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Chapter 12: 134SE (page 807)

Suppose you have developed a regression model to explain the relationship between y and x1, x2, and x3. The ranges of the variables you observed were as follows: 10 鈮� y 鈮� 100, 5 鈮� x1 鈮� 55, 0.5 鈮� x2 鈮� 1, and 1,000 鈮� x3 鈮� 2,000. Will the error of prediction be smaller when you use the least squares equation to predict y when x1 = 30, x2 = 0.6, and x3 = 1,300, or when x1 = 60, x2 = 0.4, and x3 = 900? Why?

Short Answer

Expert verified

Therefore, when predicting y values, the error of prediction will be smaller when x1= 30, x2 = 0.6, and x3 = 1300 since the values of independent variables are well within the range described in the question.

Step by step solution

01

Range of independent variables

The range of x1, x2, and x3 is given as 5 鈮� x1 鈮� 55, 0.5 鈮� x2 鈮� 1, and 1,000 鈮� x3 鈮� 2,000. When x1= 30, x2 = 0.6, and x3 = 1300, all the variables x1,x2 and x3are well within the range of values. While when x1 = 60, x2 = 0.4, and x3 = 900, x1and x2 are out of the range and x3 is within the range.

02

Conclusion

Therefore, when predicting y values, the error of prediction will be smaller when x1= 30, x2 = 0.6, and x3 = 1300.

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Most popular questions from this chapter

Production technologies, terroir, and quality of Bordeaux wine. In addition to state-of-the-art technologies, the production of quality wine is strongly influenced by the natural endowments of the grape-growing region鈥攃alled the 鈥渢erroir.鈥� The Economic Journal (May 2008) published an empirical study of the factors that yield a quality Bordeaux wine. A quantitative measure of wine quality (y) was modeled as a function of several qualitative independent variables, including grape-picking method (manual or automated), soil type (clay, gravel, or sand), and slope orientation (east, south, west, southeast, or southwest).

  1. Create the appropriate dummy variables for each of the qualitative independent variables.
  2. Write a model for wine quality (y) as a function of grape-picking method. Interpret the尾鈥檚 in the model.
  3. Write a model for wine quality (y) as a function of soil type. Interpret the尾鈥檚 in the model.
  4. Write a model for wine quality (y) as a function of slope orientation. Interpret the尾鈥檚 in the model.

Write a model relating E(y) to one qualitative independent variable that is at four levels. Define all the terms in your model.

Question: Personality traits and job performance. Refer to the Journal of Applied Psychology (January 2011) study of the relationship between task performance and conscientiousness, Exercise 12.94 (p. 766). Recall that y = task performance score (measured on a 30-point scale) was modeled as a function of x1 = conscientiousness score (measured on a scale of -3 to +3) and x2 = {1 if highly complex job, 0 if not} using the complete model

E(y)=尾0+尾1x1+尾2(x1)2+尾3x2+尾4x1x2+尾5(x1)2x2

a. Specify the null hypothesis for testing the overall adequacy of the model.

b. Specify the null hypothesis for testing whether task performance score (y) and conscientiousness score (x1) are curvilinearly related.

c. Specify the null hypothesis for testing whether the curvilinear relationship between task performance score (y) and conscientiousness score (x1) depends on job complexity (x2).

Explain how each of the tests, parts a鈥揷, should be conducted (i.e., give the forms of the test statistic and the reduced model).

Question: Shared leadership in airplane crews. Refer to the Human Factors (March 2014) study of shared leadership by the cockpit and cabin crews of a commercial airplane, Exercise 8.14 (p. 466). Recall that simulated flights were taken by 84 six-person crews, where each crew consisted of a 2-person cockpit (captain and first officer) and a 4-person cabin team (three flight attendants and a purser.) During the simulation, smoke appeared in the cabin and the reactions of the crew were monitored for teamwork. One key variable in the study was the team goal attainment score, measured on a 0 to 60-point scale. Multiple regression analysis was used to model team goal attainment (y) as a function of the independent variables job experience of purser (x1), job experience of head flight attendant (x2), gender of purser (x3), gender of head flight attendant (x4), leadership score of purser (x5), and leadership score of head flight attendant (x6).

a. Write a complete, first-order model for E(y) as a function of the six independent variables.

b. Consider a test of whether the leadership score of either the purser or the head flight attendant (or both) is statistically useful for predicting team goal attainment. Give the null and alternative hypotheses as well as the reduced model for this test.

c. The two models were fit to the data for the n = 60 successful cabin crews with the following results: R2 = .02 for reduced model, R2 = .25 for complete model. On the basis of this information only, give your opinion regarding the null hypothesis for successful cabin crews.

d. The p-value of the subset F-test for comparing the two models for successful cabin crews was reported in the article as p 6 .05. Formally test the null hypothesis using 伪 = .05. What do you conclude?

e. The two models were also fit to the data for the n = 24 unsuccessful cabin crews with the following results: R2 = .14 for reduced model, R2 = .15 for complete model. On the basis of this information only, give your opinion regarding the null hypothesis for unsuccessful cabin crews.

f. The p-value of the subset F-test for comparing the two models for unsuccessful cabin crews was reported in the article as p < .10. Formally test the null hypothesis using 伪 = .05. What do you conclude?

Question:Consider the first-order model equation in three quantitative independent variables E(Y)=2-3x1+5x2-x3

  1. Graph the relationship between Y and x3for x1=2 and x2=1
  2. Repeat part a for x1=1and x2=-2
  3. How do the graphed lines in parts a and b relate to each other? What is the slope of each line?
  4. If a linear model is first-order in three independent variables, what type of geometric relationship will you obtain when is graphed as a function of one of the independent variables for various combinations of the other independent variables?
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