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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 伪 = .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=尾0+尾1x1+尾2x2+尾3x21+尾4x22.

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: 尾3 = 尾4 = 0 while Ha: At least one of 尾 parameters are nonzero.

c. The complete and reduced model for determining whether the curvature terms can be written as y=尾0+尾1x1+尾2x2+尾3x21+尾4x22and y=尾0+尾1x1+尾2x2respectively.

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

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

f. For 伪 = 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=尾0+尾1x1+尾2x2+尾3x21+尾4x22.

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: 尾3 = 尾4 = 0while Ha: At least one of 尾 parameters are nonzero.

03

Complete and reduced model

The complete and reduced model for determining whether the curvature terms can be written as y=尾0+尾1x1+尾2x2+尾3x21+尾4x22andy=尾0+尾1x1+尾2x2 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 伪 = 0.10, the rejection region is defined as p-value > 伪. The p-value of the test is 0.000 and 0.000.

06

Conclusion

For 伪 = 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

Consider the model:

E(y)=尾0+尾1x1+尾2x2+尾3x22+尾4x3+尾5x1x22

where x2 is a quantitative model and

x1=(1receivedtreatment0didnotreceivetreatment)

The resulting least squares prediction equation is

localid="1649802968695" y铃�=2+x1-5x2+3x22-4x3+x1x22

a. Substitute the values for the dummy variables to determine the curves relating to the mean value E(y) in general form.

b. On the same graph, plot the curves obtained in part a for the independent variable between 0 and 3. Use the least squares prediction equation.

Question: Suppose the mean value E(y) of a response y is related to the quantitative independent variables x1and x2

E(y)=2+x1-3x2-x1x2

a. Identify and interpret the slope forx2.

b. Plot the linear relationship between E(y) andx2forx1=0,1,2, where.

c. How would you interpret the estimated slopes?

d. Use the lines you plotted in part b to determine the changes in E(y) for each x1=0,1,2.

e. Use your graph from part b to determine how much E(y) changes when3猢�x1猢�5and1猢�x2猢�3.

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?

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鈥檚 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: Novelty of a vacation destination. Many tourists choose a vacation destination based on the newness or uniqueness (i.e., the novelty) of the itinerary. The relationship between novelty and vacationing golfers鈥� demographics was investigated in the Annals of Tourism Research (Vol. 29, 2002). Data were obtained from a mail survey of 393 golf vacationers to a large coastal resort in the south-eastern United States. Several measures of novelty level (on a numerical scale) were obtained for each vacationer, including 鈥渃hange from routine,鈥� 鈥渢hrill,鈥� 鈥渂oredom-alleviation,鈥� and 鈥渟urprise.鈥� The researcher employed four independent variables in a regression model to predict each of the novelty measures. The independent variables were x1 = number of rounds of golf per year, x2 = total number of golf vacations taken, x3 = number of years played golf, and x4 = average golf score.

  1. Give the hypothesized equation of a first-order model for y = change from routine.
  1. A test of H0: 尾3 = 0 versus Ha: 尾3< 0 yielded a p-value of .005. Interpret this result if 伪 = .01.
  1. The estimate of 尾3 was found to be negative. Based on this result (and the result of part b), the researcher concluded that 鈥渢hose who have played golf for more years are less apt to seek change from their normal routine in their golf vacations.鈥� Do you agree with this statement? Explain.
  1. The regression results for three dependent novelty measures, based on data collected for n = 393 golf vacationers, are summarized in the table below. Give the null hypothesis for testing the overall adequacy of the first-order regression model.
  1. Give the rejection region for the test, part d, for 伪 = .01.
  1. Use the test statistics reported in the table and the rejection region from part e to conduct the test for each of the dependent measures of novelty.
  1. Verify that the p-values reported in the table support your conclusions in part f.
  1. Interpret the values of R2 reported in the table.
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