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Chapter 12: Q144SE (page 807)

Suppose you fit the regression model Ey=尾0+尾1x1+尾2x2+尾3x22+尾4x1x2+尾5x1x222 to n = 35 data points and wish to test the null hypothesis H0:尾4=尾5=0

  1. State the alternative hypothesis.

  2. Explain in detail how to compute the F-statistic needed to test the null hypothesis.

  3. What are the numerator and denominator degrees of freedom associated with the F-statistic in part b?

  4. Give the rejection region for the test if 伪 = .05.

Short Answer

Expert verified

  1. The alternate hypothesis to test the significance of interaction terms would be Ha: At least one of the parameters 尾4or 尾5is nonzero.

  2. The F-statistic to check the goodness of fit of the model can be computed by F test statistic =SSEn-(k+1).

  3. In part b, the degrees of freedom for numerator is (n-k) while the degree of freedom for denominator is [n-(k+1)].

  4. When 伪 = 0.05, the rejection region for the significance of interaction terms can be defined when the t-statistic < t0.025, n-1.

Step by step solution

01

Alternate hypothesis

The alternate hypothesis to test the significance of interaction terms would be Ha: At least one of the parameters 尾4 or 尾5 is nonzero.

02

F-statistic

The F-statistic to check the goodness of fit of the model can be computed by F test statistic = SSE.n-(k+1)

03

Degrees of freedom

In part b, the degrees of freedom for numerator is (n-k) while the degree of freedom for denominator is [n-(k+1)].

04

Rejection region


When 伪 = 0.05, the rejection region for the significance of interaction terms can be defined when the t-statistic < t0.025, n-1.

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Most popular questions from this chapter

State casket sales restrictions. Some states permit only licensed firms to sell funeral goods (e.g., caskets, urns) to the consumer, while other states have no restrictions. States with casket sales restrictions are being challenged in court to lift these monopolistic restrictions. A paper in the Journal of Law and Economics (February 2008) used multiple regression to investigate the impact of lifting casket sales restrictions on the cost of a funeral. Data collected for a sample of 1,437 funerals were used to fit the model. A simpler version of the model estimated by the researchers is E(y)=尾0+尾1x1+尾2x2+尾3x1x2, where y is the price (in dollars) of a direct burial, x1 = {1 if funeral home is in a restricted state, 0 if not}, and x2 = {1 if price includes a basic wooden casket, 0 if no casket}. The estimated equation (with standard errors in parentheses) is:

y^=1432 + 793x1- 252x2+ 261x1x2, R2= 0.78

(70) (134) (109)

  1. Calculate the predicted price of a direct burial with a basic wooden casket at a funeral home in a restricted state.

  2. The data include a direct burial funeral with a basic wooden casket at a funeral home in a restricted state that costs \(2,200. Assuming the standard deviation of the model is \)50, is this data value an outlier?

  3. The data also include a direct burial funeral with a basic wooden casket at a funeral home in a restricted state that costs \(2,500. Again, assume that the standard deviation of the model is \)50. Is this data value an outlier?

The Minitab printout below was obtained from fitting the modely=尾0+尾1x1+尾2x2+尾3x1x2+蔚to n = 15 data points.

a) What is the prediction equation?

b) Give an estimate of the slope of the line relating y to x1 when x2 =10 .

c) Plot the prediction equation for the case when x2 =1 . Do this twice more on the same graph for the cases when x2 =3 and x2 =5 .

d) Explain what it means to say that x1and x2interact. Explain why your graph of part c suggests that x1and x2interact.

e) Specify the null and alternative hypotheses you would use to test whetherx1andx2interact.

f)Conduct the hypothesis test of part e using 伪=0.01.

Question: Study of supervisor-targeted aggression. 鈥淢oonlighters鈥� are workers who hold two jobs at the same time. What are the factors that impact the likelihood of a moonlighting worker becoming aggressive toward his/her supervisor? This was the research question of interest in the Journal of Applied Psychology (July 2005). Completed questionnaires were obtained from n = 105 moonlighters, and the data were used to fit several multiple regression models for supervisor-directed aggression score 1y2. Two of the models (with R2-values in parentheses) are given below:

a. Interpret the R2-values for the models.

b. Give the null and alternative hypotheses for comparing the fits of models 1 and 2.

c. Are the two models nested? Explain.

d. The nested F-test for comparing the two models resulted in F = 42.13 and p-value < .001. What can you conclude from these results?

e. A third model was fit, one that hypothesizes all possible pairs of interactions between self-esteem, history of aggression, interactional injustice at primary job, and abusive supervisor at primary job. Give the equation of this model (model 3).

f. A nested F-test to compare models 2 and 3 resulted in a p-value > .10. What can you conclude from this result?

Suppose you fit the quadratic model E(y)=尾0+尾1x+尾2x2to a set of n = 20 data points and found R2=0.91, SSyy=29.94, and SSE = 2.63.

a. Is there sufficient evidence to indicate that the model contributes information for predicting y? Test using a = .05.

b. What null and alternative hypotheses would you test to determine whether upward curvature exists?

c. What null and alternative hypotheses would you test to determine whether downward curvature exists?

Question: Personality traits and job performance. Refer to the Journal of Applied Psychology (Jan. 2011) study of the relationship between task performance and conscientiousness, Exercise 12.54 (p. 747). Recall that the researchers used a quadratic model to relate y = task performance score (measured on a 30-point scale) to x1 = conscientiousness score (measured on a scale of -3 to +3). In addition, the researchers included job complexity in the model, where x2 = {1 if highly complex job, 0 if not}. The complete model took the form

E(y)=尾0+尾1x1+尾2x12+尾3x2+尾4x1x2+尾5x12x2herex2=1E(y)=尾0+尾1x1+尾2x12+尾3(1)+尾4x1(1)+尾5(1)2(1)E(y)=(尾0+尾3)+(尾1+尾4)x1+(尾2+尾5)(x1)2

a. For jobs that are not highly complex, write the equation of the model for E1y2 as a function of x1. (Substitute x2 = 0 into the equation.)

b. Refer to part a. What do each of the b鈥檚 represent in the model?

c. For highly complex jobs, write the equation of the model for E(y) as a function of x1. (Substitute x2 = 1 into the equation.)

d. Refer to part c. What do each of the b鈥檚 represent in the model?

e. Does the model support the researchers鈥� theory that the curvilinear relationship between task performance score (y) and conscientiousness score (x1) depends on job complexity (x2)? Explain.
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