Question

Heat treating is often used to carburize metal parts, such as gears. The thickness 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, where an actual part is cross-sectioned. 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). Table 12-6 shows the results of the pitch carbon analysis test for 32 parts.

Temp	SoakTime	SoakPct	DiffTime	DiffPct	Pitch
1650	0.58	1.1	0.25	0.9	0.013
1650	0.66	1.1	0.33	0.9	0.016
1650	0.66	1.1	0.33	0.9	0.015
1650	0.66	1.1	0.33	0.95	0.016
1600	0.66	1.15	0.33	1	0.015
1600	0.66	1.15	0.33	1	0.016
1650	1	1.1	0.5	0.8	0.014
1650	1.17	1.1	0.58	0.8	0.021
1650	1.17	1.1	0.58	0.8	0.018
1650	1.17	1.1	0.58	0.8	0.019
1650	1.17	1.1	0.58	0.9	0.021
1650	1.17	1.1	0.58	0.9	0.019
1650	1.17	1.15	0.58	0.9	0.021
1650	1.2	1.15	1.1	0.8	0.025
1650	2	1.15	1	0.8	0.025
1650	2	1.1	1.1	0.8	0.026
1650	2.2	1.1	1.1	0.8	0.024
1650	2.2	1.1	1.1	0.8	0.025
1650	2.2	1.5	1.1	0.8	0.024
1650	2.2	1.1	1.1	0.9	0.025
1650	2.2	1.1	1.1	0.9	0.027
1650	2.2	1.1	1.5	0.9	0.026
1650	3	1.15	1.5	0.8	0.029
1650	3	1.1	1.5	0.7	0.03
1650	3	1.1	1.5	0.75	0.028
1650	3	1.15	1.66	0.85	0.032
1650	3.33	1.1	1.5	0.8	0.033
1700	4	1.1	1.5	0.7	0.039
1650	4	1.1	1.5	0.7	0.04
1650	4	1.15	1.5	0.85	0.035
1700	12.5	1	1.5	0.7	0.056
1700	18.5	1	1.5	0.7	0.068

1. Fit a regression model using all five regressors. Write out the equation. Summarize this analysis by providing the equation, ?², and list the standard errors.    What does the analysis of variance indicate?
2. Check for multicollinearity. Explain your findings.
3. Describe what you notice about the p values for the regressors. Construct a t-test on each regression coefficient. What can you conclude about the variables in this model? Use an alpha = 0.05.
4. Prepare a normal probability plot of the residuals and check the adequacy of the model.

Answer 1

a)

Regression Equation

Pitch   =   -0.0302 + 0.000029 Temp + 0.002318 SoakTime - 0.00303 SoakPct + 0.00848 DiffTime
- 0.00236 DiffPct

Coefficients
Term	Coef	SE Coef	T-Value	P-Value	VIF
Constant	-0.0302	0.0618	-0.49	0.629
Temp	0.000029	0.000034	0.83	0.414	2.78
SoakTime	0.002318	0.000174	13.35	0	2.28
SoakPct	-0.00303	0.00584	-0.52	0.609	1.22
DiffTime	0.00848	0.00122	6.96	0	2
DiffPct	-0.00236	0.00808	-0.29	0.772	2.71

Analysis of variance :

ANOVA
	df	SS	MS	F	Significance F
Regression	5	0.004189391	0.000837878	158.9236	1.25397E-18
Residual	26	0.000137077	5.27221E-06
Total	31	0.004326469

Analysis of Variance (ANOVA): provides the analysis of the variance in the model, as the name suggests. Regression statistics: provide numerical information on the variation and how well the model explains the variation for the given data/observations.

From above ANOVA table F-statistic = 158.9236 >Significance F = 1.25397E-18 so we reject the null hypothesis and conclude that their is a significant relationship exists between dependent and independent variables.

b)  

Multicollinearty Assumption

Coefficients
Term	Coef	SE Coef	T-Value	P-Value	VIF
Constant	-0.0302	0.0618	-0.49	0.629
Temp	0.000029	0.000034	0.83	0.414	2.78
SoakTime	0.002318	0.000174	13.35	0	2.28
SoakPct	-0.00303	0.00584	-0.52	0.609	1.22
DiffTime	0.00848	0.00122	6.96	0	2
DiffPct	-0.00236	0.00808	-0.29	0.772	2.71

from above table we have VIF that is Variation Inflation Factor which is used detect multicollinearity

If VIF is greater than 10 then we say that their is serious multicollinearity exist we need to remove it.

But for our model all the VIF < 10 so no multicollinearty exist.

c)

Coefficients
Term	Coef	SE Coef	T-Value	P-Value	VIF
Constant	-0.0302	0.0618	-0.49	0.629
Temp	0.000029	0.000034	0.83	0.414	2.78
SoakTime	0.002318	0.000174	13.35	0	2.28
SoakPct	-0.00303	0.00584	-0.52	0.609	1.22
DiffTime	0.00848	0.00122	6.96	0	2
DiffPct	-0.00236	0.00808	-0.29	0.772	2.71

this is the output table of t-test which used to check the significance of independent variable in the regreesion model.

From above table

For Temp, SockPct, DiffPct P-value > alpha=0.05 so we accept the null hypothesis and can conclude that these variables are not significant in the model

for SoakTime & DiffTime P-value < alpha=0.05 so we reject the null hypothesis can conclude that these variables are significant in the model

d)

From above normality plot maximum points are lies near to straight line so data follows normality assumption.