STATISTICS 402 - Assignment 8
Due April 26, 2013
1. An aluminum smelting operation makes its own carbon anodes for use in their aluminum
smelting pots. The density of the anode is an important quality characteristic because it
affects the usable life of the anode. An experiment is conducted to see the effect of bake
temperature on anode density. The temperature of the oven 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 anode before it is baked. 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 several 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 work 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) Traffic engineers are interested in the effect on the average traffic speed due to erecting signs
that say “Accident Reduction Project Area” and metering the flow of vehicles onto freeway
on-ramps. Twenty similar freeway interchanges are chosen. Each interchange has a traffic
light at the on-ramp. The interchanges are spread widely around a single large metropolitan
area in Southern California. Ten intersections are chosen at random to get “Accident
Reduction Project Area” signs; the other ten intersections get no signs. The traffic lights at
each intersection can be turned off (no minimum time between vehicles) or set to require 3 or
6 seconds between entering vehicles. Average traffic speed during “rush hour” will be
measured at each interchange on three consecutive Tuesdays in June. At each interchange,
the three settings of traffic light times are assigned at random to the three Tuesdays. Below
are the data.
1
Accident Reduction
Project Area Sign?
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
No
No
No
No
No
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
Intersection
Time = 0
Time = 3
Time = 6
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
19
12
10
12
26
17
17
18
21
24
13
24
22
24
23
23
24
19
21
15
31
28
24
23
41
31
27
32
29
37
25
35
38
32
35
33
35
34
33
30
33
27
29
28
41
30
31
33
30
37
26
37
41
37
38
35
41
35
37
30
What is the response?
What are the experimental units?
What is the whole plot factor?
What is the sub plot 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 JMP output with your
assignment.
Is there an effect due to the sign? Report the appropriate F- and P- values to support your
answer.
Are there significant differences among the times? Report the appropriate F- and Pvalues to support your answer.
Compute the LSD for comparing mean strengths for the three times. What, if any, times
have statistically significant differences?
Is there a statistically significant linear relationship between time and average speed?
Support your answer with an appropriate linear contrast and test of hypothesis.
Are the results of h) and i) consistent? Explain briefly.
Comment on the interaction plot. Be sure to tell me what you see (be specific) and what
the nature of the interaction is.
Is there a statistically significant interaction between time and sign? Report the
appropriate F- and P-values to support your answer.
Comment on the residuals. Tell me what you see in the various plots (residual vs factors,
Normal quantile plots, box plots and histograms) and indicate what this tells you about
the conditions for conducting the analysis. Remember that there are two error terms in a
split plot design so be sure to indicate what set of plots goes with each error term.
2
Traffic Sign, Light Timing Experiment Residuals
4
10
3
2
Residual AveSpeed
Residuals
5
0
-5
1
0
-1
-2
-3
-10n
-4 0
y
3
6
Timing
Sign
2.33
0.95
0.9
0.8
0.67
0.0
0.98
Normal Quantile Plot
1.28
0.5
Normal Quantile Plot
1.64
1.64
1.28
0.9
0.8
0.67
0.0
0.5
-0.67
-0.67
0.2
0.2
-1.28
-1.28
-1.64
0.1
-1.64
0.1
0.05
0.02
-2.33
10
15
10
4
5
2
-10
-5
0
Residuals
5
10
-4
-3
-2
-1
0
1
2
3
4
Residual AveSpeed
3
Count
6
Count
8