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Stat 514
Homework 10
1. An engineer is interested in the eﬀects of cutting speed(
A
), tool geometry (
B
), and cutting angle (
C
)onthe
life (in hours) of a machine tool. Two levels of each factor are chose, and three replicates of a 2
3
factorial
design are run. The results follow.
Factor
Replicate
ABC
II
I
I

22
31
25
+

32
43
29

+

35
34
50
++

55
47
46
+4
44
53
8
+

03
73
6

+
+
60
50
54
+++3
94
14
7
(a) Estimate the factorial eﬀects. Which eﬀects appear to be large (signiﬁcant)?
The estimates of the factorial eﬀects are summarized in the table below.
Eﬀect
Estimate
A
0
.
3333
B
11
.
3333
C
6
.
8333
AB

1
.
6667
AC

8
.
8333
BC

2
.
8333
ABC

2
.
1667
The eﬀects
B
,
C
and
AC
appear to be large (or signiﬁcant), based on the table above.
(b) Use analysis of variance to conﬁrm your conclusions for part (a).
The results of analysis of variance in SAS are given below. Note that the eﬀects
B
,
C
,and
AC
are all
signiﬁcant.
Analysis of Variance
Sum of
Mean
Source
DF
Squares
Square
F Value
Pr > F
Model
7
1612.66667
230.38095
7.64
0.0004
Error
16
482.66667
30.16667
Corrected Total
23
2095.33333
Root MSE
5.49242
RSquare
0.7696
Dependent Mean
40.83333
Adj RSq
0.6689
Coeff Var
13.45082
Parameter Estimates
1
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View Full Document Parameter
Standard
Variable
DF
Estimate
Error
t Value
Pr > t
Intercept
1
40.83333
1.12114
36.42
<.0001
x1
1
0.16667
1.12114
0.15
0.8837
x2
1
5.66667
1.12114
5.05
0.0001
x3
1
3.41667
1.12114
3.05
0.0077
x1x2
1
0.83333
1.12114
0.74
0.4681
x1x3
1
4.41667
1.12114
3.94
0.0012
x2x3
1
1.41667
1.12114
1.26
0.2245
x1x2x3
1
1.08333
1.12114
0.97
0.3483
(c) Write down a regression model for predicting tool life (in hours) based on the results of this experiment.
Based on the results above, the suggested model for predicting tool life (in hours) is:
Y
ijk‘
=
β
0
+
β
1
A
i
+
β
2
B
j
+
β
3
C
k
+
β
13
A
i
C
k
+
±
ijk‘
,
where,
Y
ijk‘
is the tool life,
A
i
=
±
1,
B
j
=
±
1,
C
k
=
±
1,
‘
=1
,
2
,
3, and
±
ijk‘
∼
N
(0
,
1) are i.i.d.
random variables.
Using just the signiﬁcant eﬀects,
A
,
C
,and
AC
, together with the main eﬀect of
A
, the estimated
regression function is given as:
ˆ
Y
=40
.
83333 + 0
.
16667
A
+5
.
66667
B
+3
.
41667
C

4
.
41667
AC,
where
Y
is the etch rate and
A
=
±
1,
B
=
±
1, and
C
=
±
1.
since
C
is already included in the
model, whether
A
should be included or not is usually subjective.
(d) Analyze the residuals. Are there any obvious problems?
Several plots of the residuals were created. First, a plot of the residuals versus the predicted values
was generated. This plot is given here:
When the residuals are plotted against the predicted values, the evidence against the assumptions of
normality and constancy of error variance is weak.
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This note was uploaded on 12/09/2011 for the course STAT 514 taught by Professor Staff during the Fall '08 term at Purdue University.
 Fall '08
 Staff

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