Statistics for Business and Economics: Global Edition

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

Chapter 10

Two Population Hypothesis Tests - all with Video Answers

Educators

Chapter Questions

Problem 1

You have been asked to determine if two different production processes have different mean numbers of units produced per hour. Process 1 has a mean defined as $\mu_1$ and process 2 has a mean defined as $\mu_2$. The null and alternative hypotheses are as follows:
$$
\begin{aligned}
& H_0: \mu_1-\mu_2=0 \\
& H_1: \mu_1-\mu_2>0
\end{aligned}
$$
Using a random sample of 25 paired observations, the sample means are 50 and 60 for populations 1 and 2 , respectively. Can you reject the null hypothesis using a probability of Type I error $\alpha=0.05$ in each case?
a. The sample standard deviation of the difference is 20
b. The sample standard deviation of the difference is 30
c. The sample standard deviation of the difference is 15
d. The sample standard deviation of the difference is 40

Sheryl Ezze

Sheryl Ezze

Numerade Educator

Problem 2

You have been asked to determine if two different production processes have different mean numbers of units produced per hour. Process 1 has a mean defined as $\mu_1$ and process 2 has a mean defined as $\mu_2$. The null and alternative hypotheses are as follows:
$$
\begin{aligned}
& H_0: \mu_1-\mu_2 \geq 0 \\
& H_1: \mu_1-\mu_2<0
\end{aligned}
$$
Using a random sample of 25 paired observations, the standard deviation of the difference between sample means is 25 . Can you reject the null hypothesis using a probability of Type I error $\alpha=0.05$ in each case?
a. The sample means are 56 and 50
b. The sample means are 59 and 50
c. The sample means are 56 and 48
d. The sample means are 54 and 50

Rashmi Sinha

Numerade Educator

Problem 3

In a study comparing banks in Germany and Great Britain, a sample of 145 matched pairs of banks was formed. Each pair contained one bank from Germany and one from Great Britain. The pairings were made in such a way that the two members were as similar as possible in regard to such factors as size and age. The ratio of total loans outstanding to total assets was calculated for each of the banks. For this ratio, the sample mean difference (German - Great Britain) was 0.0518 , and the sample standard deviation of the differences was 0.3055 .Test, against a two-sided alternative, the null hypothesis that the two population means are equal.

Sheryl Ezze

Numerade Educator

Problem 4

You have been asked to conduct a national study of urban home selling prices to determine if there has been an increase in selling prices over time. There has been some concern that housing prices in major urban areas have not kept up with inflation over time. Your study will use data collected from Atlanta, Chicago, Dallas, and Oakland, which is contained in the data file House Selling Price. Formulate an appropriate hypothesis test and use your statistical computer package to compute the appropriate statistics for analysis. Perform the hypothesis test and indicate your conclusion.
Repeat the analysis using data from only the city of Atlanta.

Rashmi Sinha

Numerade Educator

Problem 5

An agency offers preparation courses for a graduate school admissions test to students. As part of an experiment to evaluate the merits of the course, 12 students were chosen and divided into 6 pairs in such a way that the members of any pair had similar academic records. Before taking the test, one member of each pair was assigned at random to take the preparation course, while the other member did not take a course. The achievement test scores are contained in the Student Pair data file. Assuming that the differences in scores follow a normal distribution, test, at the $5 \%$ level, the null hypothesis that the two population means are equal against the alternative that the true mean is higher for students taking the preparation course.

Rashmi Sinha

Numerade Educator

Problem 6

You have been asked to determine if two different production processes have different mean numbers of units produced per hour. Process 1 has a mean defined as $\mu_1$ and process 2 has a mean defined as $\mu_2$. The null and alternative hypotheses are as follows:
$$
\begin{aligned}
& H_0: \mu_1-\mu_2=0 \\
& H_1: \mu_1-\mu_2>0
\end{aligned}
$$
Use a random sample of 25 observations from process 1 and 28 observations from process 2 and the known variance for process 1 equal to 900 and the known variance for process 2 equal to 1,600 . Can you reject the null hypothesis using a probability of Type I error $\alpha=0.05$ in each case?
a. The process means are 50 and 60 .
b. The difference in process means is 20 .
c. The process means are 45 and 50.
d. The difference in process means is 15 .

Sheryl Ezze

Numerade Educator

Problem 7

=You have been asked to determine if two different production processes have different mean numbers of units produced per hour. Process 1 has a mean defined as $\mu_1$ and process 2 has a mean defined as $\mu_2$. The null and alternative hypotheses are as follows:
$$
\begin{aligned}
& H_0: \mu_1-\mu_2 \leq 0 \\
& H_1: \mu_1-\mu_2>0
\end{aligned}
$$
The process variances are unknown but assumed to be equal. Using random samples of 25 observations from process 1 and 36 observations from process 2 , the sample means are 56 and 50 for populations 1 and 2, respectively. Can you reject the null hypothesis using a probability of Type I error $\alpha=0.05$ in each case?
a. The sample standard deviation from process 1 is 30 and from process 2 is 28 .
b. The sample standard deviation from process 1 is 22 and from process 2 is 33 .
c. The sample standard deviation from process 1 is 30 and from process 2 is 42 .
d. The sample standard deviation from process 1 is 15 and from process 2 is 36 .

Sheryl Ezze

Numerade Educator

Problem 8

A screening procedure was designed to measure attitudes toward minorities as managers. High scores indicate negative attitudes and low scores indicate positive attitudes. Independent random samples were taken of 151 male financial analysts and 108 female financial analysts. For the former group the sample mean and standard deviation scores were 85.8 and 19.13, whereas the corresponding statistics for the latter group were 71.5 and 12.2 . Test the null hypothesis that the two population means are equal against the alternative that the true mean score is higher for male than for female financial analysts.

Sheryl Ezze

Numerade Educator

Problem 9

For a random sample of 125 British entrepreneurs, the mean number of job changes was 1.91 and the sample standard deviation was 1.32 . For an independent random sample of 86 British corporate managers, the mean number of job changes was 0.21 and the sample standard deviation was 0.53 . Test the null hypothesis that the population means are equal against the alternative that the mean number of job changes is higher for British entrepreneurs than for British corporate managers.

Sheryl Ezze

Numerade Educator

Problem 10

A political science professor is interested in comparing the characteristics of students who do and do not vote in national elections. For a random sample of 114 students who claimed to have voted in the last presidential election, she found a mean grade point average of 2.71 and a standard deviation of 0.64 . For an independent random sample of 123 students who did not vote, the mean grade point average was 2.79 and the standard deviation was 0.56. Test, against a twosided alternative, the null hypothesis that the population means are equal.

Sheryl Ezze

Numerade Educator

Problem 11

In light of a recent large corporation bankruptcy, auditors are becoming increasingly concerned about the possibility of fraud. Auditors might be helped in determining the chances of fraud if they carefully measure cash flow. To evaluate this possibility, samples of midlevel auditors from CPA firms were presented with cash-flow information from a fraud case, and they were asked to indicate the chance of material fraud on a scale from 0 to 100. A random sample of 36 auditors used the cash-flow information. Their mean assessment was 36.21 , and the sample standard deviation was 22.93. For an independent random sample of 36 auditors not using the cash-flow information, the sample mean and standard deviation were, respectively, 47.56 and 27.56. Assuming that the two population distributions are normal with equal variances, test, against a two-sided alternative, the null hypothesis that the population means are equal.

Sheryl Ezze

Numerade Educator

Problem 12

The recent financial collapse has led to considerable concern about the information provided to potential investors. The government and many researchers have pointed out the need for increased regulation of financial offerings. The study in this exercise concerns
the effect of sales forecasts on initial public offerings. Initial public offerings' prospectuses were examined. In a random sample of 70 prospectuses in which sales forecasts were disclosed, the mean debt-to-equity ratio prior to the offering issue was 3.97, and the sample standard deviation was 6.14. For an independent random sample of 51 prospectuses in which sales earnings forecasts were not disclosed, the mean debt-to-equity ratio was 2.86 , and the sample standard deviation was 4.29. Test, against a two-sided alternative, the null hypothesis that population mean debt-to-equity ratios are the same for disclosers and nondisclosers of earnings forecasts.

Sheryl Ezze

Numerade Educator

Problem 13

A publisher is interested in the effects on sales of college texts that include more than 100 data files. The publisher plans to produce 20 texts in the business area and randomly chooses 10 to have more than 100 data files. The remaining 10 are produced with at most 100 data files. For those with more than 100, first-year sales averaged 9,254, and the sample standard deviation was 2,107 . For the books with at most 100, average first-year sales were 8,167 , and the sample standard deviation was 1,681. Assuming that the two population distributions are normal with the same variance, test the null hypothesis that the population means are equal against the alternative that the true mean is higher for books with more than 100 data files.

Rashmi Sinha

Numerade Educator

Problem 14

Test the hypotheses
$$
\begin{aligned}
& H_0: P_x-P_y=0 \\
& H_1: P_x-P_y<0
\end{aligned}
$$
using the following statistics from random samples.
a.
$$
\begin{aligned}
& \hat{p}_x=0.42, n_x=500 \\
& \hat{p}_y=0.50, n_y=600
\end{aligned}
$$
b.
$$
\begin{aligned}
& \hat{p}_x=0.60, n_x=500 ; \\
& \hat{p}_y=0.64, n_y=600
\end{aligned}
$$
c.
$$
\begin{aligned}
& \hat{p}_x=0.42, n_x=500 ; \\
& \hat{p}_y=0.49, n_y=600
\end{aligned}
$$
d.
$$
\begin{aligned}
& \hat{p}_x=0.25, n_x=500 ; \\
& \hat{p}_y=0.34, n_y=600 \\
& \hat{p}_x=0.39, n_x=500 ; \\
& \hat{p}_y=0.42, n_y=600
\end{aligned}
$$
e.
$$
\begin{aligned}
& \hat{p}_x=0.39, n_x=500 \\
& \hat{p}_y=0.42, n_y=600
\end{aligned}
$$

Rashmi Sinha

Numerade Educator

Problem 15

Random samples of 900 people in the United States and in Great Britain indicated that $60 \%$ of the people in the United States were positive about the future economy, whereas $66 \%$ of the people in Great Britain were positive about the future economy. Does this provide strong evidence that the people in Great Britain are more optimistic about the economy?

Sheryl Ezze

Numerade Educator

Problem 16

A random sample of 1,556 people in country A were asked to respond to this statement: Increased world trade can increase our per capita prosperity. Of these sample members, $38.4 \%$ agreed with the statement. When the same statement was presented to a random sample of 1,108 people in country B, $52.0 \%$ agreed. Test the null hypothesis that the population proportions agreeing with this statement were the same in the two countries against the alternative that a higher proportion agreed in country $B$.

Sheryl Ezze

Numerade Educator

Problem 17

Small-business telephone users were surveyed 6 months after access to carriers other than AT\&T became available for wide-area telephone service. Of a random sample of 368 users, 92 said they were attempting to learn more about their options, as did 37 of an independent random sample of 116 users of alternative carriers. Test, at the $5 \%$ significance level against a two-sided alternative, the null hypothesis that the two population proportions are the same.

Sheryl Ezze

Numerade Educator

Problem 18

Employees of a building materials chain facing a shutdown were surveyed on a prospective employee ownership plan. Some employees pledged $$\$ 10,000$$ to this plan, putting up $$\$ 800$$ immediately, while others indicated that they did not intend to pledge. Of a random sample of 175 people who had pledged, $78 \mathrm{had}$ already been laid off, whereas 208 of a random sample of 604 people who had not pledged had already been laid off. Test, at the $5 \%$ level against a two-sided alternative, the null hypothesis that the population proportions already laid off were the same for people who pledged as for those who did not.

Sheryl Ezze

Numerade Educator

Problem 19

Of a random sample of 381 high-quality investment equity options, 191 had less than $30 \%$ debt. Of an independent random sample of 166 high-risk investment equity options, 145 had less than $30 \%$ debt. Test, against a two-sided alternative, the null hypothesis that the two population proportions are equal.

Sheryl Ezze

Numerade Educator

Problem 20

Two different independent random samples of consumers were asked about satisfaction with their computer system each in a slightly different way. The options available for answer were slightly different in the two cases. When asked how satisfied they were with their computer system, 138 of the first group of 240 sample members opted for "very satisfied." When the second group was asked how dissatisfied they were with their computer system, 128 of 240 sample members opted for very satisfied. Test, at the $5 \%$ significance level against the obvious one-sided alternative, the null hypothesis that the two population proportions are equal.

Sheryl Ezze

Numerade Educator

Problem 21

Of a random sample of 1,200 people in Denmark, 480 had a positive attitude toward car salespeople. Of an independent random sample of 1,000 people in France, 790 had a positive attitude toward car salespeople. Test, at the $1 \%$ level the null hypothesis that the population proportions are equal, against the alternative that a higher proportion of French have a positive attitude toward car salespeople.

Sheryl Ezze

Numerade Educator

Problem 22

Test the hypothesis
$$
\begin{aligned}
& H_0: \sigma_x^2=\sigma_y^2 \\
& H_1: \sigma_x^2>\sigma_y^2
\end{aligned}
$$
using the following data.
a. $s_x^2=125, n_y=45 ; s_y^2=51, n_y=41$
b. $s_x^2=125, n_y=45 ; s_y^2=235, n_y=44$
c. $s_x^2=134, n_y=48 ; s_y^2=51, n_y=41$
d. $\mathrm{s}_x^2=88, n_y=39 ; \mathrm{s}_y^2=167, n_y=25$

Rashmi Sinha

Numerade Educator

Problem 23

It is hypothesized that the more expert a group of people examining personal income tax filings, the more variable the judgments will be about the accuracy. Independent random samples, each of 30 individuals, were chosen from groups with different levels of expertise. The low-expertise group consisted of people who had just completed their first intermediate accounting course. Members of the high-expertise group had completed undergraduate studies and were employed by reputable CPA firms. The sample members were asked to judge the accuracy of personal income tax filings. For the low-expertise group, the sample variance was 451.770 , whereas for the high-expertise group, it was 1,614.208. Test the null hypothesis that the two population variances are equal against the alternative that the true variance is higher for the high-expertise group.

Sheryl Ezze

Numerade Educator

Problem 24

It is hypothesized that the total sales of a corporation should vary more in an industry with active price competition than in one with duopoly and tacit collusion. In a study of the merchant ship production industry it was found that in 4 years of active price competition, the variance of company A's total sales was 114.09 . In the following 7 years, during which there was duopoly and tacit collusion, this variance was 16.08. Assume that the data can be regarded as an independent random sample from two normal distributions. Test, at the $5 \%$ level, the null hypothesis that the two population variances are equal against the alternative that the variance of total sales is higher in years of active price competition.

Sheryl Ezze

Numerade Educator

Problem 25

In light of a number of recent large-corporation bankruptcies, auditors are becoming increasingly concerned about the possibility of fraud. Auditors might be helped in determining the chances of fraud if they carefully measure cash flow. To evaluate this possibility, samples of midlevel auditors from CPA firms were presented with cash-flow information from a fraud case, and they were asked to indicate the chance of material fraud on a scale from 0 to 100 . A random sample of 36 auditors used the cash-flow information. Their mean assessment was 36.21 , and the sample standard deviation was 22.93 . For an independent random sample of 36 auditors not using the cash-flow information, the sample mean and standard deviation were respectively 47.56 and 27.56 .
Test the assumption that population variances for assessments of the chance of material fraud were the same for auditors using cash-flow information as for auditors not using cash-flow information against a two-sided alternative hypothesis.

Sheryl Ezze

Numerade Educator

Problem 26

A publisher is interested in the effects on sales of college texts that include more than 100 data files. The publisher plans to produce 20 texts in the business area and randomly chooses 10 to have more than 100 data files. The remaining 10 are produced with at most 100 data files. For those with more than 100 , first-year sales averaged 9,254, and the sample standard deviation was 2,107 . For the books with at most 100 , average first-year sales were 8,167, and the sample standard deviation was 1,681. Assuming that the two population distributions are normal, test the null hypothesis that the population variances are equal against the alternative that the population variance is higher for books with more than 100 data files.

Rashmi Sinha

Numerade Educator

Problem 27

A university research team was studying the relationship between idea generation by groups with and without a moderator. For a random sample of four groups with a moderator, the mean number of ideas generated per group was 78.0, and the standard deviation was 24.4 . For a random sample of four groups without a moderator, the mean number of ideas generated was 63.5 , and the standard deviation was 20.2. Test the assumption that the two population variances were equal against the alternative that the population variance is higher for groups with a moderator.

Sheryl Ezze

Numerade Educator

Problem 28

A statistician tests the null hypothesis that the proportion of men favoring a tax reform proposal is the same as the proportion of women. Based on sample data, the null hypothesis is rejected at the $5 \%$ significance level. Does this imply that the probability is at least 0.95 that the null hypothesis is false? If not, provide a valid probability statement.

Sheryl Ezze

Numerade Educator

Problem 29

In a study of performance ratings of ex-smokers, a random sample of 34 ex-smokers had a mean rating of 2.21 and a sample standard deviation of 2.21. For an independent random sample of 86 long-term ex-smokers, the mean rating was 1.47 and the sample standard deviation was 1.69 . Find the lowest level of significance at which the null hypothesis of equality of the two population means can be rejected against a two-sided alternative.

Sheryl Ezze

Numerade Educator

Problem 30

Independent random samples of business managers and college economics faculty were asked to respond on a scale from 1 (strongly disagree) to 7 (strongly agree) to this statement: Grades in advanced economics are good indicators of students' analytical skills. For a sample of 70 business managers, the mean response was 4.4 and the sample standard deviation was 1.3. For a sample of 106 economics faculty the mean response was 5.3 and the sample standard deviation was 1.4.
a. Test, at the $5 \%$ level, the null hypothesis that the population mean response for business managers would be at most 4.0 .
b. Test, at the $5 \%$ level, the null hypothesis that the population means are equal against the alternative that the population mean response is higher for economics faculty than for business managers.

Sheryl Ezze

Numerade Educator

Problem 31

Independent random samples of bachelor's and master's degree holders in statistics, whose initial job was with a major actuarial firm and who subsequently moved to an insurance company, were questioned. For a sample of 44 bachelor's degree holders, the mean number of months before the first job change was 35.02 and the sample standard deviation was 18.20 . For a sample of 68 master's degree holders, the mean number of months before the first job change was 36.34 and the sample standard deviation was 18.94 . Test, at the $10 \%$ level against a two-sided alternative, the null hypothesis that the population mean numbers of months before the first job change are the same for the two groups.

Sheryl Ezze

Numerade Educator

Problem 32

A study was aimed at assessing the effects of group size and group characteristics on the generation of advertising concepts. To assess the influence of group size, groups of four and eight members were compared. For a random sample of four-member groups, the mean number of advertising concepts generated per group was 78.0 and the sample standard deviation was 24.4 . For an independent random sample of eight-member groups, the mean number of advertising concepts generated per group was 114.7 and the sample standard deviation was 14.6. (In each case, the groups had a moderator.) Stating any assumptions that you need to make, test, at the $1 \%$ level, the null hypothesis that the population means are the same against the alternative that the mean is higher for eight-member groups.

Sheryl Ezze

Numerade Educator

Problem 33

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 above 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. In this project you are to determine if the consumption of these food groups are greater in the metro compared to the non-metro counties. As part of your research you have developed the data file Food Nutrition Atlas-described in the Chapter 9 appendix - which contains a number of nutrition and population variables collected by county over all states. It is true that some counties do not report all of the variables. Perform an analysis using 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.

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Problem 34

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 there is a difference in these two obesity rates in metro versus nonmetro counties. Use the data file Food Nutrition Atlas-described in the Chapter 9 appendix-as the basis for your statisti$\mathrm{cal}$ analysis. Prepare a rigorous analysis and a short statement that reports your statistical results and your conclusions.

Lucas Finney

Numerade Educator

Problem 35

Independent random samples of business and economics faculty were asked to respond on a scale from 1 (strongly disagree) to 4 (strongly agree) to this statement. The threat and actuality of takeovers of publicly held companies provide discipline for boards and managers to maximize the value of the company to shareholders. For a sample of 202 business faculty, the mean response was 2.83 and the sample standard deviation was 0.89 . For a sample of 291 economics faculty, the mean response was 3.00 and the sample standard deviation was 0.67. Test the null hypothesis that the population means are equal against the alternative that the mean is higher for economics faculty.

Sheryl Ezze

Numerade Educator

Problem 36

Independent random samples of patients who had received knee and hip replacement were asked to assess the quality of service on a scale from 1 (low) to 7 (high). For a sample of 83 knee patients, the mean rating was 6.543 and the sample standard deviation was 0.649 . For a sample of 54 hip patients, the mean rating was 6.733 and the sample standard deviation was 0.425 . Test, against a two-sided alternative, the null hypothesis that the population mean ratings for these two types of patients are the same.

Sheryl Ezze

Numerade Educator

Problem 37

Of a random sample of 148 accounting majors, 75 rated a sense of humor as a very important trait to their career performance. This same view was held by 81 of an independent random sample of 178 finance majors.
a. Test, at the $5 \%$ level, the null hypothesis that at least one-half of all finance majors rate a sense of humor as very important.
b. Test, at the $5 \%$ level against a two-sided alternative, the null hypothesis that the population proportions of accounting and finance majors who rate a sense of humor as very important are the same.

Sheryl Ezze

Numerade Educator

Problem 38

Aimed at finding substantial earnings decreases, a random sample of 23 firms with substantial earnings decreases showed that the mean return on assets 3 years previously was 0.058 and the sample standard deviation was 0.055 . An independent random sample of 23 firms without substantial earnings decreases showed a mean return of 0.146 and a standard deviation 0.058 for the same period. Assume that the two population distributions are normal with equal standard deviations. Test, at the $5 \%$ level, the null hypothesis that the population mean returns on assets are the same against the alternative that the true mean is higher for firms without substantial earnings decreases.

Sheryl Ezze

Numerade Educator

Problem 39

Random samples of employees were drawn in fastfood restaurants where the employer provides a training program. Of a sample of 67 employees who had not completed high school, 11 had participated in a training program provided by their current employer. Of an independent random sample of 113 employees who had completed high school but had not attended college, 27 had participated. Test, at the $1 \%$ level, the null hypothesis that the participation rates are the same for the two groups against the alternative that the rate is lower for those who have not completed high school.

Sheryl Ezze

Numerade Educator

Problem 40

Of a random sample of 69 health insurance firms, 47 did public relations in-house, as did 40 of an independent random sample of 69 casualty insurance firms. Find and interpret the $p$-value of a test of equality of the population proportions against a two-sided alternative.

Sheryl Ezze

Numerade Educator

Problem 41

Independent random samples were taken of male and female clients of University Entrepreneurship Centers. These clients were considering starting a business. Of 94 male clients, 53 actually started a business venture, as did 47 of 68 female clients. Find and interpret the $p$-value of a test of equality of the population proportions against the alternative that the proportion of female clients actually starting a business is higher than the proportion of male clients.

Sheryl Ezze

Numerade Educator

Problem 42

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 there is a difference in these two obesity rates in metro versus nonmetro counties. Your analysis is restricted to counties in the following states; California, Michigan, Minnesota, and Florida. Conduct your analysis for each state. Use the data file Food Nutrition Atlas-described in the Chapter 9 appendix - as the basis for your statistical analysis. You will first need to obtain a subset of the data file using the capabilities of your statistical analysis computer program.
Prepare a rigorous analysis and a short statement that reports your statistical results and your conclusions.

Lucas Finney

Numerade Educator

Problem 43

National education officials are concerned that there may be a large number of low-income students who are eligible for free lunches in their schools. They also believe that the percentage of students eligible for free lunches is larger in rural areas. As part of a larger research study, you have been asked to determine if rural counties have a greater percentage of students eligible for free lunches compared to urban residents. As your study begins you obtain the data file Food Nutrition Atlas-described in the Chapter 9 appendix-which contains a number of health and nutrition variables measured over counties in the United States. Perform an analysis to determine if there is strong evidence to conclude that rural residents have higher rates of free-lunch eligibility and prepare a short report on your results.

Jessica Waggener

Jessica Waggener

Numerade Educator

Problem 44

You are in charge of rural economic development in a rapidly developing country that is using its newfound oil wealth to develop the entire country. As part of your responsibility you have been asked to determine if there is evidence that the new rice-growing procedures have increased output per hectare. A random sample of 27 fields was planted using the old procedure, and the sample mean output was 60 per hectare with a sample variance of 100 . During the second year the new procedure was applied to the same fields and the sample mean output was 64 per hectare, with a sample variance of 150 . The sample correlation between the two fields was 0.38 . The population variances are assumed to be equal, and that assumption should be used for the problem analysis.
a. Use a hypothesis test with a probability of Type I error $=0.05$ to determine if there is strong evidence to support the conclusion that the new process leads to higher output per hectare, and interpret the results.
b. Under the assumption that the population variances are equal, construct a $95 \%$ acceptance interval for the ratio of the sample variances. Do the observed sample variances lead us to conclude that the population variances are the same? Please explain.

Tyler Moulton

Numerade Educator

Problem 45

The president of Amalgamated Retailers International, Samiha Peterson, has asked for your assistance in studying the market penetration for the company's new cell phone. You are asked to study two markets and determine if the difference in market share remains the same. Historically, in market 1 in western Poland, Amalgamated has had a 30\% market share. Similarly, in market 2 in southern Austria, Amalgamated has had a $35 \%$ market share. You obtain a random sample of potential customers from each area. From market 1, 258 out of a total sample of 800 indicate they will purchase from Amalgamated. From market 2, 260 out of 700 indicate they will purchase from Amalgamated.
a. Using a probability of error $\alpha=0.03$, test the hypothesis that the market shares are equal versus the hypothesis that they are not equal (market 2 -market 1).
b. Using a probability of error $\alpha=0.03$, test the hypothesis that the market shares are equal versus the hypothesis that the share in market 2 is larger.

Sheryl Ezze

Numerade Educator

Problem 46

National education officials are concerned that there may be a large number of low-income students who are eligible for free lunches in their schools. They also believe that the percentage of students eligible for free lunches is larger in rural areas.
As part of a larger research study you have been asked to determine if rural counties have a greater percentage of students eligible for free lunches compared to urban residents. In this part of the study you are to answer the free-lunch-eligibility question for each of the three states, California, Texas, and Florida. For this study you will have to learn how to create subsets from large data files using your local statistical package. Assistance for that effort can be obtained from your professor, teaching assistant, the Help option in your statistical package, or similar sources. As your study begins, you obtain the data file Food Nutrition Atlasdescribed in the Chapter 9 appendix - which contains a number of health and nutrition variables measured over counties in the United States. Perform an analysis to determine if there is strong evidence to conclude that rural residents have higher rates of eligibility for free lunches and prepare a short report on your results.

Jessica Waggener

Jessica Waggener

Numerade Educator

Problem 47

You are the product manager for brand 4 in a large food company. The company president has complained that a competing brand, called brand 2, has higher average sales. The data services group has stored the latest product sales (saleb2 and saleb4) and price data (apriceb2 and apriceb4) in a file named Storet described in Chapter 10 appendix.
a. Based on a statistical hypothesis test, does the president have strong evidence to support her complaint? Show all statistical work and reasoning.
b. After analyzing the data, you note that a large outlier of value 971 is contained in the sample for brand 2. Repeat part a with this extreme observation removed. What do you now conclude about the president's complaint?

Kari Hasz

Numerade Educator

Problem 48

Joe Ortega is the product manager for Ole ice cream. You have been asked to determine if Ole ice cream has greater sales than Carl's ice cream, which is a strong competitor. The data file Ole contains weekly sales and price data for the competing brands over the year in three different supermarket chains. These sample data represent a random sample of all ice cream sales for the two brands. The variable names clearly identify the variables.
a. Design and implement an analysis to determine if there is strong evidence to conclude that Ole ice cream has higher mean sales than Carl's ice cream $(\alpha=0.05)$. Explain your procedure and show all computations. You may include Minitab output if appropriate to support your analysis. Explain your conclusions.
b. Design and implement an analysis to determine if the prices charged for the two brands are different $(\alpha=0.05)$. Carefully explain your analysis, show all computations, and interpret your results.

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Problem 49

Mary Peterson is in charge of preparing blended flour for exotic bread making. The process is to take two different types of flour and mix them together in order to achieve high-quality breads. For one of the products, flour A and flour B are mixed together. The package of flour A comes from a packing process that has a population mean weight of 8 ounces with a population variance of 0.04 . The package of flour $B$ has a population mean weight of 8 ounces and a population variance of 0.06 . The package weights have a correlation of 0.40 . The $\mathrm{A}$ and $\mathrm{B}$ packages are mixed together to obtain a 16-ounce package of special exotic flour. Every 60 minutes a random sample of four packages of exotic flour is selected from the process, and the mean weight for the four packages is computed. Prepare a $99 \%$ acceptance interval for a quality-control chart for the sample means from the sample of four packages. Show all your work and explain your reasoning. Explain how this acceptance chart would be used to ensure that the package weights continue to meet the standard.

Nick Johnson

Numerade Educator

Problem 50

A study was conducted to determine if there was a difference in humor content in British and American trade magazine advertisements. In an independent random sample of 270 American trade magazine advertisements, 56 were humorous. An independent random sample of 203 British trade magazine advertisements contained 52 humorous ads. Do these data provide evidence that there is a difference in the proportion of humorous ads in British versus American trade magazines?

Harsh Gadhiya

Numerade Educator

Problem 51

Individuals have their HEI measured on two different days with the first and second day indicated by the variable daycode. A number of researchers argue that individuals will have a higher-quality diet for the second interview because they will adjust their diet after the first interview. You are asked to perform an appropriate hypothesis test to determine if there is strong evidence to conclude that individuals have a higher $\mathrm{HEI}$ on the second day compared to the first day.

James Kiss

Numerade Educator

Problem 52

Previous research has suggested that immigrants in the United States have a stronger interest in good diet compared to the rest of the population. If true, this behavior could result from a desire for overall life improvement, historical experience from their previous country, or some other complex rationale. You have been asked to determine if immigrants (variable immigrant $=1$ ) have healthier diets compared to nonimmigrants $(=0)$. Perform an appropriate statistical test to determine if there is strong evidence to conclude that immigrants have better diets compared to natives.
You will do the analysis based first on the data from the first interview, create subsets of the data file using daycode $=1$; then a second time, using data from the second interview, create subsets of the data file using daycode $=2$. Note differences in the results between the first and second interviews.

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Problem 53

There is an increasing interest in healthier lifestyles, especially among the younger population. This is exhibited in the increased interest in exercise and a variety of emphases on eating foods that contribute to a higher-quality diet. You have been asked to determine if people who are physically active (variable activity level $=2$ or 3 ) have healthier diets compared to those who are not (variable activity level = 1). Determine if there is strong evidence for your conclusion. You will do the analysis based first on the data from the first interview and create subsets of the data file using daycode $=1$, and then a second time using data from the second interview, creating subsets of the data file using daycode $=2$. Note differences in the results between the first and second interviews.

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Problem 54

Various research studies and personal lifestyle advisers argue that increased social interaction is important for a higher quality of life. You have been asked to determine if people who are single (variable single = 1) have a healthier diet than those who are married or living with a partner. Determine if there is strong evidence for your conclusion. You will do the analysis based first on the data from the first interview, creating subsets of the data file using daycode $=1$, and a second time using data from the second interview, creating subsets of the data file using daycode $=2$. Note differences in the results between the first and second interviews.

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Problem 55

Throughout society there are various claims of behavioral differences between men and women on many different characteristics. You have been asked to conduct a comparative study of diet quality between men and women. The variable female is coded 1 for females and 0 for males. Perform an appropriate analysis to determine if men and women have different dietquality levels. You will do the analysis based first on the data from the first interview by creating subsets of the data file using daycode $=1$ and then a second time using data from the second interview, creating subsets of the data file using daycode $=2$. Note differences in the results between the first and second interviews.

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Problem 56

A recent radio commentator argued that his experience indicated that women believed that purchasing higher-cost food would improve their lifestyle. Is there evidence to conclude that women have a lower daily food cost compared to men (daily-cost)? Use an appropriate test to determine the answer. You will do the analysis based first on the data from the first interview, creating subsets of the data file using daycode $=1$, and a second time using data from the second interview, creating subsets of the data file using daycode $=2$. Note differences in the results between the first and second interviews.

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Problem 57

The food stamp program has been part of a longterm public policy to ensure that lower-income families will be provided with adequate nutrition at lower cost. Some people argue that providing food income supplements will merely encourage lower-income people to purchase more expensive food, without any improvement in their diet. Perform an analysis to determine how the nutrition level of people receiving food stamps compares with the rest of the population. Is there evidence that people who receive food stamps have a higher-quality diet compared to the rest of the population? Is there evidence that they have a lower-quality diet? Is there evidence that people who receive food stamps spend more for their food compared to the rest of the population? Is there evidence that they spend less for their food? Based on your statistical analysis, what do you conclude about the food stamp program? You will do the analysis based first on the data from the first interview, creating subsets of the data file using daycode $=1$, and a second time using data from the second interview, creating subsets of the data file using daycode $=2$. Note differences in the results between the first and second interviews.

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Problem 58

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. Does the immigrant population have a lower percentage of people that are overweight compared to the remainder of the population? Provide strong evidence to support your conclusion. You will do the analysis based first on the data from the first interview, creating subsets of the data file using daycode $=1$, and a second time using data from the second interview, creating subsets of the data file using daycode $=2$. Note differences in the results between the first and second interviews.

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