Statistics 496, Applied Statistics for Industry II
Name: _________________
Exam 2, Spring 2009
Site: ___________________
INSTRUCTIONS: You will have 1 hour and 30 minutes to complete the exam. There are
4 questions worth a total of 100 points. Not all questions have the same point value so
gauge your time appropriately. Read the questions carefully and completely. Answer
each question and show work in the space provided on the exam. Turn in the entire exam
when you are done or when time is up. For essay questions, think before you write.
1. [30] The effects of three process variables; A: Seal Temperature (o F), B: Cooling Bar
Temperature (o F), and C: % Polyethylene Additive on the seal strength (g/in) of a bread
wrapper stock are studied with a 23 factorial experiment with 4 center points. The data
are given below.
A
225
285
225
285
225
285
225
285
255
255
255
255
B
46
46
64
64
46
46
64
64
55
55
55
55
C
0.5
0.5
0.5
0.5
1.7
1.7
1.7
1.7
1.1
1.1
1.1
1.1
Strength
6.6
6.9
7.9
6.1
9.2
6.8
10.4
7.3
10.4
9.5
9.8
9.9
The estimated full effects for this experiment are as follows:
A: –1.75, B: 0.55, AB: –0.70, C: 1.55, AC: –1.00, BC: 0.30, and ABC: 0.35.
The variance at the center points is 0.14.
a) [6] Compute the standard error of an estimated full effect and the critical effect
size (use t=3). Which of the estimated full effects are statistically significant?
Explain briefly.
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b) [5] Using only those effects that are statistically significant, give the prediction
equation for seal strength. Be sure to explicitly define the variables in the
prediction equation.
c) [3] Using the prediction equation in (b), predict the seal strength for a seal
temperature of 225oF, cooling bar temperature of 64oF and polyethylene additive
of 1.7%.
d) [5] Construct a prediction interval for your prediction in (c).
e) [7] Is there evidence of curvature in the response? Be sure to include the mean of
the factorial experiment and the mean for the center points as well as the value
and interpretation of an appropriate test statistic.
f) [4] In light of your result in (e), would you say your prediction in (c) is accurate?
Explain briefly.
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2. [20 pts] Golf ball manufacturers appeal to golfers with claims that their product goes
farther, flies straighter or lands softer. Of particular interest is the driving distance of a
golf ball. Some of the factors that may affect the distance a golf ball travels are: brand,
compression, core and cover. Compression is a measure of the resiliency of the ball when
it is struck by the golf club. Golf balls have either a one-piece core or a two-piece core
(rubberized string wound around a solid or liquid inner core). The two types of cover
commonly used are a hard cover (Surlyn) and a softer cover (Balata). Besides the
characteristics of the golf ball, the golfer and weather/course conditions have a big effect
on the distance a golf ball will travel. Better golfers can hit the ball straighter and farther
than beginners. A 23 factorial experiment is run with a single golfer in a completely
randomized design with 5 replications. All golf balls have two-piece cores. The factors
are:
Factor
A: Brand
B: Compression
C: Cover
Low (–1)
Titleist
90
Balata
High (+1)
Maxfli
100
Surlyn
The summarized data are:
A
B
C
–1
+1
–1
+1
–1
+1
–1
+1
–1
–1
+1
+1
–1
–1
+1
+1
–1
–1
–1
–1
+1
+1
+1
+1
Mean, Yi
260.8
259.8
258.8
257.4
266.0
276.2
275.8
272.6
ni
5
5
5
5
5
5
5
5
Variance, s i2
136.7
81.7
120.2
170.3
166.0
113.7
171.7
104.3
a) [6] Complete the following list of estimated full effects.
Effect Name
Y
A
B
AB
C
AC
BC
ABC
Estimate
265.925
0.45
13.45
2.35
2.65
–3.25
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b) [2] Compute the MSrepError for this problem.
c) [5] Calculate the standard error of an estimated effect and give the critical effect
size, use t=3. What effect(s) is(are) significant?
d) [7] In a brief paragraph, summarize your findings. This paragraph should be
written so that it could be understood by someone with little or no knowledge of
statistics. Be sure to comment on the limitations of this experiment.
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3. [20 pts] 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. The levels of each of the 3
factors are given below.
low (–1)
50 rpm
Type 1
5o
A: cutting speed
B: tool geometry
C: cutting angle
high (+1)
100 rpm
Type 2
10o
Since the hardness of the raw material being tooled may vary, heats of raw material are
used as blocks. Below are the data.
A
–1
+1
–1
+1
–1
+1
–1
+1
B
–1
–1
+1
+1
–1
–1
+1
+1
Block Mean
C
–1
–1
–1
–1
+1
+1
+1
+1
I
22
32
35
45
43
36
50
47
38.75
Block
II
31
43
50
55
45
38
61
43
45.75
III
25
30
35
47
38
40
54
39
38.5
Mean
26
35
40
49
42
38
55
43
41.0
For this experiment SSrepError = 474.0 with 16 degrees of freedom. Then estimated full
effects are: A: 0.5, B: 11.5, AB: –2.0, C: 7.0, AC: –8.5, BC: –2.5, ABC: –2.0.
a) [4] Below is a plot of the main effects. Comment on the apparent effects of the
three factors.
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b) [5] Below are plots of the A: Speed by C: Angle interactions for Type 1 and Type
2 Geometries. What do these plots indicate about the possible interaction between
A and C? What do they indicate about the 3-way interaction?
c) [8] What is the standard error of an estimated full effect?
d) [3] What effects are statistically significant? Use t=3.
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4. [30 pts] A 24 un-replicated factorial experiment was conducted on the polymer resin,
PMR-II-50 to see the effect of four factors on weight loss. The four factors and their
levels are given below.
Factor
Oven Position
Kapton
Preprocessing
Dianhydride
Low Level (–1)
Position 1
no kapton (0)
Time 15 minutes
Type Polymer grade (1)
High Level (+1)
Position 2
with kapton (1)
120 minutes
Electronic grade (2)
All specimens, which initially had about the same mass, are baked at 600oF for 936
hours. The percent weight loss (wghtloss) is the measured response. The output of the
analysis of the 16 observations is given below.
Fractional Factorial Fit
Estimated Full Effects and Parameter Estimates for wghtloss
Term
Constant
Position
Kapton
Time
Type
Position*Kapton
Position*Time
Position*Type
Kapton*Time
Kapton*Type
Time*Type
Position*Kapton*Time
Position*Kapton*Type
Position*Time*Type
Kapton*Time*Type
Position*Kapton*Time*Type
Full
Effect
4.3375
0.2750
0.0750
–0.0500
–1.0500
0.0750
0.0500
–0.2500
–0.0500
–0.2000
–0.0250
0.0500
0.1000
0.0750
–0.0750
0.1250
Parameter
Estimate
0.1375
0.0375
–0.0250
–0.5250
0.0375
0.0250
–0.1250
–0.0250
–0.1000
–0.0125
0.0250
0.0500
0.0375
–0.0375
0.0635
Analysis of Variance for wghtloss
Source
Main Effects
2-Way Interactions
3-Way Interactions
4-Way Interactions
Residual Error
Total
DF
4
6
4
1
0
15
SS
4.74500
0.45500
0.09500
0.06250
0.00000
5.35750
MS
1.18625
0.07583
0.02375
0.06250
0.00000
On the next page is a normal probability plot of estimated full effects.
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a) [5] On the normal plot above, clearly label, with their names, the effects that you
suspect are significant.
b) [8] If we use the 3-Way and 4-Way interactions to produce an error term, which
main effect(s) and interaction(s) would be statistically significant? Justify your
answer and use t=3.
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An alternative analysis would drop Time and look at the analysis of the 23 experiment in
Position, Kapton and Type. The output of that analysis is given below.
Fractional Factorial Fit
Estimated Full Effects and Parameter Estimates for wghtloss
Term
Full
Effect
4.3375
0.2750
0.0750
–1.0500
0.0750
–0.2500
–0.2000
0.1000
Constant
Position
Kapton
Type
Position*Kapton
Position*Type
Kapton*Type
Position*Kapton*Type
Parameter
Estimate
0.1375
0.0375
–0.5250
0.0375
–0.1250
–0.1000
0.0500
Analysis of Variance for wghtloss
Source
Main Effects
2-Way Interactions
3-Way Interactions
Residual Error
Pure Error
Total
DF
3
3
1
8
8
15
SS
4.73500
0.43250
0.04000
0.15000
0.15000
5.35750
MS
1.57833
0.14417
0.04000
0.01875
0.01875
c) [7] Using this alternative analysis, which main effect(s) and interaction(s) are
statistically significant? Justify your answer and use t=3.
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d) [3] Is there another factor that can be dropped from the model? If so, which
factor?
e) [7] Give the prediction equation for the model with only significant terms. Use
this model to make recommendations on how to set all of the factors to give the
least amount of weight loss. What is the predicted weigh loss at this
recommendation?
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