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Chapter 12: 144SE (page 807)

Suppose you fit the regression model Ey=β0+β1x1+β2x2+β3x2+β4x1x2+β5x1x222 to n = 35 data points and wish to test the null hypothesisH0:β4=β5=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 α = .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 β4 or β5 is nonzero.
  2. The F-statistic to check the goodness of fit of the model can be computed byF 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 α = 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 β4 or β5 is nonzero.

02

F-statistic

The F-statistic to check the goodness of fit of the model can be computed by

F test statistic =SSEn-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 α = 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

Question: Accuracy of software effort estimates. Periodically, software engineers must provide estimates of their effort in developing new software. In the Journal of Empirical Software Engineering (Vol. 9, 2004), multiple regression was used to predict the accuracy of these effort estimates. The dependent variable, defined as the relative error in estimating effort, y = (Actual effort - Estimated effort)/ (Actual effort) was determined for each in a sample of n = 49 software development tasks. Eight independent variables were evaluated as potential predictors of relative error using stepwise regression. Each of these was formulated as a dummy variable, as shown in the table.

Company role of estimator: x1 = 1 if developer, 0 if project leader

Task complexity: x2 = 1 if low, 0 if medium/high

Contract type: x3 = 1 if fixed price, 0 if hourly rate

Customer importance: x4 = 1 if high, 0 if low/medium

Customer priority: x5 = 1 if time of delivery, 0 if cost or quality

Level of knowledge: x6 = 1 if high, 0 if low/medium

Participation: x7 = 1 if estimator participates in work, 0 if not

Previous accuracy: x8 = 1 if more than 20% accurate, 0 if less than 20% accurate

a. In step 1 of the stepwise regression, how many different one-variable models are fit to the data?

b. In step 1, the variable x1 is selected as the best one- variable predictor. How is this determined?

c. In step 2 of the stepwise regression, how many different two-variable models (where x1 is one of the variables) are fit to the data?

d. The only two variables selected for entry into the stepwise regression model were x1 and x8. The stepwise regression yielded the following prediction equation:

Give a practical interpretation of the β estimates multiplied by x1 and x8.

e) Why should a researcher be wary of using the model, part d, as the final model for predicting effort (y)?

Question:If the analysis of variance F-test leads to the conclusion that at least one of the model parameters is nonzero, can you conclude that the model is the best predictor for the dependent variable ? Can you conclude that all of the terms in the model are important for predicting ? What is the appropriate conclusion?

Question: Shopping on Black Friday. Refer to the International Journal of Retail and Distribution Management (Vol. 39, 2011) study of shopping on Black Friday (the day after Thanksgiving), Exercise 6.16 (p. 340). Recall that researchers conducted interviews with a sample of 38 women shopping on Black Friday to gauge their shopping habits. Two of the variables measured for each shopper were age (x) and number of years shopping on Black Friday (y). Data on these two variables for the 38 shoppers are listed in the accompanying table.

  1. Fit the quadratic model, E(y)=β0+β1x+β2x2, to the data using statistical software. Give the prediction equation.
  2. Conduct a test of the overall adequacy of the model. Use α=0.01.
  3. Conduct a test to determine if the relationship between age (x) and number of years shopping on Black Friday (y) is best represented by a linear or quadratic function. Use α=0.01.

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

Personality traits and job performance. When attempting to predict job performance using personality traits, researchers typically assume that the relationship is linear. A study published in the Journal of Applied Psychology (Jan. 2011) investigated a curvilinear relationship between job task performance and a specific personality trait—conscientiousness. Using data collected for 602 employees of a large public organization, task performance was measured on a 30-point scale (where higher scores indicate better performance) and conscientiousness was measured on a scale of -3 to +3 (where higher scores indicate a higher level of conscientiousness).

a. The coefficient of correlation relating task performance score to conscientiousness score was reported as r = 0.18. Explain why the researchers should not use this statistic to investigate the curvilinear relationship between task performance and conscientiousness.

b. Give the equation of a curvilinear (quadratic) model relating task performance score (y) to conscientiousness score (x).

c. The researchers theorized that task performance increases as level of conscientiousness increases, but at a decreasing rate. Draw a sketch of this relationship.

d. If the theory in part c is supported, what is the expected sign ofβ2in the model, part b?

e. The researchers reportedβ^2=0.32with an associated p-value of less than 0.05. Use this information to test the researchers’ theory atα=0.05
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