568 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORSMIND-EXPANDING EXERCISES14-55. Consider an unreplicated 2kfactorial, and sup-
pose that one of the treatment combinations is missing.
One logical approach to this problem is to estimate the
missing value with a number that makes the highest or-
der interaction estimate zero. Apply this technique to the
data in Example 14-5, assuming that abis missing.
Compare the results of the analysis of these data with the
results in Example 14-5.
14-56. What blocking scheme would you recommend
if it were necessary to run a 2^4 design in four blocks of
four runs each?
14-57. Consider a 2^2 design in two blocks with
ABconfounded with blocks. Prove algebraically that
SSABSSBlocks.
14-58. Consider a 2^3 design. Suppose that the largest
number of runs that can be made in one block is four, but
we can afford to perform a total of 32 observations.
(a) Suggest a blocking scheme that will provide some
information on all interactions.
(b) Show an outline (source of variability, degrees of
freedom only) for the analysis of variance for this
design.
14-59. Construct a 2^5 ^1 design. Suppose that it is
necessary to run this design in two blocks of eight runs
each. Show how this can be done by confounding a two-
factor interaction (and its aliased three-factor interac-
tion) with blocks.
14-60. Construct a design. Show how this
design may be confounded in four blocks of eight runs
each. Are any two-factor interactions confounded with
blocks?
14-61. Construct a design. Show how this de-
sign can be confounded in two blocks of eight runs each
without losing information on any of the two-factor
interactions.
14-62. Set up a design using DAB, EAC,
FBC, and GABCas the design generators. Ignore
all interaction above the two factors.
(a) Verify that each main effect is aliased with three
two-factor interactions.
(b) Suppose that a second design with generators
DAB, EAC, FBC, and G ABCis
run. What are the aliases of the main effects in this
design?
(c) What factors may be estimated if the two sets of
factor effect estimates above are combined?To work Exercises 14-63 through 14-67 you will need to
read Section 14.6 on the CD.
14-63. Consider the experiment described in
Example 14-4. Suppose that both factors were random.
(a) Analyze the data and draw appropriate conclusions.
(b) Estimate the variance components.
14-64. For the breaking strength data in Table S14-1,
suppose that the operators were chosen at random, but
machines were a fixed factor. Does this influence the
analysis or your conclusions?
14-65. A company employs two time-study engi-
neers. Their supervisor wishes to determine whether the
standards set by them are influenced by an interaction
between engineers and operators. She selects three oper-
ators at random and conducts an experiment in which
the engineers set standard times for the same job. She
obtains the data shown here:Operator
Engineer 1 2 3
1 2.59 2.38 2.40
2.78 2.49 2.72
2 2.15 2.85 2.66
2.86 2.72 2.87(a) State the appropriate hypotheses.
(b) Use the analysis of variance to test these hypotheses
with 0.05.
(c) Graphically analyze the residuals from this exper-
iment.
(d) Estimate the appropriate variance components.
14-66. Consider the experiment on baked anode den-
sity described in Exercise 14-4. Suppose that positions
on the furnace were chosen at random and temperature
is a fixed factor.
(a) State the appropriate hypotheses.
(b) Use the analysis of variance to test these hypotheses
with 0.05.
(c) Estimate the variance components.
14-67. Consider the experiment described in Exercise
14-63. How does the analysis (and conclusions) change
if both factors are random? Use 0.05.
To work Exercises 14-68 and 14-69 you will need to read
Section 14-7.4 on the CD.(^27) III^4
(^27) III^4
(^27) IV^3
(^27) IV^2
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