For a study of a high pressure inlet fogging method for a gas turbine engine, th

Question

For a study of a high pressure inlet fogging method for a gas turbine engine, the heat rate (HEATRATE, kilojoules per kilowatt per hour) was measured for each in a sample of 67 gas turbines. In addition, several other variables were measured, including cycle speed (X1- RPM, revolutions per minute), inlet temperature (INLTTEMP, 0C), exhaust gas temperature (X2 - EXHTEMP, 0C), and cycle pressure ratio (X3 - CPRATIO). [Dataset: Gasturbine.xls]

e. Fit a first-order linear multiple regression model (reduced model) for heat rate (y) as a function of speed, exhaust gas temperature and cycle pressure ratio. Provide both the ANOVA table and the parameter-estimate table. [Hint: predictor variables: X1, X2, X3]

f. At = 0.05, use partial-F test to test whether the full model in part (a) is better than the reduced model.

The data file can be accessed in this google drive link:
https://drive.google.com/open?id=0B59_85LvhUgob0ZGWUlPSXNxdjg

Explanation / Answer

Here dependent variable is heart rate (y) and there are three independent variables as speed (X1), exhaust gas temperature (X2) and cycle pressure ratio (X3).

This is the problem of multiple regression.

We have to fit regression of y on X1, X2 and X3.

We can fit regression in MINITAB.

steps :

ENTER data into MINITAB sheet --> STAT --> Regression --> Regression --> Response : y --> Predictors : X1,X2 and X3 --> Results : select second option --> ok --> ok

————— 02-11-2017 18:04:12 ————————————————————

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Regression Analysis: HEATRATE versus RPM, EXHTEMP, CPRATIO

The regression equation is
HEATRATE = 15346 + 0.139 RPM - 5.85 EXHTEMP - 156 CPRATIO

Predictor Coef SE Coef T P
Constant 15346.4 952.5 16.11 0.000
RPM 0.13905 0.01179 11.79 0.000
EXHTEMP -5.851 1.646 -3.55 0.001
CPRATIO -156.34 19.36 -8.08 0.000

S = 581.3 R-Sq = 87.3% R-Sq(adj) = 86.7%

Analysis of Variance

Source DF SS MS F P
Regression 3 146611957 48870652 144.65 0.000
Residual Error 63 21285252 337861
Total 66 167897208

The regression equation is,

HEATRATE = 15346 + 0.139 RPM - 5.85 EXHTEMP - 156 CPRATIO

Here we can test the two hypothesis.

i) Overall significance :

Here we have to test the hypothesis that,

H0 : Bj = 0 Vs H1 : Bj not= 0

where Bj is population slope for jth independent variable.

Assume alpha = level of significance = 5% = 0.05

Here test statistic follows F-distribution.

Test statistic = 144.65

P-value = 0.000

P-value < alpha

Reject H0 at 5% level of significance

Conclusion : Atleast one of the slope is differ than 0.

ii) Individual significance :

Here we have to test the hypothesis that,

H0: B = 0 Vs H1 : B not= 0

where B is population slope for independent variable.

Assume alpha = level of significance = 5% = 0.05

Here test statistic follows t-distribution.

Decision rule :

If P-value < alpha then that variable is significant otherwise that variable is insignificant.

SO we can see that all th three variables are significant since P-value < alpha.

Significant variables we included in the model.

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