14-8 BLOCKING AND CONFOUNDING IN THE 2kDESIGN 543(b) Conduct an analysis of variance to confirm your findings
for part (a).
(c) Construct a normal probability plot of the residuals. Is the
plot satisfactory?
(d) Plot the residuals versus the predicted yields and versus
each of the five factors. Comment on the plots.
(e) Interpret any significant interactions.
(f) What are your recommendations regarding process oper-
ating conditions?
(g) Project the 2^5 design in this problem into a 2rfor r 5
design in the important factors. Sketch the design and
show the average and range of yields at each run. Does
this sketch aid in data interpretation?
14-20. Consider the data from Exercise 14-13. I suppose
that the data from the second replicate was not available.
Analyze the data from replicate I only and comment on your
findings.
14-21. An experiment has run a single replicate
of a 2^4 design and calculated the following factor
effects:A 80.25 AB53.25 ABC 2.95
B65.50 AC11.00 ABD 8.00
C9.25 AD 9.75 ACD 10.25
D20.50 BC18.36 BCD 7.95
BD15.10 ABCD6.25
CD1.25(a) Construct a normal probability plot of the effects.
(b) Identify a tentative model, based on the plot of effects in
part (a).
(c) Estimate the regression coefficients in this model, assum-
ing that
14-22. A2^4 factorial design was run in a chemical
process. The design factors areAtime,Bconcentration,
Cpressure, andDtemperature. The response variable isy400.(a) Estimate the factor effects. Based on a normal probability
plot of the effect estimates, identify a model for the data
from this experiment.
(b) Conduct an ANOVA based on the model identified in part
(a). What are your conclusions?
(c) Analyze the residuals and comment on model adequacy.
(d) Find a regression model to predict yield in terms of the ac-
tual factor levels.
(e) Can this design be projected into a 2^3 design with two
replicates? If so, sketch the design and show the average
and range of the two yield values at each cube corner.
Discuss the practical value of this plot.Yield Factor Levels
Run ABCD(pounds)
1 12 A(hours) 25 3
2 18 B (%) 14 18
3 13 C (psi) 60 80
4 16 D (C) 200 250
5 17
6 15
7 20
8 15
9 10
10 25
11 13
12 24
13 19
14 21
15 17
16 2314-8 BLOCKING AND CONFOUNDING IN THE 2kDESIGNIt is often impossible to run all the observations in a 2kfactorial design under homogeneous
conditions. Blocking is the design technique that is appropriate for this general situation.
However, in many situations the block size is smaller than the number of runs in the complete
replicate. In these cases, confoundingis a useful procedure for running the 2kdesign in 2p
blocks where the number of runs in a block is less than the number of treatment combinations
in one complete replicate. The technique causes certain interaction effects to be indistinguish-
able from blocks or confounded with blocks.We will illustrate confounding in the 2kfactorial
design in 2pblocks, where p k.
Consider a 2^2 design. Suppose that each of the 2^2 4 treatment combinations requires
four hours of laboratory analysis. Thus, two days are required to perform the experiment. If
days are considered as blocks, we must assign two of the four treatment combinations to
each day.yield. The data follows:c 14 .qxd 5/9/02 7:54 PM Page 543 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH112 FIN L: