STATISTICS 402B
Spring 2016
Homework Set#6 Solution
1. .
(a) Fill out the following table first:
Source of Variation d.f.
SS
Treatment
7 597.06
Week
1
1.21
Error
7
19.35
Total
15 617.62
(b) Breakdown the treatment SS in the above anova into single degree of freedom SS corresponding to
each of the effects in this experiment:
Source of Variation
d.f.
SS
Temperature
1 309.76
Catalyst
1
6.76
Temperature × Catalyst
1
73.96
pH
1 196.00
Temperature × pH
1
9.61
Catalyst × pH
1
0.16
Temperature × Catalyst × pH
1
0.81
(c) .
Source of Variation
d.f.
SS
MS
F
p-value
Temperature
1 309.76 309.76 112.0579 < 0.0001
Catalyst
1
6.76
6.76
2.4455
0.1618
Temperature × Catalyst
1
73.96
73.96
26.7556
0.0013
pH
1 196.00 196.00
70.9044 < 0.0001
Temperature × pH
1
9.61
9.61
3.4765
0.1045
Catalyst × pH
1
0.16
0.16
0.0579
0.8168
Temperature × Catalyst × pH
1
0.81
0.81
0.2930
0.6051
Week
1
1.21
1.21
Error
7
19.35
2.764
The significant effects are Temperature, pH, and Temperature × Catalyst Interaction.
(d) Let A denote the factor Temperature; B denote the factor Catalyst; C denote the factor pH.
Treatment
Combination
(1)
a
b
ab
c
ac
bc
abc
Contrast
Divisor for Estimate
Estimate of Effect
Observed
Total
122.5
146.5
129
134
133.9
162.3
139.4
152.4
I
+
+
+
+
+
+
+
+
1120
16
70
A
+
+
+
+
70.4
8
8.8
1
Factorial
B
AB
+
+
+
+
+
+
+
+
-10.4 -34.4
8
8
-1.3
-4.3
Effect
C AC
+
+
+
+
+
+
+
+
56 12.4
8
8
7 1.55
BC
+
+
+
+
1.6
8
0.2
ABC
+
+
+
+
3.6
8
0.45
(e)
2 × sE
=
s.e.(E) =
2 × 2k
√
2.764
= 0.83, t0.975,8 = 2.31, t0.975,8 s.e.(E) = 1.92
2
The approximate 95% CI’s are:
Effects
A
B
C
AB
AC
BC
ABC
95% C.I.
8.8±1.92
-1.3±1.92
-4.3±1.92
7±1.92
1.55±1.92
0.2±1.92
0.45±1.92
(f) Effect A: Effect A is positive and increase the mean by 8.8
Effect C: Effect C is positive and increase the mean by 7
B
A
130
150
1
64.10
77.20
2
67.10
71.60
At the low level of A, effect of B is to increase mean by 3. At the low level of B, effect of A is to
increase mean by 13.1.
2
2. .
(a) .
Effect
A
B
A*B
C
A*C
B*C
A*B*C
D
A*D
B*D
A*B*D
C*D
A*C*D
B*C*D
A*B*C*D
DF Sum of Squares
1
0.00002500
1
0.11902500
1
0.00010000
1
0.03802500
1
0.00040000
1
0.02250000
1
0.00122500
1
0.18922500
1
0.00040000
1
0.00040000
1
0.00022500
1
0.00010000
1
0.00002500
1
0.00002500
1
0.00040000
(b) .
0.25
D
0.2
0.15
0.1
C
0.05
0
-0.05
B*C
-0.1
-0.15
B
-0.2
-2
-1.5
-1
-0.5
0
z-score
(c) Important Effects:
Effect
B
C
BC
D
Estimate
-.1725
0.0750
-0.0750
0.2175
SS
0.119025
0.038025
0.022500
0.189225
(d) .
3
0.5
1
1.5
2
B*C
Least Squares Means Table
Level
Least Sq
Mean
0.49750000
0.67000000
0.40000000
0.42250000
-1,-1
-1,1
1,-1
1,1
Std Error
.
.
.
.
LS Means Plot
0.8
-1
0.7
1
0.6
0.5
0.4
0.3
0.2
-1
1
C
D it is observed that impurity is higher (.67) at the low level of B and
From the BC interaction plot,
Least Squares Means Table
high level of C. However, since
the investigator
is interested
in keeping
the mean impurity small, it is
Level
Least Sq
Std Error
Mean
observed that they are smaller at the Mean
high level of B (and smallest at the low level of C).
-1
0.38875000
.
0.388750
(e) It is observed clearly that Main
Effect
A and all involving factor
A are very small and therefore can
1
0.60625000
. 0.606250
be considered negligible. Thus we ignore factor A and analyze the data as a 23 facorial with two
LS Means Plot
0.8
replications of all treatment combinations
of B, C and D. This is easily accomplished by using the
same JMP data table as in part 0.7(a) and doing a full factorial of the factors B, C, and D. The following
0.6
ANOVA table is the result:
0.5
Source
DF
Treatment
B
C
B*C
D
B*D
C*D
B*C*D
Error
Total
7
1
1
1
1
1
1
1
8
15
0.4
Sum
of
0.3
Squares
0.2
0.369300
0.119025
0.038025
0.022500
0.189225
0.000400
0.000100
0.000025
0.002800
0.372100
-1
Mean
Square
0.052757
D
0.119025
0.038025
0.022500
0.189225
0.000400
0.000100
0.000025
0.000350
1
F Ratio
p-value
150.7347
340.0714
108.6429
64.2857
540.6429
1.1429
0.2857
0.0714
< .0001∗
< .0001∗
< .0001∗
< .0001∗
0.3162
0.6075
0.7960
Clearly, effects B,C, D and BC are the only significat effects.
(f) The recommendation is that the level of factor B (concentration of NaOH) be increased to 45% to
reduce the mean impurity level, while keeping the other facors at the current levels.
4