Question

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)

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.State the problem—what are you trying to find out?

2.Fit a regression model using all of the five regressors. Summarize this analysis by providing the equation, ??², and list the standard errors. What do you notice about the p values for the regressors? What does the analysis of variance indicate?

3.Use the model to predict PITCH when TEMP=1650, SOAKTIME=1, SOAKPCT=1.1, DIFFTIME=1 and DIFFPCT=.8.

4.Do you have any problems with multicollinearity?

5.Construct a t-test on each regression coefficient. What can you conclude about the variables in this model? Use an alpha = 0.05.

6.Find a 95% confidence interval on each of the coefficients that were identified to be significant. Also construct a confidence interval and prediction interval on the future observation for the data given in Q3.

7.Prepare a normal probability plot of the residuals and check the adequacy of the model.

8.Utilize stepwise regression to identify a model. How does this model differ from the first one you created? Construct a normal probability plot on the residuals to check model adequacy. Write out the equation.

9.What are the possible benefits of using a model that has fewer regressors?

Answer 1

solution:

1.our problem is to test whether do these regresion varibles really affect the response pitch

2. Output through miinitab

Regression Equation

pitch = -0.0302 + 0.000029 temp + 0.002318 soaktime - 0.00303 soakpct + 0.00848 difftime - 0.00236 diffpct

Regression Analysis: pitch versus temp, soaktime, soakpct, difftime, diffpct

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Regression	5	0.004189	0.000838	158.92	0
temp	1	0.000004	0.000004	0.69	0.414
soaktime	1	0.000939	0.000939	178.17	0
soakpct	1	0.000001	0.000001	0.27	0.609
difftime	1	0.000256	0.000256	48.47	0
diffpct	1	0	0	0.09	0.772
Error	26	0.000137	0.000005
Lack-of-Fit	19	0.000127	0.000007	4.6	0.023
Pure Error	7	0.00001	0.000001
Total	31	0.004326

?conclusion: through ANOVA table we can observe that Pvlue for Soaktime and diff time is 0 ,ths implies that there is no contributio of these two variables in variablity of pitch

3) pitch=-0.0302+0.000029*1650+0.00231*1-0.00303*1.1+0.00848*1-0.00236*0.8

    pitch=0.023219
4) we can observe for all variables is greater than 1 ,hence multicollinearity is present

5) t-test on each regression coefficient

?# result of indidvidual t test

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

??conclusion: we can observe that Pvlue for Soaktime and diff time is 0 ,ths implies that there is no contributio of these two variables in variablity of pitch

6) Confidence interval

regressor	lower limit	upper limit
temp	1.722E-05	4.07804E-05
soakpct	-0.00505346	-0.001006543
diffpct	-0.00515958	0.000439577