In an experiment conducted by the Mining Engineering Department at Virginia Tech

Question

In an experiment conducted by the Mining Engineering Department at Virginia Tech to study a particular filtering system for coal, a coagulant was added to a solution in a tank containing coal and sludge, which was then placed in a recirculation system in order that the coal could be washed. Three factors were varied in the experimental process:

Factor A:     percent solids circulated initially in the overflow

Factor B:     flow rate of the polymer

Factor C:      pH of the tank

The amount of solids in the underflow of the cleansing system determines how clean the coal has become. Two levels of each factor were used and two experimental runs were made for each combination. The response is the measurements in percent solids by weight in the underflow of the circulation system. Data is provided below.

Use MINITAB to do a complete analysis of the data. Use p-values in your conclusions.

percent solid flow rate Underflow 4.65 21.42 12.66 18.27 7.93 13.18 6.51 18.23 5.81 21.35 12.56 16.62 7.88 12.87 6.26 17.83 1 1 1 1 1 1 1 1

Explanation / Answer

Factorial Regression: Underflow versus percent solid, flow rate, pH

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
Model 7 490.235 70.034 254.43 0.000
Linear 3 400.413 133.471 484.90 0.000
percent solid 1 356.360 356.360 1294.65 0.000
flow rate 1 11.989 11.989 43.56 0.000
pH 1 32.064 32.064 116.49 0.000
2-Way Interactions 3 10.211 3.404 12.37 0.002
percent solid*flow rate 1 5.748 5.748 20.88 0.002
percent solid*pH 1 4.463 4.463 16.21 0.004
flow rate*pH 1 0.001 0.001 0.00 0.967
3-Way Interactions 1 79.611 79.611 289.23 0.000
percent solid*flow rate*pH 1 79.611 79.611 289.23 0.000
Error 8 2.202 0.275
Total 15 492.437

Model Summary

S R-sq R-sq(adj) R-sq(pred)
0.524649 99.55% 99.16% 98.21%

Coded Coefficients

Term Effect Coef SE Coef T-Value P-Value VIF
Constant 12.752 0.131 97.22 0.000
percent solid 9.439 4.719 0.131 35.98 0.000 1.00
flow rate 1.731 0.866 0.131 6.60 0.000 1.00
pH -2.831 -1.416 0.131 -10.79 0.000 1.00
percent solid*flow rate -1.199 -0.599 0.131 -4.57 0.002 1.00
percent solid*pH -1.056 -0.528 0.131 -4.03 0.004 1.00
flow rate*pH 0.011 0.006 0.131 0.04 0.967 1.00
percent solid*flow rate*pH 4.461 2.231 0.131 17.01 0.000 1.00

Regression Equation in Uncoded Units

Underflow = 12.752 + 4.719 percent solid + 0.866 flow rate - 1.416 pH
- 0.599 percent solid*flow rate - 0.528 percent solid*pH + 0.006 flow rate*pH
+ 2.231 percent solid*flow rate*pH

Alias Structure

Factor Name

A percent solid
B flow rate
C pH

Aliases

I
A
B
C
AB
AC
BC
ABC

Fits and Diagnostics for Unusual Observations

Obs Underflow Fit Resid Std Resid
4 18.270 17.445 0.825 2.22 R
12 16.620 17.445 -0.825 -2.22 R

The intraction effect flow rate*pH is not significant as p =  0.967 > 0.05.

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