6. Use Minitab to do a complete analysis of the following problem. Heat treating

Question

6. Use Minitab to do a complete analysis of the following problem. Heat treating is often used to carburize metal parts such as gears. The of the carburized layer is considered a crucial feature of the gear and contributes to the overall reliability of the part. Because of the critical nature of this feature, two different lab tests are performed on each furnace load. One test is run on a sample pin that accompanies each load. The other test is a destructive test that cross-sections an actual part. This test involves running a carbon analysis on the surface of both the gear pitch (top of the gear tooth) and the gear root (between the gear teeth). Data is in excel file HW 6-Q6 Heat Treating.xlsx, shows the results of the pitch carbon analysis test for 32 parts. The regressors are furnace temperature (TEMP), carbon concentration and duration of the carburizing cycle (SOAK-PCT, SOAKTIME), and carbon concentration and duration of the diffuse cycle (DIFFPCT, DIFFTIME)

a. Provide a Matrix plot

b. Fit a MLR model relating the results of the pitch carbon analysis test (PITCH) to the five regressor variables

c. Estimate 2.

d. Find the standard errors se(j)

e. Use the model in part (a) to predict PITCH when TEMP = 1650, SOAKTIME = 1.00, SOAKPCT = 1.10, DIFFTIME = 1.00 and DIFFPCT = 0.80.

f. Test the regression model for significance of regression. Using = 0.05, find the P-value for the test and draw conclusions.

g. Evaluate the contribution of each regressor to the model using the t-test with = 0.05.

h. Give the best “final” model

i. Check for assumptions of the model

Temp SoakTime SoakPct DiffTime DiffPct Pitch 1650 1650 1650 1650 1600 1600 0.58 0.25 0.33 0.33 0.9 0.66 0.66 0.66 0.66 0.95 0.010 0.33 0.33 1.15 1 0.016 1650 0.58 0.58 0.58 0.58 0.58 0.58 () 0.021 1650 1650 1650 1650 1650 1650 1650 1650 1650 1650 1650 1650 1650 1650 1650 0.9 0.9 0.021 0.019 0.021 0.025 0.025 ()..(J26 0.024 1.17 1.15 1.15 0.8 2 0. 0.024 1.15 1.50.9 0.02 003 (),()/ 1700 3.33 1.15.5 0.70.029 1650 1650 10) 1700 1650 1650 1650 1650 1.21.5 0.8 0.026 0.75 0.85 0.032 0.033 0.7 0.039 1.5 4 4 4 12.5 18.5 0.7 1.5 0.85 0.035 0.7 0.056 0.7 0.068 1.5

Explanation / Answer

a.

b. Regression Analysis: Pitch versus Temp, SoakTime, DiffTime, DiffPct

The regression equation is
Pitch = - 0.141 + 0.000095 Temp + 0.00208 SoakTime - 0.00279 SoakPct
+ 0.00893 DiffTime - 0.00265 DiffPct

S = 0.00215956 R-Sq = 97.2% R-Sq(adj) = 96.7%

Here, R-Sq is near to 1 so model fitting is good.

Source DF Seq SS
Temp 1 0.00260497
SoakTime 1 0.00118484
SoakPct 1 0.00000590
DiffTime 1 0.00040876
DiffPct 1 0.00000073

c.  2= RSS/(n-(k+1)) = 0.00000466/(32-(5+1))=1.79231E-07

Obs Temp Pitch Fit SE Fit Residual St Resid
7 1650 0.014000 0.018045 0.000796 -0.004045 -2.02R
19 1650 0.024000 0.024776 0.002071 -0.000776 -1.27 X
28 1700 0.039000 0.038236 0.002013 0.000764 0.98 X
29 1650 0.040000 0.033463 0.000907 0.006537 3.34R
32 1700 0.068000 0.068608 0.001882 -0.000608 -0.57 X

Analysis of Variance

Source DF SS MS F P
Regression 5 0.00420521 0.00084104 180.34 0.000
Residual Error 26 0.00012126 0.00000466
Total 31 0.00432647

Here, Pvalue is less than 0.05 therefore, the regression model is significant.

Predictor Coef SE Coef T P
Constant -0.14081 0.07944 -1.77 0.088
Temp 0.00009546 0.00004673 2.04 0.051
SoakTime 0.0020754 0.0002096 9.90 0.000
SoakPct -0.002788 0.005449 -0.51 0.613
DiffTime 0.008927 0.001171 7.62 0.000
DiffPct -0.002655 0.006710 -0.40 0.696

Here, pvalue for furnace temperature (TEMP), carbon concentration (SOAK-PCT) and DiffPct is greater than 0.05 so they are insignificant regressors and p value for SOAKTIME and DIFFTIME is less than 0.05 so they are significant regressores.

h. Give the best “final” model

Best Subsets Regression: Pitch versus Temp, SoakTime, ...

Response is Pitch

  
Mallows m m c m c
Vars R-Sq R-Sq(adj) C-p S p e t e t
1 87.5 87.1 87.9 0.0042442 X
1 60.2 58.9 341.1 0.0075752 X
2 96.6 96.4 5.5 0.0022512 X X
2 90.9 90.2 58.7 0.0036904 X X
3 97.2 96.8 2.4 0.0020981 X X X
3 96.7 96.3 6.7 0.0022600 X X X
4 97.2 96.8 4.2 0.0021256 X X X X
4 97.2 96.7 4.3 0.0021298 X X X X
5 97.2 96.7 6.0 0.0021596 X X X X X

Here, p=k+1=No. of Regresoors+1=5+1=6, Sp Mallows Cp for full model is 6.0 which is near to p, so full model i.e. model with all regressors can be condoderd as a best model.

i. Check for assumptions of the model

From NPP, all points fall near to stright line so normality assumption is followed by the model. From the graph of residual vs fitted value assumption of constant variance is also satisfied.  

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