Stat 512 – 2
Solutions to Homework #9
Dr. Simonsen
Due November 2, 2005 by 4:30pm
1.
KNNL 19.4: In a twofactor study, the treatment means
μ
ij
are as follows:
Factor B
Factor A
B
B
1
B
B
2
B
B
3
A
1
34
23
36
A
2
40
29
42
a.
Obtain the factor A level means.
1
2
34
23
36
40
29
42
ˆ
ˆ
31,
37
3
3
+
+
+
+
μ
=
=
μ
=
=
i
i
b.
Obtain the main effects of factor A.
1
1
2
2
34
23
36
40
29
42
ˆ
34
6
ˆ
ˆ
ˆ
31
34
3
ˆ
ˆ
ˆ
37
34
3
+
+
+
+
+
μ =
=
α = μ
−μ =
−
= −
α
= μ
−μ =
−
= +
i
i
c.
Does the fact that
μ
12
–
μ
11
= –11 while
μ
13
–
μ
12
= 13 imply that factors A and B
interact?
Explain.
No.
An interaction effect cannot be established by considering only a single level of factor A.
This
difference is completely attributable to the main effect of factor B.
d.
Prepare a treatment means plot and determine whether the two factors interact.
What do you find?
The lines are completely parallel.
The treatment means plot indicates no interaction.
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2.
Recall the Filling Machines data set used in Homework #78. In that data set, there were
three columns (Y , machine, and carton). Is carton (with levels 120) a second treatment
factor that one could have included in the model? Explain. (HINT: How many cartons are
really used in this experiment, 20 or 120? Would that column be a meaningful variable to
include in your analysis?).
No.
There are actually 120 cartons used in this experiment.
The “carton” label is not actually a
factor.
It is simply a label to identify which of the 20 randomly selected cartons for a particular
machine is referred to.
However, carton 6 (say) from machine 1 has no relationship to carton 6
from machine 2.
For the remaining questions use the Hay Fever Relief dataset from problem 19.14 described on
page 868 of KNNL.
3.
Give a table of sample sizes, means, and standard deviations for the nine different treatment
combinations.
Level of
Level of
relief
A
B
N
Mean
Std Dev
1
1
4
2.4750000
0.17078251
1
2
4
4.6000000
0.29439203
1
3
4
4.5750000
0.17078251
2
1
4
5.4500000
0.26457513
2
2
4
8.9250000
0.17078251
2
3
4
9.1250000
0.30956959
3
1
4
5.9750000
0.22173558
3
2
4
10.2750000
0.33040379
3
3
4
13.2500000
0.20816660
4.
Write the factor effects model for this analysis, and estimate the parameters of this model
under the zerosum constraint system (i.e. the one described on page 816 of the text).
Also
demonstrate that your estimates do in fact satisfy these constraints.
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 Linear Regression, Variance, main effect, PROC GLM

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