14-60
Supplemental Exercises
14-51
a)
Estimated Effects and Coefficients for var_1 (coded units)
Term
Effect
Coef
SE Coef
T
P
Constant
191.563
1.158
165.49
0.000
factor_A (PH)
5.875
2.937
1.158
2.54
0.026
factor_B (CC)
-0.125
-0.062
1.158
-0.05
0.958
factor_A*factor_B
11.625
5.812
1.158
5.02
0.000
Analysis of Variance for var_1 (coded units)
Source
DF
Seq SS
Adj SS
Adj MS
F
P
Main Effects
2
138.125
138.125
69.06
3.22
0.076
2-Way Interactions
1
540.562
540.562
540.56
25.22
0.000
Residual Error
12
257.250
257.250
21.44
Pure Error
12
257.250
257.250
21.44
Total
15
935.938
The main effect of pH and the interaction of pH and Catalyst Concentration (CC) are significant at the 0.05
level of significance. The model used is viscosity = 191.563 + 2.937x
1
−
0.062x
2
+ 5.812x
12
b)
The interaction plot shows that there is a strong interaction. When Factor A is at its low level, the mean
response is large at the low level of B and is small at the high level of B. However, when A is at its high
level, the results are opposite.
c) The plots of the residuals show that the equality of variance assumption is reasonable.
However, there is a large gap in the middle of the normal probability plot.
Sometimes, this can
indicate that there is another variable that has an effect on the response but which is not included
in the experiment.
For example, in this experiment, note that the replicates in each cell have two
pairs of values that are very similar, but there is a rather large difference in the mean values of the
two pairs. (Cell 1 has 189 and 192 as one pair and 198 and 199 as the other.)
-1
1
185
190
195
200
factor_B
Mean
Interaction Plot (data means) for var_1
A high
A low

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14-61
14-52
a)
Factor
Type Levels Values
Gear Typ
fixed
3 20 24 28
Time
fixed
2
90 120
Analysis of Variance for flatness, using Adjusted SS for Tests
Source
DF
Seq SS
Adj SS
Adj MS
F
P
Gear Typ
2
0.0007591
0.0007591
0.0003796
3.90
0.082
Time
1
0.0024941
0.0024941
0.0024941
25.66
0.002
Gear Typ*Time
2
0.0000505
0.0000505
0.0000253
0.26
0.779
Error
6
0.0005833
0.0005833
0.0000972
Total
11
0.0038870
Term
Coef
SE Coef
T
P
Constant
0.056500
0.002846
19.85
0.000
Gear Typ
20
-0.011125
0.004025
-2.76
0.033
24
0.007000
0.004025
1.74
0.133
Time
90
-0.014417
0.002846
-5.07
0.002
Gear Typ*Time
20
90
-0.000708
0.004025
-0.18
0.866
24
90
-0.002083
0.004025
-0.52
0.623
There is weak evidence that flatness distortion is different for the different gear types (
p
= 0.082). Gear type
is significant at
α
= 0.1, but not at
α
= 0.05. Also, the gear type 20 coefficient has a
p
-value = 0.033. Heat-
treating time affects the flatness distortion (
p
= 0.002). There is no evidence that factors interact (
p
= 0.779).
185
190
195
200
-5
0
5
Fitted Value
Residual
Residuals Versus the Fitted Values
(response is var_1)
-1
0
1
-5
0
5
factor_B
Residual
Residuals Versus factor_B
(response is var_1)
5
0
-5
2
1
0
-1
-2
Normal Score
Residual
Normal Probability Plot of the Residuals
(response is var_1)
-1
0
1
-5
0
5
factor_A
Residual
Residuals Versus factor_A
(response is var_1)