use $\alpha=0.05$.

Nick Johnson
Nick Johnson
Numerade Educator
01:29

Problem 36

A random sample of 202 business faculty members was asked if there should be a required foreign language course for business majors. Of these sample members, 140 felt there was a need for a foreign language course. Test the hypothesis that at least $75 \%$ of all business faculty members hold this view. Use $\alpha=0.05$.

Nick Johnson
Nick Johnson
Numerade Educator
03:49

Problem 37

Consider a problem with the hypothesis test
$$
\begin{aligned}
& H_0: \mu=5 \\
& H_1: \mu>5
\end{aligned}
$$
and the following decision rule:
$$
\begin{aligned}
& \text { reject } H_0 \text { if } \frac{\bar{x}-5}{0.1 / \sqrt{16}}>1.645 \text { or } \\
& \bar{x}>5+1.645(0.1 / \sqrt{16})=5.041
\end{aligned}
$$
Compute the probability of Type II error and the power for the following true population means.
a. $\mu=5.10$
b. $\mu=5.03$
c. $\mu=5.15$
d. $\mu=5.07$

Nick Johnson
Nick Johnson
Numerade Educator
09:50

Problem 38

Consider Example 9.6 with the null hypothesis
$$
H_0: P=P_0=0.50
$$
and the alternative hypothesis
$$
H_0: P \neq 0.50
$$
The decision rule is
$$
\begin{gathered}
\frac{\hat{p}_x-0.50}{\sqrt{0.50(1-0.50) / 600}}<-1.96 \text { or } \\
\frac{\hat{p}_x-0.50}{\sqrt{0.50(1-0.50) / 600}}>1.96
\end{gathered}
$$
with a sample size of $n=600$. What is the probability of Type II error if the actual population proportion is each of the following?
a. $P=0.52$
b. $P=0.58$
c. $P=0.53$
d. $P=0.48$
e. $P=0.43$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
00:01

Problem 39

A company that receives shipments of batteries tests a random sample of nine of them before agreeing to take a shipment. The company is concerned that the true mean lifetime for all batteries in the shipment should be at least 50 hours. From past experience it is safe to conclude that the population distribution of lifetimes is normal with a standard deviation of 3 hours. For one particular shipment the mean lifetime for a sample of nine batteries was 48.2 hours.
a. Test, at the $10 \%$ level, the null hypothesis that the population mean lifetime is at least 50 hours.
b. Find the power of a $10 \%$-level test when the true mean lifetime of batteries is 49 hours.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:03

Problem 40

A pharmaceutical manufacturer is concerned that the impurity concentration in pills does not exceed $3 \%$. It is known that from a particular production run, impurity concentrations follow a normal distribution with standard deviation $0.4 \%$. A random sample of 64 pills from a production run was checked, and the sample mean impurity concentration was found to be $3.07 \%$.
a. Test, at the $5 \%$ level, the null hypothesis that the population mean impurity concentration is $3 \%$ against the alternative that it is more than $3 \%$.
b. Find the probability of a $5 \%$-level test rejecting the null hypothesis when the true mean impurity concentration is $3.10 \%$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:15

Problem 41

A random sample of 1,562 undergraduates enrolled in management ethics courses was asked to respond, on a scale from 1 (strongly disagree) to 7 (strongly agree), to this proposition: Senior corporate executives are interested in social justice. The sample mean response was

Nick Johnson
Nick Johnson
Numerade Educator
00:01

Problem 42

A random sample of 802 supermarket shoppers determined that 378 shoppers preferred generic-brand items. Test at the $10 \%$ level the null hypothesis that at least one-half of all shoppers preferred genericbrand items against the alternative that the population proportion is less than one-half. Find the power of a $10 \%$-level test if, in fact, $45 \%$ of the supermarket shoppers preferred generic brands.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:01

Problem 43

In a random sample of 340 export managers in Malaysia, 61 of the sample members indicated some measure of disagreement with this statement: The most important export market for Malaysian manufacturers in 10 years' time will be Europe.
a. Test, at the $5 \%$ level, the null hypothesis that at least $25 \%$ of all members of this population would disagree with this statement.
b. Find the probability of rejecting the null hypothesis with a $5 \%$ level test if, in fact, $20 \%$ of all members of this population would disagree with the statement.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:01

Problem 44

Of a random sample of 199 auditors, 104 indicated some measure of agreement with this statement: Cash flow is an important indication of profitability.
a. Test, at the $10 \%$ significance level against a two-sided alternative, the null hypothesis that one-half of the members of this population would agree with this statement. Also find and interpret the $p$-value of this test.
b. Find the probability of accepting the null hypothesis with a $10 \%$-level test if, in fact, $60 \%$ of all auditors agree that cash flow is an important indicator of profitability.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
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Problem 45

Each day, a fast-food chain tests that the average weight of its "two-pounders" is at least 32 ounces. The alternative hypothesis is that the average weight is less than 32 ounces, indicating that new processing procedures are needed. The weights of two-pounders can be assumed to be normally distributed, with a standard deviation of 3 ounces. The decision rule adopted is to reject the null hypothesis if the sample mean weight is less than 30.8 ounces.
a. If random samples of $n=36$ two-pounders are selected, what is the probability of a Type I error, using this decision rule?
b. If random samples of $n=9$ two-pounders are selected, what is the probability of a Type I error, using this decision rule? Explain why your answer differs from that in part $\mathrm{a}$.
c. Suppose that the true mean weight is 31 ounces. If random samples of 36 two-pounders are selected, what is the probability of a Type II error, using this decision rule?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
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Problem 46

A wine producer claims that the proportion of its customers who cannot distinguish its product from frozen grape juice is, at most, 0.09 . The producer decides to test this null hypothesis against the alternative that the true proportion is more than 0.09 . The decision rule adopted is to reject the null hypothesis if the sample proportion of people who cannot distinguish between these two flavors exceeds 0.14 .
a. If a random sample of 100 customers is chosen, what is the probability of a Type I error, using this decision rule?
b. If a random sample of 400 customers is selected, what is the probability of a Type I error, using this decision rule? Explain, in words and graphically, why your answer differs from that in part $\mathrm{a}$.
c. Suppose that the true proportion of customers who cannot distinguish between these flavors is 0.20 . If a random sample of 100 customers is selected, what is the probability of a Type II error?
d. Suppose that, instead of the given decision rule, it is decided to reject the null hypothesis if the sample proportion of customers who cannot distinguish between the two flavors exceeds 0.16 . A random sample of 100 customers is selected.
i. Without doing the calculations, state whether the probability of a Type I error will be higher than, lower than, or the same as in part a.
ii. If the true proportion is 0.20 , will the probability of a Type II error be higher than, lower than, or the same as in part $c$ ?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:01

Problem 47

Test the hypotheses
$$
\begin{aligned}
& H_0: \sigma^2 \leq 100 \\
& H_1: \sigma^2>100
\end{aligned}
$$
using the following results from the following random samples.
a. $s^2=165 ; n=25$
c. $s^2=159 ; n=25$
b. $s^2=165 ; n=29$
d. $s^2=67 ; n=38$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:01

Problem 48

At the insistence of a government inspector, a new safety device is installed in an assembly-line operation. After the installation of this device, a random sample of 8 days' output gave the following results for numbers of finished components produced:
$\begin{array}{llllllll}618 & 660 & 638 & 625 & 571 & 598 & 639 & 582\end{array}$
Management is concerned about the variability of daily output and views any variance above 500 as undesirable. Test, at the $10 \%$ significance level, the null hypothesis that the population variance for daily output does not exceed 500 .

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:45

Problem 49

Plastic sheets produced by a machine are periodically monitored for possible fluctuations in thickness. If the true variance in thicknesses exceeds 2.25 square millimeters, there is cause for concern about product quality. Thickness measurements for a random sample of 10 sheets produced in a particular shift were taken, giving the following results (in millimeters):
$\begin{array}{llllllll}226 & 226 & 232 & 227 & 225 & 228 & 225 & 228\end{array}$
$229 \quad 230$
a. Find the sample variance.
b. Test, at the $5 \%$ significance level, the null hypothesis that the population variance is at most 2.25 .

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 50

One way to evaluate the effectiveness of a teaching assistant is to examine the scores achieved by his or her students on an examination at the end of the course. Obviously, the mean score is of interest. However, the variance also contains useful informationsome teachers have a style that works very well with more-able students but is unsuccessful with less-able or poorly motivated students. A professor sets a standard examination at the end of each semester for all sections of a course. The variance of the scores on this test is typically very close to 300 . A new teaching assistant has a class of 30 students whose test scores had a variance of 480 . Regarding these students' test scores as a random sample from a normal population, test, against a two-sided alternative, the null hypothesis that the population variance of their scores is 300 .

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:46

Problem 51

A company produces electric devices operated by a thermostatic control. The standard deviation of the temperature at which these controls actually operate should not exceed $2.0^{\circ} \mathrm{F}$. For a random sample of 20 of these controls, the sample standard deviation of operating temperatures was $2.36^{\circ} \mathrm{F}$. Stating any assumptions you need to make, test, at the $5 \%$ level, the null hypothesis that the population standard deviation is 2.0 against the alternative that it is larger.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:01

Problem 52

An instructor has decided to introduce a greater component of independent study into an intermediate microeconomics course as a way of motivating students to work independently and think more carefully about the course material. A colleague cautions that a possible consequence may be increased variability in student performance. However, the instructor responds that she would expect less variability. From her records she found that in the past, student scores on the final exam for this course followed a normal distribution with standard deviation 18.2 points. For a class of 25 students using the new approach, the standard deviation of scores on the final exam was 15.3 points. Assuming that these 25 students can be viewed as a random sample of all those who might be subjected to the new approach, test the null hypothesis that the population standard deviation is at least 18.2 points against the alternative that it is lower.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
07:49

Problem 53

Explain carefully the distinction between each of the following pairs of terms.
a. Null and alternative hypotheses
b. Simple and composite hypotheses
c. One-sided and two-sided alternatives
d. Type I and Type II errors
e. Significance level and power

Rashmi Sinha
Rashmi Sinha
Numerade Educator
00:02

Problem 54

Carefully explain what is meant by the $p$-value of a test, and discuss the use of this concept in hypothesis testing.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:01

Problem 55

A random sample of 10 students contains the following observations, in hours, for time spent studying in the week before final exams:
$\begin{array}{llllllllll}28 & 57 & 42 & 35 & 61 & 39 & 55 & 46 & 49 & 38\end{array}$
Assume that the population distribution is normal.
a. Find the sample mean and standard deviation.
b. Test, at the $5 \%$ significance level, the null hypothesis that the population mean is 40 hours against the alternative that it is higher.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:37

Problem 56

State whether each of the following is true or false.
a. The significance level of a test is the probability that the null hypothesis is false.
b. A Type I error occurs when a true null hypothesis is rejected.
c. A null hypothesis is rejected at the 0.025 level but is not rejected at the 0.01 level. This means that the $p$-value of the test is between 0.01 and 0.025 .
d. The power of a test is the probability of accepting a null hypothesis that is true.
e. If a null hypothesis is rejected against an alternative at the $5 \%$ level, then using the same data, it must be rejected against that alternative at the $1 \%$ level.
f. If a null hypothesis is rejected against an alternative at the $1 \%$ level, then using the same data, it must be rejected against the alternative at the $5 \%$ level.
g. The $p$-value of a test is the probability that the null hypothesis is true.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
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Problem 57

An insurance company employs agents on a commission basis. It claims that in their first-year agents will earn a mean commission of at least $$\$ 40,000$$ and that the population standard deviation is no more than $$\$ 6,000$$. A random sample of nine agents found for commission in the first year,
$$
\sum_{i=1}^9 x_i=333 \text { and } \sum_{i=1}^9\left(x_i-\bar{x}\right)^2=312
$$
where $x_i$ is measured in thousands of dollars and the population distribution can be assumed to be normal. Test, at the $5 \%$ level, the null hypothesis that the population mean is at least $$\$ 40,000$$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:30

Problem 58

Supporters claim that a new windmill can generate an average of at least 800 kilowatts of power per day. Daily power generation for the windmill is assumed to be normally distributed with a standard deviation of 120 kilowatts. A random sample of 100 days is taken to test this claim against the alternative hypothesis that the true mean is less than 800 kilowatts. The claim will not be rejected if the sample mean is 776 kilowatts or more and rejected otherwise.
a. What is the probability $\alpha$ of a Type I error using the decision rule if the population mean is, in fact, 800 kilowatts per day?
b. What is the probability $\beta$ of a Type II error using this decision rule if the population mean is, in fact, 740 kilowatts per day?
c. Suppose that the same decision rule is used, but with a sample of 200 days rather than 100 days.
i. Would the value of $\alpha$ be larger than, smaller than, or the same as that found in part $a$ ?
ii. Would the value of $\beta$ be larger than, smaller than, or the same as that found in part b?
d. Suppose that a sample of 100 observations was taken, but that the decision rule was changed so that the claim would not be rejected if the sample mean was at least 765 kilowatts.
i. Would the value of $\alpha$ be larger than, smaller than, or the same as that found in part a?
ii. Would the value of $\beta$ be larger than, smaller than, or the same as that found in part b?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:01

Problem 59

In a random sample of 545 accountants engaged in preparing county operating budgets for use in planning and control, 117 indicated that estimates of cash flow were the most difficult element of the budget to derive.
a. Test at the $5 \%$ level the null hypothesis that at least $25 \%$ of all accountants find cash flow estimates the most difficult estimates to derive.
b. Based on the procedure used in part a, what is the probability that the null hypothesis would be rejected if the true percentage of those finding cash flow estimates most difficult was each of the following?
i. $20 \%$
ii. $25 \%$
iii. $30 \%$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:10

Problem 60

A random sample of 104 marketing vice presidents from large Fortune 500 corporations was questioned on future developments in the business environment. Of those sample members, 50 indicated some measurement of agreement with this statement: Firms will concentrate their efforts more on cash flow than on profits. What is the lowest level of significance at which the null hypothesis, which states that the true proportion of all such executives who would agree with this statement is one-half, can be rejected against a two-sided alternative?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
00:03

Problem 61

Of a random sample of 95 small-business owners in Rome, Italy 54 said they liked statistical work. Test the null hypothesis that one-half of all members of this population like statistics against the alternative that the population proportion is bigger than one-half.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:05

Problem 62

In a random sample of 150 business graduates 50 agreed or strongly agreed that businesses should focus their efforts on innovative e-commerce strategies. Test at the $5 \%$ level the null hypothesis that at most $25 \%$ of all business graduates would be in agreement with this assertion.

Nick Johnson
Nick Johnson
Numerade Educator
00:58

Problem 63

Of a random sample of 142 admissions counselors on college campuses 39 indicated that, on average, they spent 15 minutes or less studying each résumé. Test the null hypothesis that at most $20 \%$ of all admissions counselors spend this small amount of time studying résumés.

Nick Johnson
Nick Johnson
Numerade Educator
02:34

Problem 64

Northeastern Franchisers, Ltd., has a number of clients that use their process for producing exotic Norwegian dinners for customers throughout New England. The operating cost for the franchised process has a fixed cost of $$\$ 1,000$$ per week plus $$\$ 5$$ for $\mathrm{ev}$ ery unit produced. Recently, a number of restaurant owners using the process have complained that the cost model is no longer valid and, in fact, the weekly costs are higher. Your job is to determine if there is strong evidence to support the owners' claim. To do so, you obtain a random sample of $n=25$ restaurants and determine their costs. You also find that the number of units produced in each restaurant is normally distributed with a mean of $\mu=400$ and a variance of $\sigma^2=625$. The random sample mean $(n=25)$ for weekly costs was $\$ 3,050$. Prepare and implement an analysis to determine if there is strong evidence to conclude that costs are greater than those predicted by the cost model.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:59

Problem 65

Prairie Flower Cereal, Inc., has asked you to study the variability of the weights of cereal bags produced in plant 2, located in rural Malaysia. The package weights are known to be normally distributed. Using a random sample of $n=71$, you find that the sample mean weight is 40 and the sample variance is 50 .
The marketing vice president claims that there is a very small probability that the population mean weight is less than 39 . Use an appropriate statistical analysis and comment on his claim.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 66

You have been hired by the National Nutrition Council to study nutrition practices in the United States. In particular they want to know if their nutrition guidelines are being met by people in the United States. These guidelines indicate that per capita consumption of fruits and vegetables should be more than 170 pounds per year, per capita consumption of snack foods should be less than 114 pounds, per capita consumption of soft drinks should be less than 65 gallons, and per capita consumption of meat should be more than 70 pounds. As part of your research you have developed the data file Food Nutrition Atlas, which contains a number of nutrition and population variables collected by county over all states. Variable descriptions are located in the chapter appendix. It is true that some counties do not report all the variables. Perform an analysis of the available data and prepare a short report indicating how well the nutrition guidelines are being met. Your conclusions should be supported by rigorous statistical analysis.

Check back soon!
03:40

Problem 67

A recent report from a study of health concerns indicated that there is strong evidence of a nation's overall health decay if the percent of obese adults exceeds $28 \%$. In addition, if the low-income preschool obesity rate exceeds $13 \%$, there is great concern about long-term health. You are asked to conduct an analysis to determine if the U.S. population exceeds that rate. Use the data file Food Nutrition Atlas as the basis for your statistical analysis. Variable descriptions are located in the chapter appendix. Prepare a rigorous analysis and a short statement that reports your statistical results and your conclusions.

Lucas Finney
Lucas Finney
Numerade Educator
01:54

Problem 68

Big River, Inc., a major Alaskan fish processor, is attempting to determine the weight of salmon in the northwest Green River. A random sample of salmon was obtained and weighed. The data are stored in the file labeled Bigfish. Use a classical hypothesis test to determine if there is strong evidence to conclude that the population mean weight for the fish is greater than 40 . Use a probability of Type I error equal to 0.05 .
Prepare a power curve for the test. (Hint: Determine the population mean values for $\beta=0.50, \beta=0.25$, $\beta=0.10$, and $\beta=0.05$, and plot those means versus the power of the test.)

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
00:37

Problem 69

A process produces cable for the local telephone company. When the process is operating correctly, cable diameter follows a normal distribution with mean 1.6 inches and standard deviation 0.05 inch. A random sample of 16 pieces of cable found diameters with a sample mean of 1.615 inches and a sample standard deviation of 0.086 inch.
a. Assuming that the population standard deviation is 0.05 inch, test, at the $10 \%$ level against a two-sided alternative, the null hypothesis that the population mean is 1.6 inches. Find also the lowest level of significance at which this null hypothesis can be rejected against the two-sided alternative.
b. Test, at the $10 \%$ level, the null hypothesis that the population standard deviation is 0.05 inch against the alternative that it is bigger.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:01

Problem 70

When operating normally, a manufacturing process produces tablets for which the mean weight of the active ingredient is 5 grams, and the standard deviation is $0.025 \mathrm{gram}$. For a random sample of 12 tablets the following weights of active ingredient (in grams) were found:
$\begin{array}{llllll}5.01 & 4.69 & 5.03 & 4.98 & 4.98 & 4.95 \\ 5.00 & 5.00 & 5.03 & 5.01 & 5.04 & 4.95\end{array}$
a. Without assuming that the population variance is known, test the null hypothesis that the population mean weight of active ingredient per tablet is 5 grams. Use a two-sided alternative and a $5 \%$ significance level. State any assumptions that you make.
b. Stating any assumptions that you make, test the null hypothesis that the population standard deviation is 0.025 gram against the alternative hypothesis that the population standard deviation exceeds 0.025 gram. Use a $5 \%$ significance level.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:33

Problem 71

An insurance company employs agents on a commission basis. It claims that, in their first year, agents will earn a mean commission of at least $$\$ 40,000$$ and that the population standard deviation is no more than $$\$ 6,000$$. A random sample of nine agents found for commission in the first year,
$$
\sum_{i=1}^9 x_i=333,000 \text { and } \sum_{i=1}^9\left(x_i-\bar{x}\right)^2=312,000,000
$$
measured in thousands of dollars. The population distribution can be assumed to be normal. Test, at the $10 \%$ level, the null hypothesis that the population standard deviation is at most $$\$ 6,000$$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:40

Problem 72

A recent report from a health-concerns study indicated that there is strong evidence of a nation's overall health decay if the percent of obese adults exceeds $28 \%$. In addition, if the low-income preschool obesity rate exceeds $13 \%$, there is great concern about long-term health. You are asked to conduct an analysis to determine if the U.S. population exceeds that rate. Your analysis is restricted to those counties where the adult participation in physical activity exceeds $64.3 \%$. To do this you will first need to obtain a subset of the data file using the capabilities of your statistical analysis computer program. Use the data file Food Nutrition Atlas as the basis for your statistical analysis. Variable descriptions are located in the chapter appendix. Prepare a rigorous analysis and a short statement that reports your statistical results and your conclusions.

Lucas Finney
Lucas Finney
Numerade Educator
03:40

Problem 73

A recent report from a health-concerns study indicated that there is strong evidence of a nation's overall health decay if the percent of obese adults exceeds $28 \%$. In addition, if the low-income preschool obesity rate exceeds $13 \%$, there is great concern about long-term health. You are asked to conduct an analysis to determine if the U.S. population exceeds that rate. Your analysis is restricted to those counties in the following states: California, Michigan, Minnesota, and Florida. Conduct your analysis for each state. To do this, you will first need to obtain a subset of the data file using the capabilities of your statistical analysis computer program. Use the data file Food Nutrition Atlas as the basis for your statistical analysis. Variable descriptions are located in the chapter appendix. Prepare a rigorous analysis and a short statement that reports your statistical results and your conclusions.
Nutrition Research-Based Exercises
The Economic Research Service (ERS), a prestigious think tank research center in the U.S. Department of Agriculture, is conducting a series of research studies to determine the nutrition characteristics of people in the United States. This research is used for both nutrition education and government policy designed to improve personal health. See for example, Carlson, A, et al. 2010.
The data file HEI Cost Data Variable Subset contains considerable information on randomly selected individuals who participated in an extended interview and medical examination. There are two observations for each person in the study. The first observation, identified by daycode $=1$, contains data from the first interview and the second observation, daycode $=2$, contains data from the second interview. This data file contains the data for the following exercises. The variables are described in the data dictionary in the Chapter 10 appendix.

Lucas Finney
Lucas Finney
Numerade Educator
05:21

Problem 74

The body mass index (variable BMI) provides an indication of a person's level of body fat as follows: healthy weight, 20-25; overweight, >25-30; obese, greater than 30 . Excess body weight is, of course, related to diet, but, in turn, what we eat depends on who we are in terms of culture and our entire life experience. Based on an analysis, can you conclude that based on mean weight, men are not obese? Can you conclude that based on mean weight, women are not obese? You will do the analysis based first on the data from the first interview, create a subset from the data file using daycode $=1$, and a second time using data from the second interview, create a subset from the data file using daycode $=2$. Note differences in the results between the first and second interviews.

Sheryl Ezze
Sheryl Ezze
Numerade Educator

Problem 75

The body mass index (variable BMI) provides an indication of a person's level of body fat as follows: healthy weight, 20-25; overweight, $>25-30$; obese, greater than 30 . Excess body weight is, of course, related to diet, but, in turn, what we eat depends on who we are in terms of culture and our entire life experience. Based on an analysis can you conclude that based on mean weight, immigrants are not obese? You will do the analysis based first on the data from the first interview, create a subset from the data file using daycode $=1$, and a second time using data from the second interview, create a subset from the data file using daycode $=2$. Note differences in the results

Check back soon!
05:21

Problem 76

The body mass index (variable BMI) provides an indication of a person's level of body fat as follows: healthy weight, 20-25; overweight, $>25-30$; obese, greater than 30 . Excess body weight is, of course, related to diet, but, in turn, what we eat depends on who we are in terms of culture and our entire life experience. Based on an analysis using mean weight, can you conclude that white people have a healthy weight? Can you conclude that based on mean weight, white people are overweight? You will do the analysis based first on the data from the first interview, create a subset from the data file using daycode $=1$, and a second time using data from the second interview, create a subset from the data file using daycode $=2$. Note that there are differences in the responses between the first and second interviews.

Sheryl Ezze
Sheryl Ezze
Numerade Educator

Problem 77

The body mass index (variable BMI) provides an indication of a person's level of body fat as follows: healthy weight, 20-25; overweight, $>25-30$; obese, greater than 30 . Excess body weight, is of course, related to diet, but, in turn, what we eat depends on who we are in terms of culture and our entire life experience. Based on an analysis using mean weight, can you conclude that Hispanic people have a healthy weight? Can you conclude that based on mean weight, Hispanic people are overweight? You will do the analysis based first on the data from the first interview, create a subset from the data file using daycode $=1$, and a second time using data from the second interview, create a subset from the data file using daycode $=2$. Note differences in the results between the first and second interviews.

Check back soon!
07:11

Problem 78

The body mass index (variable BMI) provides an indication of a person's level of body fat as follows: healthy weight, 20-25; overweight, >25-30; obese, greater than 30 . Excess body weight, is of course, related to diet, but, in turn, what we eat depends on who we are in terms of culture and our entire life experience. Based on an analysis using mean weight,
can you conclude that people who have been diagnosed with high blood pressure have a healthy weight? Can you conclude that using mean weight, people who have been diagnosed with high blood pressure are obese? You will do the analysis based first on the data from the first interview, create a subset from the data file using daycode $=1$, and a second time using data from the second interview, create a subset from the data file using daycode $=2$. Note differences in the results between the first and second interviews.

Sid Wan
Sid Wan
University of Louisville