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87E

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Statistics For Business And Economics

Found in: Page 501

Book edition 13th

Author(s) James T. McClave, P. George Benson, Terry Sincich

Pages 888 pages

ISBN 9780134506593

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Short Answer

Shopping vehicle and judgment. Refer to the Journal of Marketing Research (December 2011) study of shopping cart design, Exercise 2.85 (p. 112). Recall that design engineers want to know whether the mean choice of the vice-over-virtue score is higher when a consumer’s arm is flexed (as when carrying a shopping basket) than when the consumer’s arm is extended (as when pushing a shopping cart). The average choice score for the n1 = 11 consumers with a flexed arm was $\bar{x} 1$ = 59, while the average for the n2 = 11 Consumers with an extended arm was $\bar{x} 2$ = 43. In which scenario is the assumption required for a t-test to compare means more likely to be violated, $S 1$ = 4 and $S_{2}$ = 2, or, $S_{1}$ = 10 and $S_{2}$ = 15? Explain.

The answer can be reduced from the following steps.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given information

Referring to Exercises 8.8 and 8.9, there is no difference between the n1 and n2 consumers' average scores. The sample mean of consumers with a flexed arm and extended arm was different, that is $\bar{x} 1$ = 59, $\bar{x} 2$ = 43. The standard deviation in consumers with a flexed arm and consumers with an extended are dissimilar.

Step 2: Explaining the t-test

A statistical test called a t-test is employed to contrast the means of two clusters. It is frequently employed in hypothesis testing to establish whether one procedure or treatment affects the target group or even if two groups vary.

The two different kinds of t-tests are as follows.

Use a two-tailed t-test if the only thing that matters is how the two populations vary from each other.
Use a one-tailed t-test to determine if one population mean is higher or lower compared to the other.

Step 3: Validation of the t-test's premise

t-test hypotheses are as follows

$H 0 : μ 1 = μ 2 H 1 : μ 1 \neq μ 2$

The application of test statistics is as follows.

$t = \frac{(\bar{x} 1 - \bar{x} 2) - (μ 1^{} - μ 2)}{\sqrt{\frac{S 1^{2}}{n 1} + \frac{S 2^{2}}{n 2}}}$

Following is test statistics with the degree of freedom.

$d f = n 1 - 1$ and $d f = n_{2} - 1$ are similar because of the mean of consumers with a flexed arm and extended arm are similar. As $H_{0} : μ 1 = μ 2$ and $n_{1} = n_{2}$ , reject the claim that the value of t is higher. Therefore, the hypothesis will most likely be rejected if $S_{1}$ and $S_{2}$ are lower.

Case-1: If $S_{1} = 4$ and $S_{2} = 2$

The test statistics is as follows.

$t = \frac{59 - 43}{\sqrt{\frac{4^{2}}{11} + \frac{2^{2}}{11}}} t = \frac{16}{\sqrt{\frac{20}{11}}} t = \sqrt[8]{\frac{11}{5}} t = 11.866$

Case-2: If $S_{1} = 10$ and $S_{2} = 15$

$t = \frac{59 - 43}{\sqrt{\frac{10^{2}}{11} + \frac{15^{2}}{11}}} t = \sqrt[16]{\frac{11}{325}} t = 2.946$

The p-value for case 1 will be higher than case 2 as t increases. In case 1, it is more likely that the hypotheses will be rejected.

Most popular questions for Math Textbooks

Question: Refer to the Journal of Business Logistics (Vol. 36, 2015) study of the factors that lead to successful performance-based logistics projects, Exercise 2.45 (p. 95). Recall that the opinions of a sample of Department of Defense (DOD) employees and suppliers were solicited during interviews. Data on years of experience for the 6 commercial suppliers interviewed and the 11 government employees interviewed are listed in the accompanying table. Assume these samples were randomly and independently selected from the populations of DOD employees and commercial suppliers. Consider the following claim: “On average, commercial suppliers of the DOD have less experience than government employees.”

a. Give the null and alternative hypotheses for testing the claim.

b. An XLSTAT printout giving the test results is shown at the bottom of the page. Find and interpret the p-value of the test user .

c. What assumptions about the data are required for the inference, part b, to be valid? Check these assumptions graphically using the data in the PBL file.

A random sample of size n = 121 yielded $\hat{p}$ = .88.

a. Is the sample size large enough to use the methods of this section to construct a confidence interval for p? Explain.

b. Construct a 90% confidence interval for p.

c. What assumption is necessary to ensure the validity of this confidence interval?

Question: A company sent its employees to attend two different English courses. The company is interested in knowing if there is any difference between the two courses attended by its employees. When the employees returned from the courses, the company asked them to take a common test. The summary statistics of the test results of each of the two English courses are recorded in the following table:

a. Identify the parameter(s) that would help the company determine the difference between the two courses.

b. State the appropriate null and alternative hypotheses that the company would like to test.

c. After conducting the hypothesis test at the significance level, the company found the p-value . Interpret this result for the company.

A paired difference experiment produced the following results:

$n_{d} = 38, {\bar{x}}_{1} = 92, {\bar{x}}_{2} = 95.5, \bar{d} {= -3.5, s}_{d}^{2} =21$

a. Determine the values $z$ for which the null hypothesis $μ_{1} - μ_{2} = 0$ would be rejected in favor of the alternative hypothesis $μ_{1} - μ_{2} < 0$ Use .role="math" localid="1652704322912" $α = .10$

b. Conduct the paired difference test described in part a. Draw the appropriate conclusions.

c. What assumptions are necessary so that the paired difference test will be valid?

d. Find a $90 %$ confidence interval for the mean difference $μ_{d}$ .

e. Which of the two inferential procedures, the confidence interval of part d or the test of the hypothesis of part b, provides more information about the differences between the population means?

The gender diversity of a large corporation’s board of directors was studied in Accounting & Finance (December 2015). In particular, the researchers wanted to know whether firms with a nominating committee would appoint more female directors than firms without a nominating committee. One of the key variables measured at each corporation was the percentage of female board directors. In a sample of $491$ firms with a nominating committee, the mean percentage was $7.5 %$ ; in an independent sample of $501$ firms without a nominating committee, the mean percentage was role="math" localid="1652702402701" $4.3 %$ .

a. To answer the research question, the researchers compared the mean percentage of female board directors at firms with a nominating committee with the corresponding percentage at firms without a nominating committee using an independent samples test. Set up the null and alternative hypotheses for this test.

b. The test statistic was reported as $z = 5.1$ with a corresponding p-value of $0.0001$ . Interpret this result if $α = 0.05$ .

c. Do the population percentages for each type of firm need to be normally distributed for the inference, part b, to be valid? Why or why not?

d. To assess the practical significance of the test, part b, construct a $95 %$ confidence interval for the difference between the true mean percentages at firms with and without a nominating committee. Interpret the result.

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