STAT 496, Spring 2011
Homework Assignment #5, Due by Thursday, March 10
1. A factorial experiment is performed to determine the importance of four factors on
the efficiency of a distillation column that is used to separate two chemical
compounds. The four factors are: A – Relative Volatility that is used to compare the
vapor pressures of the components of a liquid mixture and is used to express how
easy a chemical mixture can be distilled, B – Bottom Composition that indicates the
ratio of one compound to the total volume of the two components at the bottom of the
distillation column, C – top Composition that indicates the ratio of one compound to
the total volume of the two components at the top of the distillation column, and D –
Feed Composition which is the ratio of the two components where they enter the
system. The sixteen combinations of these four factors, each at two levels, are run in
a completely randomized design. The factors and levels are given below.
Factor
A – Relative Volatility
B – Bottom Composition
C – Top Composition
D – Feed Composition
Low (–1)
1.1
0.005
0.70
0.30
High (+1)
2.0
0.200
0.99
0.60
The response is the efficiency of the distillation column, %. The data are given below.
Note that coded levels of the factors are given in the data table and that a value above
100 is possible.
Trmt. Comb.
1
a
b
ab
c
ac
bc
abc
d
ad
bd
abd
cd
acd
bcd
abcd
XA
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
XB
–1
–1
+1
+1
–1
–1
+1
+1
–1
–1
+1
+1
–1
–1
+1
+1
XC
–1
–1
–1
–1
+1
+1
+1
+1
–1
–1
–1
–1
+1
+1
+1
+1
XD
–1
–1
–1
–1
–1
–1
–1
–1
+1
+1
+1
+1
+1
+1
+1
+1
Efficiency
88
11
31
4
138
18
80
11
84
12
31
5
140
19
85
11
Run order
11
2
6
4
10
14
13
8
16
5
15
12
9
7
1
3
1
a) For each factor, calculate the mean efficiency when that factor is at the high level
and at the low level. Display these means in four main effects plots, one for each
factor. Comment on the apparent size of each main effect.
b) Construct interaction plots for the four factors. Pay particular attention to the
combinations of A – Relative Volatility with B – Bottom Composition, A –
Relative Volatility with C – Top Composition, and B – Bottom Composition with
C – Top Composition. Comment on the apparent presence or absence of
interaction for these three combinations.
c) Compute the overall grand sample mean and the 15 estimated effects. Be sure to
indicate whether you are computing estimated full effects of half effects.
d) Construct a normal probability plot for the 15 estimated full effects. Label on your
plot effects that appear to be significant.
e) You should be able to pick up some pseudo replication by eliminating a factor and
all its interactions with the other three factors. What factor can you eliminate from
the full model?
f) By eliminating the factor you identified in e) you should have a 23 experiment
with 2 replicates. With this reduced model, compute MSrepError and the standard
error of an estimated full effect.
g) Which of the remaining estimated effects in the reduced model are statistically
significant? Not statistically significant? Be sure to support your answer.
h) Give the most parsimonious (include only significant terms) prediction equation.
Be sure to explicitly define each variable in the equation.
i) In order to maximize the efficiency for this process, what settings of the four
factors would you recommend?
j) Give a predicted value, and prediction interval, for the efficiency at the choice of
factor levels you give in i). Be sure to indicate what you are using for the estimate
of error variation and why.
k) Using the prediction equation in h) compute residuals and predicted values.
Construct plots of residuals versus predicted values and residuals versus run
order. Are there any discernible patterns in these plots? If so, what do these
patterns indicate about the predictions in j)?
Note: You may use JMP, or another computer program, to assist you with any of the
graphing or calculations needed to answer the questions above.
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