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# Solucionario Walpole 8 ED

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```28
.0
28
.5
29
.0
D
y
2
2
\u22121 1
A
1
2
\u22121
1
1
126
.0
27
.0
28
.0
29
.0
BC Interaction
C
y
2
2
\u22121 1
B
1
2
\u22121
1
15.4 The ANOVA table is displayed.
Source of Degrees of Computed
Variation Freedom f P -value
A
B
AB
C
AC
BC
ABC
D
BD
ABD
CD
ACD
BCD
ABCD
Error
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
16
57.85
7.52
6.94
127.86
7.08
10.96
1.26
44.72
4.85
4.85
1.14
6.52
1.72
1.20
0.87
< 0.0001
0.0145
0.0180
< 0.0001
0.0171
0.0044
0.2787
< 0.0001
0.0427
0.0427
0.3017
0.0213
0.2085
0.2900
0.3651
Total 31
Solutions for Exercises in Chapter 15 239
All main effects and two-way interactions are significant, while all higher order inter-
actions are insignificant.
15.5 The ANOVA table is displayed.
Source of Degrees of Computed
Variation Freedom f P -value
A
B
C
D
AB
AC
BC
BD
CD
Error
1
1
1
1
1
1
1
1
1
1
5
9.98
0.20
6.54
0.02
1.83
0.20
0.57
19.03
1.83
0.02
0.0251
0.6707
0.0508
0.8863
0.2338
0.6707
0.4859
0.0073
0.2338
0.8863
Total 15
One two-factor interaction BC, which is the interaction of Blade Speed and Condition
of Nitrogen, is significant. As of the main effects, Mixing time (A) and Nitrogen
Condition (C) are significant. Since BC is significant, the insignificant main effect B,
the Blade Speed, cannot be declared insignificant. Interaction plots for BC at different
levels of A are given here.
1
1
15
.9
16
.1
16
.3
16
.5
Speed and Nitrogen Interaction (Time=1)
Nitrogen Condition
y
2
2
1 2
Speed
1
2 1 1
15
.6
15
.7
15
.8
15
.9
16
.0
Speed and Nitrogen Interaction (Time=2)
Nitrogen Condition
y
2
2
1 2
Speed
1
2
1
2
15.6 (a) The three effects are given as
wA =
301 + 304\u2212 269\u2212 292
4
= 11, wB =
301 + 269\u2212 304\u2212 292
4
= \u22126.5,
wAB =
301\u2212 304\u2212 269 + 292
4
= 5.
There are no clear interpretation at this time.
(b) The ANOVA table is displayed.
240 Chapter 15 2k Factorial Experiments and Fractions
Source of Degrees of Computed
Variation Freedom f P -value
Concentration
Feed Rate
Interaction
Error
1
1
1
4
35.85
12.52
7.41
0.0039
0.0241
0.0529
Total 7
The interaction between the Feed Rate and Concentration is closed to be signifi-
cant at 0.0529 level. An interaction plot is given here.
1
113
5
14
0
14
5
15
0
Feed Rate
y
2
2
\u22121 1
Concentration
1
2
\u22121
1
The mean viscosity does not change much at high level of concentration, while it
changes a lot at low concentration.
(c) Both main effects are significant. Averaged across Feed Rate a high concentration
of reagent yields significantly higher viscosity, and averaged across concentration
a low level of Feed Rate yields a higher level of viscosity.
15.7 Both AD and BC interaction plots are shown in Exercise 15.3. Here is the interaction
plot of AB.
1
1
26
.5
27
.0
27
.5
28
.0
28
.5
AB Interaction
A
y
2
2
\u22121 1
B
1
2
\u22121
1
For AD, at the high level of A, Factor D essentially has no effect, but at the low level
of A, D has a strong positive effect. For BC, at the low level of B, Factor C has a
strong negative effect, but at the high level of B, the negative effect of C is not as
pronounced. For AB, at the high level of B, A clearly has no effect. At the low level
of B, A has a strong negative effect.
15.8 The two interaction plots are displayed.
Solutions for Exercises in Chapter 15 241
1
1
26
27
28
29
30
A
y
2
2
\u22121 1
D
1
2
\u22121
1
1
1
25
.0
25
.5
26
.0
26
.5
27
.0
27
.5
A
y
2
2
\u22121 1
D
1
2
\u22121
1
It can be argued that when B = 1 that there is essentially no interaction between A
and D. Clearly when B = \u22121, the presence of a high level of D produces a strong
negative effect of Factor A on the response.
15.9 (a) The parameter estimates for x1, x2 and x1x2 are given as follows.
Variable Degrees of Freedom Estimate f P -value
x1
x2
x1x2
1
1
1
5.50
\u22123.25
2.50
5.99
\u22123.54
2.72
0.0039
0.0241
0.0529
(b) The coefficients of b1, b2, and b12 are wA/2, wB/2, and wAB/2, respectively.
(c) The P -values are matched exactly.
15.10 The effects are given here.
A B C D AB AC AD BC
\u22120.2625 \u22120.0375 0.2125 0.0125 \u22120.1125 0.0375 \u22120.0625 0.3625
BD CD ABC ABD ACD BCD ABCD
0.1125 0.0125 \u22120.1125 0.0375 \u22120.0625 0.1125 \u22120.0625
The normal probability plot of the effects is displayed.
\u22121 0 1
\u2212
0.
2
0.
0
0.
1
0.
2
0.
3
Normal Q\u2212Q Plot
Theoretical Quantiles
Sa
m
pl
e
Qu
an
tile
s
A
AB ABC
B
D CD
AC ABD
BD BCD
C
BC
(a) It appears that all three- and four-factor interactions are not significant.
(b) From the plot, it appears that A and BC are significant and C is somewhat
significant.
242 Chapter 15 2k Factorial Experiments and Fractions
15.11 (a) The effects are given here and it appears that B, C, and AC are all important.
A B C AB AC BC ABC
\u22120.875 5.875 9.625 \u22123.375 \u22129.625 0.125 \u22121.125
(b) The 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
0.11
4.79
12.86
1.58
12.86
0.00
0.18
0.7528
0.0600
0.0071
0.2440
0.0071
0.9640
0.6861
Total 15
1
1
30
35
40
45
50
AC Interaction
A
y
2
2
\u22121 1
C
1
2
\u22121
1
(c) Yes, they do agree.
(d) For the low level of Cutting Angle, C, Cutting Speed, A, has a positive effect on
the life of a machine tool. When the Cutting Angle is large, Cutting Speed has a
negative effect.
15.12 A is not orthogonal to BC, B is not orthogonal to AC, and C is not orthogonal to AB.
If we assume that interactions are negligible, we may use this experiment to estimate
the main effects. Using the data, the effects can be obtained as A: 1.5; B: \u22126.5;
C: 2.5. Hence Factor B, Tool Geometry, seems more significant than the other two
factors.
15.13 Here is the block arrangement.
Block Block Block
1 2 1 2 1 2
(1)
c
ab
abc
a
b
ac
bc
(1)
c
ab
abc
a
b
ac
bc
(1)
c
ab
abc
a
b
ac
bc
Replicate 1 Replicate 2 Replicate 3
AB Confounded AB Confounded AB Confounded
Solutions for Exercises in Chapter 15 243
Analysis of Variance
Source of Variation Degrees of Freedom
Blocks
A
B
C
AC
BC
ABC
Error
5
1
1
1
1
1
1
12
Total 23
15.14 (a) ABC is confounded with blocks in the first replication and ABCD is confounded
with blocks in second replication.
(b) Computing the sums of squares by the contrast method yields the following
ANOVA table.
Source of Degrees of Mean Computed
Variation Freedom Square f P -value
Blocks
A
B
C
D
AB
AC
BC
BD
CD
ABC
ABD
ACD
BCD
ABCD
Error
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
13
2.32
2.00
0.50
4.50
8.00
0.50
0.32
0.50
0.72
0.32
0.18
1.16
0.32
0.02
0.18
0.53
0.60
3.34
0.83
7.51
13.36
0.83
0.53
0.83
1.20
0.53
0.30
1.93
0.53
0.03
0.30
0.88
0.0907
0.3775
0.0168
0.0029
0.3775
0.4778
0.3775
0.2928
0.4778
0.5928
0.1882
0.4778
0.8578
0.5928
0.3659
Total 31
Only the main effects C and D are significant.
15.15 L1 = \u3b31 + \u3b32 + \u3b33 and L2 = \u3b31 + \u3b32 + \u3b34. For treatment combination (1) we find
L1 (mod 2) = 0. For treatment combination a we find L1 (mod 2) = 1 and L2 (mod 2) =
1. After evaluating L1 and L2 for all 16 treatment combinations we obtain the following
blocking scheme:
244 Chapter 15 2k Factorial Experiments and Fractions
Block 1 Block 2 Block 3 Block 4
(1)
ab
acd
bcd
c
abc```