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Chapter 12: Q29E (page 733)

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). Recall that you fit a first-order model for heat rate (y) as a function of speed (x1) , inlet temperature (x2) , exhaust temperature (x3) , cycle pressure ratio (x4) , and airflow rate (x5) . A Minitab printout with both a 95% confidence interval for E(y) and prediction interval for y for selected values of the x鈥檚 is shown below.

a. Interpret the 95% prediction interval for y in the words of the problem.

b. Interpret the 95% confidence interval forE(y)in the words of the problem.

c. Will the confidence interval for E(y) always be narrower than the prediction interval for y? Explain.

Short Answer

Expert verified

(a) Given the variable setting, it can be concluded with 95% accuracy that the value of y will lie within the interval (12157.9, 13107.1).

(b) Given the variable setting, it can be concluded with 95% accuracy that the value of E(y) will lie within the interval (13599.6, 13665.5).

(c) Prediction interval must also include the uncertainty in estimating the mean plus the variation in estimating an individual value, the prediction interval is always wider than the confidence interval.

Step by step solution

01

Confidence interval for y

95% confidence interval for y here is (12157.9, 13107.1) where the variable setting set at speed = 7500, inlet temperature = 1000, exhaust temperature = 525, cycle pressure ratio = 13.5, and airflow = 10. Given the variable setting, it can be concluded with 95% accuracy that the value of y will lie within the interval (12157.9, 13107.1).

02

Certainty of interval for E(y)

95% confidence interval for E(y) here is (13599.6, 13665.5) where the variable setting set at speed = 7500, inlet temperature = 1000, exhaust temperature = 525, cycle pressure ratio = 13.5, and airflow = 10. Given the variable setting, it can be concluded with 95% accuracy that the value of E(y) will lie within the interval (13599.6, 13665.5).

03

Fortitude interval for E(y) and prediction interval for y

A prediction interval for y predicts the range in which the individual value of y will lie. While the confidence interval shows the range of values for E(y). Since the prediction interval must also include the uncertainty in estimating the mean plus the variation in estimating an individual value, the prediction interval is always wider than the confidence interval.

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Most popular questions from this chapter

Consider fitting the multiple regression model

E(y)= 尾0+尾1x1+ 尾2x2+尾3x3+ 尾4x4 +尾5x5

A matrix of correlations for all pairs of independent variables is given below. Do you detect a multicollinearity problem? Explain

Consider fitting the multiple regression model

Ey=尾0+尾1x1+尾2x2+尾3x3+尾4x4+尾5x5

A matrix of correlations for all pairs of independent variables is given below. Do you detect a multicollinearity problem? Explain.

Question: Personality traits and job performance. Refer to the Journal of Applied Psychology (January 2011) study of the relationship between task performance and conscientiousness, Exercise 12.94 (p. 766). Recall that y = task performance score (measured on a 30-point scale) was modeled as a function of x1 = conscientiousness score (measured on a scale of -3 to +3) and x2 = {1 if highly complex job, 0 if not} using the complete model

E(y)=尾0+尾1x1+尾2(x1)2+尾3x2+尾4x1x2+尾5(x1)2x2

a. Specify the null hypothesis for testing the overall adequacy of the model.

b. Specify the null hypothesis for testing whether task performance score (y) and conscientiousness score (x1) are curvilinearly related.

c. Specify the null hypothesis for testing whether the curvilinear relationship between task performance score (y) and conscientiousness score (x1) depends on job complexity (x2).

Explain how each of the tests, parts a鈥揷, should be conducted (i.e., give the forms of the test statistic and the reduced model).


Factors that impact an auditor鈥檚 judgment. A study was conducted to determine the effects of linguistic delivery style and client credibility on auditors鈥� 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鈥檚 financial statement. The researchers gave the auditors different information on the client鈥檚 credibility and linguistic delivery style of the client鈥檚 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鈥攃redibility (x1), linguistic delivery style (x2) , and likelihood (y) 鈥攚ere all measured on a numerical scale. Regression analysis was used to fit the interaction model,y=尾0+尾1x1+尾2x2+尾3x1x2+蔚 . 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鈥搇inguistic 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鈥搇inguistic 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: 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.
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