An appliance manufacturer is interested in determining whether any of three fact
ID: 3292054 • Letter: A
Question
An appliance manufacturer is interested in determining whether any of three factors (brand of laundry detergent, water temperature, and type of detergent) affects the amount of dirt removed (measured in milligrams). A 23 factorial design with two replicates was employed with the results presented in the table below:
Brand (A)
Water Temp (B)
Warm
Hot
Detergent Type (C)
Detergent Type (C)
Powder
Liquid
Powder
Liquid
X
14.8
15.0
19.9
20.3
17.8
18.2
22.6
23.2
Y
18.4
18.8
21.7
21.9
20.7
21.1
23.4
24.6
Draw the corresponding cube with the design points. Label the points with the appropriate notation.
Produce a complete test matrix which includes calculations for contrast, effect and sum of squares.
Construct the ANOVA table. Which effects are significant at a = 0.05?
Brand (A)
Water Temp (B)
Warm
Hot
Detergent Type (C)
Detergent Type (C)
Powder
Liquid
Powder
Liquid
X
14.8
15.0
19.9
20.3
17.8
18.2
22.6
23.2
Y
18.4
18.8
21.7
21.9
20.7
21.1
23.4
24.6
Explanation / Answer
we have write data as below and did analysis by Minitab
y A B C
14.8 -1 -1 -1
19.9 -1 -1 1
17.8 -1 1 -1
22.6 -1 1 1
15.0 -1 -1 -1
20.3 -1 -1 1
18.2 -1 1 -1
23.2 -1 1 1
18.4 1 -1 -1
21.7 1 -1 1
20.7 1 1 -1
23.4 1 1 1
18.8 1 -1 -1
21.9 1 -1 1
21.1 1 1 -1
24.6 1 1 1
result :
Factorial Fit: y versus A, B, C
Estimated Effects and Coefficients for y (coded units)
Term Effect Coef SE Coef T P
Constant 20.1500 0.09922 203.09 0.000
A 2.3500 1.1750 0.09922 11.84 0.000
B 2.6000 1.3000 0.09922 13.10 0.000
C 4.1000 2.0500 0.09922 20.66 0.000
A*B -0.3500 -0.1750 0.09922 -1.76 0.116
A*C -0.9500 -0.4750 0.09922 -4.79 0.001
B*C -0.1000 -0.0500 0.09922 -0.50 0.628
A*B*C 0.0500 0.0250 0.09922 0.25 0.807
S = 0.396863 PRESS = 5.04
R-Sq = 98.97% R-Sq(pred) = 95.86% R-Sq(adj) = 98.06%
Analysis of Variance for y (coded units)
Source DF Seq SS Adj SS Adj MS F P
Main Effects 3 116.370 116.370 38.7900 246.29 0.000
A 1 22.090 22.090 22.0900 140.25 0.000
B 1 27.040 27.040 27.0400 171.68 0.000
C 1 67.240 67.240 67.2400 426.92 0.000
2-Way Interactions 3 4.140 4.140 1.3800 8.76 0.007
A*B 1 0.490 0.490 0.4900 3.11 0.116
A*C 1 3.610 3.610 3.6100 22.92 0.001
B*C 1 0.040 0.040 0.0400 0.25 0.628
3-Way Interactions 1 0.010 0.010 0.0100 0.06 0.807
A*B*C 1 0.010 0.010 0.0100 0.06 0.807
Residual Error 8 1.260 1.260 0.1575
Pure Error 8 1.260 1.260 0.1575
Total 15 121.780
Unusual Observations for y
Obs StdOrder y Fit SE Fit Residual St Resid
12 12 23.4000 24.0000 0.2806 -0.6000 -2.14R
16 16 24.6000 24.0000 0.2806 0.6000 2.14R
R denotes an observation with a large standardized residual.
the result of which effect combinations are significant from the anova table we can note that tratment AB , BC, and ABC has p-value > 0.05 so that we can comment that these treatment combinations have insignificant effect we the design we have at 5% significance.
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