STATISTICS 402 - Assignment 7
Due April 20, 2012
1. We wish to design an experiment involving the baking of carbon anodes for use in
aluminum smelting operations. Specifically, the temperature of the oven used to
bake the anodes is a variable of interest. A second variable of interest is the
thickness of a coating (mm) that may prolong the usable life of the anode. The
coating is put on an unbaked anode. The usable life of an anode can be predicted
by subjecting the anode to accelerated (higher temperature and stress) conditions
and measuring the time to failure under the accelerated conditions. There are four
baking temperatures; 500, 525, 550 and 575 oC. There are five thicknesses of
coating; 0 mm, 1 mm, 2 mm, 3 mm and 4 mm. The oven used to bake the anodes
is big enough to hold five anodes at a time. Anodes must be baked for four hours
before they can be removed from the oven, cooled and tested.
a) What are the response, conditions and experimental material?
b) How many treatment combinations are there?
c) If we have 5 replications of each treatment combination in a completely
randomized design, what is the size of the difference in temperature level
means that can be detected with alpha = 0.05 and beta = 0.05?
d) If we have 5 replications of each treatment combination in a completely
randomized design, what is the size of the difference in coating thickness
means that can be detected with alpha = 0.05 and beta = 0.05?
e) Why might 5 replications of each treatment combination in a completely
randomized design be impractical for this experiment? Hint: Remember the
maxim: “Time is money.”
f) Explain how you can use a split plot (repeated measures) design to more
efficiently conduct the experiment to examine the two factors of interest. In
your explanation you must answer the following questions. Assume you have
100 unbaked anodes to use in the experiment.
i. What are the “whole plots”?
ii. What is the “whole plot” factor?
iii. What are the “sub plots”?
iv. What is the “sub plot” factor?
v. How will randomization be used in the split plot experiment? Be specific.
g) Construct a partial ANOVA table indicating sources of variation and degrees
of freedom. Also indicate how to construct the appropriate F tests for
determining the statistical significance of the model effects.
2. In order to increase the strength, refine the grain and homogenize the structure of
steel, steel is heated above a critical temperature, quenched in water and then aircooled. An experiment is performed to determine the effect of temperature and
heat treatment time on the strength of steel. Two temperatures and three times are
selected. The experiment is performed by heating the oven to a randomly selected
temperature and inserting 3 specimens. After 10 minutes one specimen is chosen
at random and removed, after another 10 minutes a second specimen is chosen at
random and removed, and after another 10 minutes the third specimen is removed.
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The data, steel strength KSI, arising from this split plot (repeated measures)
design are given below. The run gives the randomized order of the temperature
settings.
Run
3
8
1
6
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
Temperature
1500o F
Time
10 min 20 min 30 min
52
56
61
50
52
59
58
64
71
48
54
59
Run
7
2
4
5
Temperature
1600o F
Time
10 min 20 min 30 min
89
91
62
72
80
69
73
81
69
88
92
64
What is the response?
What are the experimental units?
What is the whole plot?
What is the whole plot, between subjects, factor?
What is the sub plot?
What is the sub plot, within subjects, factor?
Use JMP to analyze these data keeping in mind that this is a split plot
(repeated measures) design. Be sure to include plots of main effects and an
interaction plot as well as an analysis of the distribution of residuals. Turn in
the computer output with your assignment.
Are the two temperatures significantly different in terms of mean strength?
Report the appropriate F- and P- values to support your answer.
Are there statistically significant differences among the times? Report the
appropriate F- and P-values to support your answer.
Compute the LSD for comparing mean strengths for the three times. What, if
any, times have statistically significant differences in mean strengths?
Is there a statistically significant linear relationship between time and
strength? Support your answer with an appropriate linear contrast and test of
hypothesis.
Comment on the interaction plot. Be sure to tell me what you see (be specific)
and what this tells you about interaction.
Is there a statistically significant interaction between time and temperature?
Report the appropriate F- and P-values to support your answer.
Comment on the residuals remembering that there are two sets of residuals for
a split plot design. Tell me what you see in the various plots (residuals vs.
factor levels, Normal quantile plot, box plot and histogram) and indicate what
this tells you about the Fisher condition equal standard deviations and
normally distributed errors.
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