STATISTICS 402 - Assignment 7
Due April 26, 2010
1. On homework 2 this semester one of the problems discussed an experiment involving the
baking of carbon anodes for use in aluminum smelting operations. Specifically, the
temperature of the oven was 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 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 of erecting signs that say “Accident
Reduction Project Area” and metering the flow of vehicles onto freeway on-ramps on the
average traffic speed. 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 this means.
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
Residuals
Residual AveSpeed
3