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Chapter 12: Q98E (page 777)

Question: Suppose you fit the regression modelE(y)=β0+β1x1+β2x2+β3x1+β4x12+β5x22to n = 30 data points and wish to test H0: β3 = β4 = β5 = 0

a. State the alternative hypothesis Ha.

b. Give the reduced model appropriate for conducting the test.

c. What are the numerator and denominator degrees of freedom associated with the F-statistic?

d. Suppose the SSE’s for the reduced and complete models are SSER = 1,250.2 and SSEC = 1,125.2. Conduct the hypothesis test and interpret the results of your test. Test using α = .05.

Short Answer

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Answer

a. The alternate hypothesis would be at least one of the β parameters under test is nonzero.

b. The reduced model under consideration here is y=β0+β1x1+β2x2.

c. In the numerator there’s (k-g) and in the denominator there are [n-(k+1)] degrees of freedom where (k – g) = Number of b parameters specified in H0 (i.e., number of β parameters tested), k + 1 = Number of β parameters in the complete model (including β0), and n = Total sample size.

d. At 95% confidence interval there is enough evidence to not reject H0. Hence, at least one of β parameters are nonzero.

Step by step solution

01

Alternate hypothesis

The alternate hypothesis would be at least one of the β parameters under test is nonzero.

02

Reduced model

The reduced model under consideration here is y=β0+β1x1+β2x2.

03

Degrees of freedom

In the numerator there’s (k-g) and in the denominator there are [n-(k+1)] degrees of freedom

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Most popular questions from this chapter

The Minitab printout below was obtained from fitting the modely=β0+β1x1+β2x2+β3x1x2+εto 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 α=0.01.

When a multiple regression model is used for estimating the mean of the dependent variable and for predicting a new value of y, which will be narrower—the confidence interval for the mean or the prediction interval for the new y-value? Why?


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=β0+β1x1+β2x2+β3x1x2+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: 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: Cooling method for gas turbines. Refer to the Journal of Engineering for Gas Turbines and Power (January 2005) study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 12.19 (p. 726). Consider a model for heat rate (kilojoules per kilowatt per hour) of a gas turbine as a function of cycle speed (revolutions per minute) and cycle pressure ratio. The data are saved in the file.

a. Write a complete second-order model for heat rate (y).

b. Give the null and alternative hypotheses for determining whether the curvature terms in the complete second-order model are statistically useful for predicting heat rate (y).

c. For the test in part b, identify the complete and reduced model.

d. The complete and reduced models were fit and compared using SPSS. A summary of the results are shown in the accompanying SPSS printout. Locate the value of the test statistic on the printout.

e. Find the rejection region for α = .10 and locate the p-value of the test on the printout.

f. State the conclusion in the words of the problem.
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