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# Solucionario Walpole 8 ED

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flour and P -value = 0.6059 for \u201cCake\u201d flour.
We also have the interaction plot which shows that sweetener at 100% concen-
tration yielded a specific gravity significantly lower than the other concentrations
for all-purpose flour. For cake flour, however, there were no big differences in the
effect of sweetener concentration.
1 1
1
1
0.
80
0.
82
0.
84
0.
86
0.
88
Sweetener
At
tri
bu
te
2
2
2
2
0 50 75 100
Flour
1
2
all\u2212pupose
cake
14.35 (a) The ANOVA table is given.
Source of Sum of Degrees of Mean Computed
Variation Squares Freedom Square f P -value
Sauce
Fish
Sauce*Fish
Error
1, 031.3603
16, 505.8640
724.6107
3, 381.1480
1
2
2
24
1, 031.3603
8, 252.9320
362.3053
140.8812
7.32
58.58
2.57
0.0123
< 0.0001
0.0973
Total 21, 642.9830 29
Interaction effect is not significant.
(b) Both P -values of Sauce and Fish Type are all small enough to call significance.
14.36 (a) The ANOVA table is given here.
Source of Sum of Degrees of Mean Computed
Variation Squares Freedom Square f P -value
Plastic Type
Humidity
Interaction
Error
142.6533
143.7413
133.9400
50.5950
2
3
6
12
71.3267
47.9138
22.3233
4.2163
16.92
11.36
5.29
0.0003
0.0008
0.0070
Total 470.9296 23
The interaction is significant.
(b) The SS for AB with only Plastic Type A and B is 24.8900 with 3 degrees of
freedom. Hence f = 24.8900/3
4.2163
= 1.97 with P -value = 0.1727. Hence, there is no
significant interaction when only A and B are considered.
Solutions for Exercises in Chapter 14 233
(c) The SS for the single-degree-of-freedom contrast is 143.0868. Hence f = 33.94
with P -value < 0.0001. Therefore, the contrast is significant.
(d) The SS for Humidity when only C is considered in Plastic Type is 2.10042. So,
f = 0.50 with P -value = 0.4938. Hence, Humidity effect is insignificant when
Type C is used.
14.37 (a) The ANOVA table is given here.
Source of Sum of Degrees of Mean Computed
Variation Squares Freedom Square f P -value
Environment
Stress
Interaction
Error
0.8624
40.8140
0.0326
0.6785
1
2
2
12
0.8624
20.4020
0.0163
0.0565
15.25
360.94
0.29
0.0021
< 0.0001
0.7547
Total 42.3875 17
The interaction is insignificant.
(b) The mean fatigue life for the two main effects are all significant.
14.38 The ANOVA table is given here.
Source of Sum of Degrees of Mean Computed
Variation Squares Freedom Square f P -value
Sweetener
Flour
Interaction
Error
1.26893
1.77127
0.14647
2.55547
3
1
3
16
0.42298
1.77127
0.04882
0.15972
2.65
11.09
0.31
0.0843
0.0042
0.8209
Total 5.74213 23
The interaction effect is insignificant. The main effect of Sweetener is somewhat in-
significant, since the P -value = 0.0843. The main effect of Flour is strongly significant.
14.39 The ANOVA table is given here.
Source of Sum of Degrees of Mean Computed
Variation Squares Freedom Square f P -value
A
B
C
AB
AC
Error
1133.5926
26896.2593
40.1482
216.5185
1.6296
2.2963
2.5926
1844.0000
2
2
2
4
4
4
8
27
566.7963
13448.1296
20.0741
54.1296
0.4074
0.5741
0.3241
68.2963
8.30
196.91
0.29
0.79
0.01
0.01
0.00
0.0016
< 0.0001
0.7477
0.5403
0.9999
0.9999
1.0000
Total 30137.0370 53
234 Chapter 14 Factorial Experiments (Two or More Factors)
All the two-way and three-way interactions are insignificant. In the main effects, only
A and B are significant.
14.40 (a) Treating Solvent as a class variable and Temperature and Time as continuous
variable, only three terms in the ANOVA model show significance. They are (1)
Intercept; (2) Coefficient for Temperature and (3) Coefficient for Time.
(b) Due to the factor that none of the interactions are significant, we can claim that
the models for ethanol and toluene are equivalent apart from the intercept.
(c) The three-way interaction in Exercise 14.23 was significant. However, the general
patterns of the gel generated are pretty similar for the two Solvent levels.
14.41 The ANOVA table is displayed.
Source of Sum of Degrees of Mean Computed
Variation Squares Freedom Square f P -value
Surface
Pressure
Interaction
Error
2.22111
39.10778
112.62222
565.72000
2
2
4
9
1.11056
19.55389
28.15556
62.85778
0.02
0.31
0.45
0.9825
0.7402
0.7718
Total 719.67111 17
All effects are insignificant.
14.42 (a) This is a two-factor fixed-effects model with interaction.
yijk = µ+ \u3b1i + \u3b2j+)\u3b1\u3b2)ij + \u1ebijk,\u2211
i
\u3b1i = 0,
\u2211
j
\u3b2j = 0,
\u2211
i
(\u3b1\u3b2)ij =
\u2211
j
(\u3b1\u3b2)ij = 0, \u1ebijk \u223c n(x; 0, \u3c3)
(b) The ANOVA table is displayed.
Source of Sum of Degrees of Mean Computed
Variation Squares Freedom Square f P -value
Time
Temperature
Interaction
Error
0.16668
0.27151
0.03209
0.01370
3
2
6
12
0.05556
0.13575
0.00535
0.00114
48.67
118.91
4.68
< 0.0001
< 0.0001
0.0111
Total 0.48398 23
The interaction is insignificant, while two main effects are significant.
(c) It appears that using a temperature of \u221220\u25e6C with drying time of 2 hours would
speed up the process and still yield a flavorful coffee. It might be useful to try
some additional runs at this combination.
Solutions for Exercises in Chapter 14 235
14.43 (a) Since it is more reasonable to assume the data come from Poisson distribution,
it would be dangerous to use standard analysis of variance because the normality
assumption would be violated. It would be better to transform the data to get at
least stable variance.
(b) The ANOVA table is displayed.
Source of Sum of Degrees of Mean Computed
Variation Squares Freedom Square f P -value
Teller
Time
Interaction
Error
40.45833
97.33333
8.66667
25.50000
3
2
6
12
13.48611
48.66667
1.44444
2.12500
6.35
22.90
0.68
0.0080
< 0.0001
0.6694
Total 171.95833 23
The interaction effect is insignificant. Two main effects, Teller and Time, are all
significant.
(c) The ANOVA table using a squared-root transformation on the response is given.
Source of Sum of Degrees of Mean Computed
Variation Squares Freedom Square f P -value
Teller
Time
Interaction
Error
0.05254
1.32190
0.11502
0.35876
3
2
6
12
0.17514
0.66095
0.01917
0.02990
5.86
22.11
0.64
0.0106
< 0.0001
0.6965
Total 2.32110 23
Same conclusions as in (b) can be reached. To check on whether the assumption
of the standard analysis of variance is violated, residual analysis may used to do
diagnostics.
Chapter 15
2k Factorial Experiments and
Fractions
15.1 Either using Table 15.5 (e.g., SSA = (\u221241+51\u221257\u221263+67+54\u221276+73)
2
24
= 2.6667) or running
an analysis of variance, we can get the Sums of Squares for all the factorial effects.
SSA = 2.6667, SSB = 170.6667, SSC = 104.1667, SS(AB) = 1.500.
SS(AC) = 42.6667, SS(BC) = 0.0000, SS(ABC) = 1.5000.
15.2 A simplified ANOVA table is given.
Source of Degrees of Computed
Variation Freedom f P -value
A
B
AB
C
AC
BC
ABC
Error
1
1
1
1
1
1
1
8
1294.65
43.56
20.88
116.49
16.21
0.00
289.23
< 0.0001
0.0002
0.0018
< 0.0001
0.0038
0.9668
< 0.0001
Total 15
All the main and interaction effects are significant, other than BC effect. However,
due to the significance of the 3-way interaction, the insignificance of BC effect cannot
be counted. Interaction plots are given.
237
238 Chapter 15 2k Factorial Experiments and Fractions
1
1
5
10
15
20
C=\u22121
A
y
2
2
\u22121 1
B
1
2
\u22121
1
1
1
6
8
10
12
14
16
18
C=1
A
y
2
2
\u22121 1
B
1
2
\u22121
1
15.3 The AD and BC interaction plots are printed here. The AD plot varies with levels of
C since the ACD interaction is significant, or with levels of B since ABD interaction
is significant.
1
1
26
.5
27
.0
27
.5