STAT 496, Spring 2011
Homework Assignment #6, Due by Thursday, March 24
1. An engineer is interested in the effects of cutting speed (A), tool geometry (B), and
cutting angle (C) on the life (in hours) of a machine tool. Because the hardness of the
raw material being tooled may vary, heats of raw material are used as blocks. The
eight treatment combinations are run in a random order within each block (heat). The
levels of each of the 3 factors are given below along with the data.
A: Cutting Speed
B: Tool Geometry
C: Cutting Angle
XA
–1
+1
–1
+1
–1
+1
–1
+1
XB
–1
–1
+1
+1
–1
–1
+1
+1
XC
–1
–1
–1
–1
+1
+1
+1
+1
Low (–1)
50 rpm
Type 1
5o
Heat I
22
32
35
45
43
36
50
47
High (+1)
100 rpm
Type 2
10o
Heat II
31
43
50
55
45
38
61
43
Heat III
25
30
35
47
38
40
54
39
a) Compute means for the low and high levels of each of the three factors. Display
these graphically and comment on the apparent effects of each of the factors.
b) Construct an interaction plot for factors A and C using the data from Type 1 tool
geometry. Construct a second interaction lot for factors A and C using the data
from the Type 2 tool geometry. Comment on the plots. What do they indicate
about the possible interaction between factors A and C? What do they indicate
about the 3-way interaction?
c) Compute the overall sample mean and estimate the full effects for each factor, 2way and 3-way interaction.
d) Compute a value of the SSrepError for this experiment.
e) Compute means for each heat (block). Use these to compute a value of SSBlock.
How many degrees of freedom are associated with this sum of squares?
f) Compute the SSError, associated degrees of freedom and the critical effect size (use
t=3).
g) According to the critical effect size in f), what effects are statistically significant?
What effects are not statistically significant?
h) Give the reduced model prediction equation that includes only those effects that
are found to be statistically significant in g). Be sure to define explicitly any
variables you use in the equation.
i) Based on the prediction equation in h), if long tool life is the goal, what levels of
the three factors would you choose? What is the predicted life and a 95%
prediction interval for you optimal choice?
1
j) Your prediction in i) will be appropriate for the average hardness of heats of raw
material. How would you change the prediction equation, and your prediction for
optimal choice, so that it would predict more accurately the tool wear for heats
with average hardness similar to Heat II?
2. An experiment is performed on a new piece of drilling equipment. The response
variable is the rate of advance of the drill. The four factors (each with a low (–1) and
high (+1) level) are A: Load on the drill, B: Flow Rate through the drill, C: Speed of
Rotation, D: Amount of Mud used in drilling. The experiment is run as a single
replicate full factorial with five center points. The data are given below. Refer to the
JMP output for drilling experiment.
Trmt. Comb.
1
a
b
ab
c
ac
bc
abc
d
ad
bd
abd
cd
acd
bcd
abcd
A
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
–1
+1
0
0
0
0
0
B
–1
–1
+1
+1
–1
–1
+1
+1
–1
–1
+1
+1
–1
–1
+1
+1
0
0
0
0
0
C
–1
–1
–1
–1
+1
+1
+1
+1
–1
–1
–1
–1
+1
+1
+1
+1
0
0
0
0
0
D
–1
–1
–1
–1
–1
–1
–1
–1
+1
+1
+1
+1
+1
+1
+1
+1
0
0
0
0
0
Rate
1.68
1.98
3.28
3.44
4.98
5.70
9.93
9.07
2.07
2.44
4.09
4.53
7.77
8.43
11.75
12.30
4.54
5.18
5.53
4.49
5.01
a) Give the overall sample mean and the estimated full effects for the factorial
portion of the experiment.
b) Give the mean at the center points and the sample variance at the center points.
c) Use the sample variance at the center points to determine the critical effect size
for a factor or interaction term. Use t=3.
d) Use the critical effect size in c) to determine which factors and/or interactions are
statistically significant.
e) Is there significant curvature? Support your answer by performing the appropriate
test of hypothesis.
f) According to JMP, which factors and/or interactions are statistically significant?
Use t=3.
g) Why are the results in d) different from the results in f)?
2
JMP Output for Drilling Experiment
Parameter Estimates
Estimated
Term
Half Effect
Intercept
5.84
A
0.14625
B
1.45875
A*B
–0.11000
C
2.90125
A*C
–0.01250
B*C
0.56250
A*B*C
–0.10125
D
0.83250
A*D
0.10625
B*D
0.03625
A*B*D
0.10500
C*D
0.48875
A*C*D
0.06250
B*C*D
–0.09500
A*B*C*D
0.07875
Center Points
Mean
Std Dev
N
4.95
0.4394883
5
Response: Rate of Advance of the Drill
Summary of Fit: Full Factorial in A, B, C, and D
Rsquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
Analysis of Variance
Source
DF
Model
15
Error
5
C. Total
20
0.980451
0.921802
0.870646
5.628095
21
Sum of Squares
190.08320
3.79012
193.87332
Mean Square
12.6722
0.7580
F Ratio
16.7174
Prob > F
0.0028
3
Lack Of Fit
Source
Lack Of Fit
Pure Error
Total Error
DF
1
4
5
Sum of Squares
3.0175238
0.7726000
3.7901238
Parameter Estimates
Term
Estimate
Intercept
5.6280952
A
0.14625
B
1.45875
A*B
–0.11
C
2.90125
A*C
–0.0125
B*C
0.5625
A*B*C
–0.10125
D
0.8325
A*D
0.10625
B*D
0.03625
A*B*D
0.105
C*D
0.48875
A*C*D
0.0625
B*C*D
–0.095
A*B*C*D
0.07875
Effect Tests
Source
A
B
A*B
C
A*C
B*C
A*B*C
D
A*D
B*D
A*B*D
C*D
A*C*D
B*C*D
A*B*C*D
Nparm
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
DF
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Std Error
0.189991
0.217662
0.217662
0.217662
0.217662
0.217662
0.217662
0.217662
0.217662
0.217662
0.217662
0.217662
0.217662
0.217662
0.217662
0.217662
Mean Square
3.01752
0.19315
t Ratio
29.62
0.67
6.70
-0.51
13.33
-0.06
2.58
-0.47
3.82
0.49
0.17
0.48
2.25
0.29
-0.44
0.36
Sum of Squares
0.34222
34.04722
0.19360
134.67602
0.00250
5.06250
0.16403
11.08890
0.18063
0.02103
0.17640
3.82203
0.06250
0.14440
0.09922
F Ratio
15.6227
Prob > F
0.0168
Max RSq
0.9960
Prob>|t|
<.0001
0.5314
0.0011
0.6348
<.0001
0.9564
0.0492
0.6614
0.0123
0.6461
0.8743
0.6499
0.0747
0.7855
0.6807
0.7323
F Ratio
0.4515
44.9157
0.2554
177.6671
0.0033
6.6785
0.2164
14.6287
0.2383
0.0277
0.2327
5.0421
0.0825
0.1905
0.1309
Prob > F
0.5314
0.0011
0.6348
<.0001
0.9564
0.0492
0.6614
0.0123
0.6461
0.8743
0.6499
0.0747
0.7855
0.6807
0.7323
4