STATISTICS 402
Unreplicated and fractionally replicated experiments
1. A chemical product is manufactured in a pressurized vessel. A key quality characteristic of the
product is its filtration rate. Higher filtration rates are desired so as to increase throughput for
the process. A factorial experiment is conducted to investigate the relationships of four factors
with the filtration rate. The four factors are: temperature (T), pressure (P), concentration
of formaldehyde (C), and stirring rate (S). Each factor has two levels, and the data obtained
from a single replicate of the 24 experiment are shown below. The 16 runs are made in random
order but the data are presented in Yates’s standard order in the table. The process engineer
is interested in maximizing the filtration rate. Current process conditions give filtration rates
of around 75 gal/hr. The process also currently uses the concentration of formaldehyde ,
factor C, at the high level. The engineer would like to reduce the formaldehyde concentration
as much as possible but has been unable to do so because it always has resulted in a lower
filtration rate.
Treatment
Combination
(1)
t
p
tp
c
tc
pc
tpc
s
ts
ps
tps
cs
tcs
pcs
tpcs
T
−1
+1
−1
+1
−1
+1
−1
+1
−1
+1
−1
+1
−1
+1
−1
+1
Factor
P
C
−1 −1
−1 −1
+1 −1
+1 −1
−1 +1
−1 +1
+1 +1
+1 +1
−1 −1
−1 −1
+1 −1
+1 −1
−1 +1
−1 +1
+1 +1
+1 +1
S
−1
−1
−1
−1
−1
−1
−1
−1
+1
+1
+1
+1
+1
+1
+1
+1
Filtration
Rate (gal/h)
45
71
48
65
68
60
80
65
43
100
45
104
75
86
70
96
You should use JMP or another statistical software package to perform the calculations.
However, I will not look at computer output that is simply attached. Answer the questions
posed and support your answer with the statistical analysis (you may cut and paste from the
JMP output).
(a) Give the overall sample mean and the 15 estimated full effects.
(b) Give a normal plot of the 15 estimated full effects. Label effects that appear to be
important with their names (e.g. T, TP, etc.).
(c) Project the data into an experiment with fewer factors by identifying and dropping a
factor that does not appear to have any significant influence on the response. Analyze
the experiment with fewer factors and indicate which factors and/or interactions are
significant.
(d) Calculate factor level means and construct main effect plots for the significant factors
identified in (c).
(e) Calculate tables of means and construct interaction plots for the significant interactions
identified in (c).
(f) Instead of dropping a factor as suggested in (c), use higher order interactions (3-way
and 4-way) to create an “Error” sum of squares. Using this “Error” term, what factors
and/or interactions are significant? Do these differ from those you found in (c)?
(g) Based on your analysis, make a recommendation for the levels of Temperature, Pressure,
Concentration of formaldehyde and Stirring rate that will yield the highest filtration
rate. Will these recommended levels achieve a high filtration rate with a low level of
formaldehyde?
2. An experiment was conducted by students at the Queensland University of Technology to
determine the factors that may affect the time it takes to swim 25 meters. There were four
variables (listed below) but only 8 observations. The levels of the fourth, D, variable were set
using the ± levels of the 3-way interaction, ABC.
Factor
Flippers
Goggles
Shirt
Start Deep End
A:Shirt
−1
+1
−1
+1
−1
+1
−1
+1
B:Goggles
−1
−1
+1
+1
−1
−1
+1
+1
No
−1
−1
−1
−1
C:Flippers
−1
−1
−1
−1
+1
+1
+1
+1
Yes
+1
+1
+1
+1
D:End
−1
+1
+1
−1
+1
−1
−1
+1
Time
22.3
24.2
20.3
21.5
16.9
17.5
16.0
17.5
Analyze the data from this experiment. Some things you should discuss in your analysis are:
What is the defining relation for this experiment? What are the values of the estimated full
effects? According to a normal plot of estimated effects, what factors and/or interactions
appear to be important? Can you project this experiment into an experiment with fewer
factors by eliminating unimportant factors? Will you be able to pick up pseudo-replication
by eliminating unimportant factors? Are you be able to use 3-way and 4-way interactions as
a substitute for error variability for these data? If you want to have the fastest swimming
time, what would you recommend? What is the predicted time for your recommendation?