STATISTICS 402 - Assignment 8
Due April 25, 2014
1. Traffic engineers are interested in the effect on the average traffic speed due to
erecting signs that say “Accident Reduction Project Area” and regulating the flow of
vehicles onto freeway on-ramps. The on-ramps are spread widely around a single
large metropolitan area in Southern California. Each on-ramp will either get an
“Accident Reduction Project Area” sign or will have no sign. Each on-ramp has a
traffic light that regulates the flow of vehicles. The traffic lights can be turned off (no
minimum time between vehicles) or set to require 3 or 6 seconds between entering
vehicles. Average traffic speed on the highway will be measured at each on-ramp for
all three traffic light settings.
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 sign 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 traffic light setting means
that can be detected with Alpha = 0.05 and Beta = 0.05?
e) There are only 20 on-ramps available for the experiment. If we want to use all 20
on-ramps why will it be impossible to run a balanced completely randomized
design?
f) Explain how you can use a split plot (repeated measures) design to conduct the
experiment to examine the two factors of interest. In your explanation, you must
answer the following questions. Assume you have 20 on-ramps to use in the
experiment.
i. What are the “whole plots”?
ii. What is the “whole plot” factor?
iii. How will random assignment be used for the whole plot factor? Be specific.
iv. What are the “sub plots”?
v. What is the “sub plot” factor?
vi. How will random assignment be used for the sub plot factor? 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. Each specimen
is measured and the yield strength (ksi) is obtained. Then the temperature is reset and
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the process is repeated. Each temperature is replicated 4 times in a completely
random order.
The data 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
1200o 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
1000o 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. 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 given on the next page. Remember that there are
two sets of residuals; one set for the whole plot and one set for the subplot.
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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Heat Treatment of Steel Experiment Residuals
Residuals
Residual Strength
3