Question

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

Answer

a. A second-order model equation in 2 independent variables can be written as$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x^2}_1+{\beta }_4{x_2}^2$.

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 H_a: At least one of β parameters are nonzero.

c. The complete and reduced model for determining whether the curvature terms can be written as $y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x^2}_1+{\beta }_4{x_2}^2$and $y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2$respectively.

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

Second-order model equation

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

as $y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x^2}_1+{\beta }_4{x_2}^2$.

 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 H_a: At least one of β parameters are nonzero.

Complete and reduced model

The complete and reduced model for determining whether the curvature terms can be written as $y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x^2}_1+{\beta }_4{x_2}^2$and$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2$ respectively.

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.

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.

Conclusion

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