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Chapter 12: Q103E (page 773)

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.


Short Answer

Expert verified

Answer

a. A second-order model equation in 2 independent variables can be written asy=β0+β1x1+β2x2+β3x21+β4x22.

b. The null and alternate hypothesis to test whether the complete model contributes more information for the prediction of y than the reduced model can be written as H0: β3 = β4 = 0 while Ha: At least one of β parameters are nonzero.

c. The complete and reduced model for determining whether the curvature terms can be written as y=β0+β1x1+β2x2+β3x21+β4x22and y=β0+β1x1+β2x2respectively.

d. For complete and reduced models, the value of the test statistic are 118.303 and 9.353 from the SPSS printout.

e. For α = 0.10, the rejection region is defined as p-value > α. The p-value of the test is 0.000 and 0.000.

f. For α = 0.10, the hypothesis testing will conclude if the models: complete and reduced are significant and explained by the variables.

Step by step solution

01

Second-order model equation

A second-order model equation in 2 independent variables can be written

as y=β0+β1x1+β2x2+β3x21+β4x22.

02

 Step 2: Hypotheses

The null and alternate hypothesis to test whether the complete model contributes more information for the prediction of y than the reduced model can be written as

H0: β3 = β4 = 0while Ha: At least one of β parameters are nonzero.

03

Complete and reduced model

The complete and reduced model for determining whether the curvature terms can be written as y=β0+β1x1+β2x2+β3x21+β4x22andy=β0+β1x1+β2x2 respectively.

04

Value of the test statistic

For complete and reduced models, the value of the test statistic is 118.303 and 9.353 from the SPSS printout.

05

Rejection region and p-value

For α = 0.10, the rejection region is defined as p-value > α. The p-value of the test is 0.000 and 0.000.

06

Conclusion

For α = 0.10, the hypothesis testing will conclude if the models: complete and reduced are significant and explained by the variables.

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Most popular questions from this chapter

To model the relationship between y, a dependent variable, and x, an independent variable, a researcher has taken one measurement on y at each of three different x-values. Drawing on his mathematical expertise, the researcher realizes that he can fit the second-order model Ey=β0+β1x+β2x2 and it will pass exactly through all three points, yielding SSE = 0. The researcher, delighted with the excellent fit of the model, eagerly sets out to use it to make inferences. What problems will he encounter in attempting to make inferences?

Reality TV and cosmetic surgery. Refer to the Body Image: An International Journal of Research (March 2010) study of the impact of reality TV shows on a college student’s decision to undergo cosmetic surgery, Exercise 12.17 (p. 725). Recall that the data for the study (simulated based on statistics reported in the journal article) are saved in the file. Consider the interaction model, , where y = desire to have cosmetic surgery (25-point scale), = {1 if male, 0 if female}, and = impression of reality TV (7-point scale). The model was fit to the data and the resulting SPSS printout appears below.

a.Give the least squares prediction equation.

b.Find the predicted level of desire (y) for a male college student with an impression-of-reality-TV-scale score of 5.

c.Conduct a test of overall model adequacy. Use a= 0.10.

d.Give a practical interpretation of R2a.

e.Give a practical interpretation of s.

f.Conduct a test (at a = 0.10) to determine if gender (x1) and impression of reality TV show (x4) interact in the prediction of level of desire for cosmetic surgery (y).

Question: Revenues of popular movies. The Internet Movie Database (www.imdb.com) monitors the gross revenues for all major motion pictures. The table on the next page gives both the domestic (United States and Canada) and international gross revenues for a sample of 25 popular movies.

  1. Write a first-order model for foreign gross revenues (y) as a function of domestic gross revenues (x).
  2. Write a second-order model for international gross revenues y as a function of domestic gross revenues x.
  3. Construct a scatterplot for these data. Which of the models from parts a and b appears to be the better choice for explaining the variation in foreign gross revenues?
  4. Fit the model of part b to the data and investigate its usefulness. Is there evidence of a curvilinear relationship between international and domestic gross revenues? Try usingα=0.05.
  5. Based on your analysis in part d, which of the models from parts a and b better explains the variation in international gross revenues? Compare your answer with your preliminary conclusion from part c.

Service workers and customer relations. A study in Industrial Marketing Management (February 2016) investigated the impact of service workers’ (e.g., waiters and waitresses) personal resources on the quality of the firm’s relationship with customers. The study focused on four types of personal resources: flexibility in dealing with customers(x1), service worker reputation(x2), empathy for the customer(x3), and service worker’s task alignment(x4). A multiple regression model was employed used to relate these four independent variables to relationship quality (y). Data were collected for n = 220 customers who had recent dealings with a service worker. (All variables were measured on a quantitative scale, based on responses to a questionnaire.)

a) Write a first-order model for E(y) as a function of the four independent variables. Refer to part

Which β coefficient measures the effect of flexibility(x1)on relationship quality (y), independently of the other

b) independent variables in the model?

c) Repeat part b for reputation(x2), empathy(x3), and task alignment(x4).

d) The researchers theorize that task alignment(x4)“moderates” the effect of each of the other x’s on relationship quality (y) — that is, the impact of eachx, x1,x2, orx3on y depends on(x4). Write an interaction model for E(y) that matches the researchers’ theory.

e) Refer to part d. What null hypothesis would you test to determine if the effect of flexibility(x1)on relationship quality (y) depends on task alignment(x4)?

f) Repeat part e for the effect of reputation(x2)and the effect of empathy(x3).

g) None of the t-tests for interaction were found to be “statistically significant”. Given these results, the researchers concluded that their theory was not supported. Do you agree?

Question: Determine which pairs of the following models are “nested” models. For each pair of nested models, identify the complete and reduced model.

a.E(y)=β0+β1x1+β2x2b.E(y)=β0+β1x1c.E(y)=β0+β1x1+β2x12d.E(y)=β0+β1x1+β2x2+β3x1x2e.E(y)=β0+β1x1+β2x2+β3x1x2+β4x21+β5x22
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