Question

A 2 ^3 full factorial screening experimental design was performed to identify the important factors affecting the percent conversion in a chemical process. Two factors, temperature in C (x1) and reactant concentration (x2) were identified as the key process input variables and the method of steepest ascent was applied to identify the maximum percent conversion area. Create and analyze a central composite design using the data in the following table (in standard run order). Test the residuals assumption and comment on the adequacy of the model. Identify the temperature and reactant concentration setting that maximizes percent conversion.

Std. Run	Temp	Reactant %	Conversion
1	200	15	53
2	250	15	88
3	200	25	79
4	250	25	83
5	189.65	20	58
6	260.35	20	88
7	225	12.93	75
8	225	27.07	84
9	225	20	86
10	225	20	89
11	225	20	83
12	225	20	81

Answer 1

Sol:

Regression Equation can be calculated from anova table and Correlation Coeficient table which is calculated through Excel functions:

ANOVA
	df	SS	MS	F	Significance F
Regression	2	970.9791	485.4895	8.183364	0.009438
Residual	9	533.9376	59.3264
Total	11	1504.917
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-29.5602	26.9144	-1.0983	0.300584	-90.4448	31.3244	-90.4448	31.3244
X Variable 1	0.407161	0.108936	3.737618	0.004643	0.160731	0.653592	0.160731	0.653592
X Variable 2	0.843277	0.54468	1.548206	0.15598	-0.38888	2.07543	-0.38888	2.07543

Regression Equation can be written as:

Conversion = 29.5 +0.40 Temp +0.8432 Reactant

For maximum prediction there are two sets these are (260.35,20) and (250,25)

Residuals can be calculated as follows:

Std. Run	Temp	Reactant%	Conversion	Prediction	Residual
1	200	15	53	64.5078	11.5078
2	250	15	88	84.8628	-3.1372
3	200	25	79	72.9398	-6.0602
4	250	25	83	93.2948	10.2948
5	189.65	20	58	64.51032	6.510315
6	260.35	20	88	93.29229	5.292285
7	225	12.93	75	72.93988	-2.06012
8	225	27.07	84	84.86272	0.862724
9	225	20	86	78.9013	-7.0987
10	225	20	89	78.9013	-10.0987
11	225	20	83	78.9013	-4.0987
12	225	20	81	78.9013	-2.0987

The assumption of regression equation is That the regression residuals must be normal distributed

But here mean = -0.01 and Median = 2

So mean and Median are not equal so we can say that residuals are not Normally Distributed