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Statistics For Business And Economics

James T. McClave, P. George Benson, Terry Sincich

$Math Studyset Vaia Explanations$ Math

13th Edition · 888 pages

ISBN: 9780134506593

Chapter 12: 121E (page 802)

Consider fitting the multiple regression model
$E (y) = β_{0} + β_{1} x_{1} + β_{2} x_{2} + β_{3} x_{3} + β_{4} x_{4} + β_{5} x_{5}$
A matrix of correlations for all pairs of independent variables is given below. Do you detect a multicollinearity problem? Explain.

Short Answer

Expert verified

In this question, x₄ and x₂has a correlation of 0.93 and x₄ and x₅ has a correlation of 0.86. These correlation numbers are very high indicating a strong positive relationship between x₄ and x₂and x₄ and x₅ respectively. Thus, the problem of multicollinearity exists in the model.

Step by step solution

01

Multicollinearity check

Multicollinearity is checked by checking the correlation amongst the independent variables. If there is high correlation amongst any two independent variables, it is said that the problem of multicollinearity exists in the model.

02

Application of multicollinearity check

In this question, x₄ and x₂has a correlation of 0.93 and x₄ and x₅ has a correlation of 0.86. These correlation numbers are very high indicating a strong positive relationship between x₄ and x₂and x₄ and x₅ respectively. Thus, the problem of multicollinearity exists in the model.

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Most popular questions from this chapter

Factors that impact an auditor’s judgment. A study was conducted to determine the effects of linguistic delivery style and client credibility on auditors’ judgments (Advances in Accounting and Behavioural Research, 2004). Two hundred auditors from Big 5 accounting firms were each asked to perform an analytical review of a fictitious client’s financial statement. The researchers gave the auditors different information on the client’s credibility and linguistic delivery style of the client’s explanation. Each auditor then provided an assessment of the likelihood that the client-provided explanation accounted for the fluctuation in the financial statement. The three variables of interest—credibility (x₁), linguistic delivery style (x₂) , and likelihood (y) —were all measured on a numerical scale. Regression analysis was used to fit the interaction model, $y = β_{0} + β_{1} x_{1} + β_{2} x_{2} + β_{3} x_{1} x_{2} + ε$ . The results are summarized in the table at the bottom of page.

a) Interpret the phrase client credibility and linguistic delivery style interact in the words of the problem.

b) Give the null and alternative hypotheses for testing the overall adequacy of the model.

c) Conduct the test, part b, using the information in the table.

d) Give the null and alternative hypotheses for testing whether client credibility and linguistic delivery style interact.

e) Conduct the test, part d, using the information in the table.

f) The researchers estimated the slope of the likelihood–linguistic delivery style line at a low level of client credibility 1x1 = 222. Obtain this estimate and interpret it in the words of the problem.

g) The researchers also estimated the slope of the likelihood–linguistic delivery style line at a high level of client credibility 1x1 = 462. Obtain this estimate and interpret it in the words of the problem.

Question: Job performance under time pressure. Time pressure is common at firms that must meet hard and fast deadlines. How do employees working in teams perform when they perceive time pressure? And, can this performance improve with a strong team leader? These were the research questions of interest in a study published in the Academy of Management Journal (October, 2015). Data were collected on n = 139 project teams working for a software company in India. Among the many variables recorded were team performance (y, measured on a 7-point scale), perceived time pressure (, measured on a 7-point scale), and whether or not the team had a strong and effective team leader (x₂ = 1 if yes, 0 if no). The researchers hypothesized a curvilinear relationship between team performance (y) and perceived time pressure (), with different-shaped curves depending on whether or not the team had an effective leader (x₂). A model for E(y) that supports this theory is the complete second-order model: $E (y) = β_{0} + β_{1} x_{1} + β_{2} {x_{1}}^{2} + β_{3} x_{2} + β_{4} x_{1} x_{2} + β_{5} {x_{1}}^{2} x_{2}$

a. Write the equation for E(y) as a function of x₁ when the team leader is not effective (x₂= 0).

b. Write the equation for E(y) as a function ofwhen the team leader is effective (x₂= 1).

c. The researchers reported the following b-estimates:.

$\hat{β_{0}} = 4.5, \hat{β_{1}} = 0.13, \hat{β_{3}} = 0.15, \hat{β_{4}} = 0.15 a n d \hat{β_{5}} = 0.29$ Use these estimates to sketch the two equations, parts a and b. What is the nature of the curvilinear relationship when the team leaders is not effective? Effective?

Question: Do blondes raise more funds? Refer to the Economic Letters (Vol. 100, 2008) study of whether the color of a female solicitor’s hair impacts the level of capital raised, Exercise 12.75 (p. 756). Recall that 955 households were contacted by a female solicitor to raise funds for hazard mitigation research. In addition to the household’s level of contribution (in dollars) and the hair color of the solicitor (blond Caucasian, brunette Caucasian, or minority female), the researcher also recorded the beauty rating of the solicitor (measured quantitatively, on a 10-point scale).

Write a first-order model (with no interaction) for mean contribution level, E(y), as a function of a solicitor’s hair color and her beauty rating.
Refer to the model, part a. For each hair color, express the change in contribution level for each 1-point increase in a solicitor’s beauty rating in terms of the model parameters.
Write an interaction model for mean contribution level, E(y), as a function of a solicitor’s hair color and her beauty rating.
Refer to the model, part c. For each hair color, express the change in contribution level for each 1-point increase in a solicitor’s beauty rating in terms of the model parameters.
Refer to the model; part c. Illustrate the interaction with a graph.

Question: Workplace bullying and intention to leave. Workplace bullying has been shown to have a negative psychological effect on victims, often leading the victim to quit or resign. In Human Resource Management Journal (October 2008), researchers employed multiple regression to examine whether perceived organizational support (POS) would moderate the relationship between workplace bullying and victims’ intention to leave the firm. The dependent variable in the analysis, intention to leave (y), was measured on a quantitative scale. The two key independent variables in the study were bullying (, measured on a quantitative scale) and perceived organizational support (measured qualitatively as “low,” “neutral,” or “high”).

Set up the dummy variables required to represent POS in the regression model.
Write a model for E(y) as a function of bullying and POS that hypothesizes three parallel straight lines, one for each level of POS.
Write a model for E(y) as a function of bullying and POS that hypothesizes three non-parallel straight lines, one for each level of POS.
The researchers discovered that the effect of bullying on intention to leave was greater at the low level of POS than at the high level of POS. Which of the two models, parts b and c, support these findings?

Consider the model:

$E (y) = β_{0} + β_{1} x_{1} + β_{2} x_{2} + β_{3} {x_{2}}^{2} + β_{4} x_{3} + β_{5} x_{1} {x_{2}}^{2}$

where x₂ is a quantitative model and

$x_{1} = (\begin{matrix} 1 r e c e i v e d t r e a t m e n t \\ 0 d i d n o t r e c e i v e t r e a t m e n t \end{matrix})$

The resulting least squares prediction equation is

localid="1649802968695" $\overset{⏜}{y} = 2 + x_{1} - 5 x_{2} + 3 {x_{2}}^{2} - 4 x_{3} + x_{1} {x_{2}}^{2}$

a. Substitute the values for the dummy variables to determine the curves relating to the mean value E(y) in general form.

b. On the same graph, plot the curves obtained in part a for the independent variable between 0 and 3. Use the least squares prediction equation.

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