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Solutions to Homework #8
Dr. Simonsen
Due October 26, 2005 by 4:30pm
The first three problems use the Filling Machines dataset from Problem 16.11 of KNNL described
on page 725, and continue the analysis begun on Homework #7.
1.
Use the Tukey multiple comparison method to determine which pairs of machines differ
significantly.
Summarize the results.
The Tukey comparison method shows that machines 3 and 4 are significantly different from
machines 1, 2, 5, and 6 (on a pairwise basis).
The GLM Procedure
Tukey's Studentized Range (HSD) Test for wtdev
NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher
Type II error rate than REGWQ.
Alpha
0.05
Error Degrees of Freedom
114
Error Mean Square
0.03097
Critical Value of Studentized Range
4.09949
Minimum Significant Difference
0.1613
Means with the same letter are not significantly different.
Tukey Grouping
Mean
N
machine
A
0.46000
20
3
A
A
0.36550
20
4
B
0.19050
20
2
B
B
0.15150
20
6
B
B
0.12500
20
5
B
B
0.07350
20
1
2.
Suppose you want to compare the average of the first two machines with the average of the
last four. Use the
ESTIMATE
and
CONTRAST
statements in
PROC GLM
to test the
appropriate hypothesis. Report the estimated value of this contrast with its standard error;
state the null and alternative hypotheses, the test statistic with degrees of freedom, the P-
value and your conclusion.
The contrast is
()
3456
12
24
L
μ+μ+μ+μ
μ+μ
=−
.
The estimated value of this contrast is
with standard error
ˆ
0.1435
L
{ }
ˆ
0.03408
sL
=
.
We test
the null hypothesis
H
0
:
0
−=
vs H
a
:
( )
( )
0
−
≠
.
[These could also be correctly written as
H
0
:
L = 0 vs. Ha:
L
≠
0 or

H
0
:
()
3456
12
24
μ+μ+μ+μ
μ+μ
=
vs. Ha:
( )
( )
μ +μ +μ +μ
≠
].
The test statistic is either F = 17.73 with 1,114 df or t = –4.21 with 114 df, and the P-value is P <
0.0001.
We reject H
0
and conclude that the mean for the first two machines is not the same as the
mean for the last four machines.
The GLM Procedure
Dependent Variable: wtdev
Contrast
DF
Contrast SS
Mean Square
F Value
Pr > F
prob2
1
0.54912667
0.54912667
17.73
<.0001
Standard
Parameter
Estimate
Error
t Value
Pr > |t|
prob2
-0.14350000
0.03407905
-4.21
<.0001
3.
Check assumptions using the residuals. Turn in the plots/output you used to check the
assumptions and state your conclusions.
resid
-0.4
-0.3
-0.2
-0.1
0. 0
0. 1
0. 2
0. 3
0. 4
machi ne
123456
The assumptions of normality and constant variance appear to be satisfied, as shown by the residual
plot and qqplot.
[Grader:
the residual plot may be plotted vs. machine or vs. predicted value.
Only
one is necessary, plus the qqplot.]

The remaining problems use the Helicopter Service dataset from Problem 18.15 on page 804 of
KNNL.
Helicopter service
.
An operations analyst in a sheriff’s department studied how frequently
their emergency helicopter was used during a recent 20-day period, by time of day (shift 1:
2 am – 8 am; shift 2:
8 am – 2 pm; shift 3:
2 pm – 8 pm;
shift 4:
8 pm – 2 am).
The data
follow (in time order).
Since the data are counts, the analyst was concerned about
the
normality and equal variances assumptions of ANOVA model (16.2).

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