Question

Build the analysis of variance table for the following 2k factorial using

a) Yates Method

b) standard computational formulas

c) software (Minitab) with steps

A completely randomized experiment is run to determine the influence of three factors, A, B, and C, in a plant on the ppm of a particular byproduct. Each of the three factors are considered at two levels. Analyze according to the instructions above.

A1				A2
B1		B2		B1		B2
C1	C2	C1	C2	C1	C2	C1	C2
1595	1745	1835	1838	1573	2184	1700	1717
1578	1689	1823	1614	1592	1538	1815	1806

Answer 1

Step 1:

Step 2:

Put Number of factors:=3,

Insert the data and analyze the design

MINITAB OUTPUT:

General Factorial Regression: Response versus A, B, C

Factor Information

Factor Levels Values
A 2 0, 1
B 2 0, 1
C 2 0, 1

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
Model 7 141531 20218.7 0.66 0.704
Linear 3 53063 17687.7 0.57 0.648
A 1 2783 2782.6 0.09 0.771
B 1 26488 26487.6 0.86 0.381
C 1 23793 23793.1 0.77 0.405
2-Way Interactions 3 88078 29359.2 0.95 0.460
A*B 1 7877 7876.6 0.26 0.627
A*C 1 16066 16065.6 0.52 0.491
B*C 1 64136 64135.6 2.08 0.187
3-Way Interactions 1 390 390.1 0.01 0.913
A*B*C 1 390 390.1 0.01 0.913
Error 8 246345 30793.2
Total 15 387876

Model Summary

S R-sq R-sq(adj) R-sq(pred)
175.480 36.49% 0.00% 0.00%

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant 1727.8 43.9 39.38 0.000
A
0 -13.2 43.9 -0.30 0.771 1.00
B
0 -40.7 43.9 -0.93 0.381 1.00
C
0 -38.6 43.9 -0.88 0.405 1.00
A*B
0 0 -22.2 43.9 -0.51 0.627 1.00
A*C
0 0 31.7 43.9 0.72 0.491 1.00
B*C
0 0 -63.3 43.9 -1.44 0.187 1.00
A*B*C
0 0 0 4.9 43.9 0.11 0.913 1.00

Regression Equation

Response = 1727.8 - 13.2 A_0 + 13.2 A_1 - 40.7 B_0 + 40.7 B_1 - 38.6 C_0 + 38.6 C_1
- 22.2 A*B_0 0 + 22.2 A*B_0 1 + 22.2 A*B_1 0 - 22.2 A*B_1 1 + 31.7 A*C_0 0
- 31.7 A*C_0 1 - 31.7 A*C_1 0 + 31.7 A*C_1 1 - 63.3 B*C_0 0 + 63.3 B*C_0 1
+ 63.3 B*C_1 0 - 63.3 B*C_1 1 + 4.9 A*B*C_0 0 0 - 4.9 A*B*C_0 0 1 - 4.9 A*B*C_0 1
0 + 4.9 A*B*C_0 1 1 - 4.9 A*B*C_1 0 0 + 4.9 A*B*C_1 0 1 + 4.9 A*B*C_1 1 0
- 4.9 A*B*C_1 1 1

Fits and Diagnostics for Unusual Observations

Obs Response Fit Resid Std Resid
12 2184 1861 323 2.60 R
14 1538 1861 -323 -2.60 R

Conclusion:

Since the p values for all the main as well as interation factors are greater than 0.05 so we can conclude that none of the factor have significant effect on the response.