Statistics 496, Applied Statistics for Industry II
Name: _________________
Final Exam, Spring 2009
Site: ___________________
INSTRUCTIONS: You will have 2 hours to complete the exam. There are 5 questions
worth a total of 125 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. [35 pts] An experiment on the fuel economy of a particular make and model of an
automobile was designed using five factors, each at 2 levels.
Factor
A: Engine Size (L)
B: Fuel Octane
C: Tire Pressure (psi)
D: Driving Speed (mph)
E: Air Conditioning
Low Level
4.0
87
22
45
Off
High Level
4.5
93
28
55
On
The experiment was run as a 25–1 Resolution V fractional factorial with the levels of E
set according to the plus/minus coding of the ABCD interaction. Below are the data,
in standard Yates order, and estimated full effects.
Treatment
Combination
e
a
b
abe
c
ace
bce
abc
d
ade
bde
abd
cde
acd
bcd
abcde
Fuel Economy
(mpg)
25
21
29
23
29
20
29
29
27
19
26
26
27
22
31
25
Estimated Full
Effect
25.50
–4.75
3.50
1.75
2.00
–0.25
0.50
0.25
–0.25
0.00
–0.25
0.00
–0.25
–0.50
–0.25
–2.50
Effect Name
Alias
Mean
A
B
AB
C
AC
BC
ABC
D
AD
BD
ABD
CD
ACD
BCD
ABCD
a) [4] Fill in the aliases.
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b) [4] On the normal plot of effects given below identify, by name and alias, those
effects that appear to be significant.
c) [5] Can you reduce this to a smaller experiment in fewer factors? If so, what are
those factors? Will you pick up pseudo-replication?
d) [7] Using an MSError = 0.20, which of the estimated effects are significant. Use
t=3 to compute the critical effect size.
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e) [5] Give a prediction equation using only those factors that are found to be
significant in d). Explicitly define any variables used in your equation.
f) [5] What factor levels, do NOT give me plus/minus codes but the actual levels,
would you recommend in order to maximize the fuel economy?
g) [5] What are the predicted value and prediction interval for the fuel economy at
the recommended levels in f)?
2. [15 pts] Statistical thinking consists of the recognition that variability always is, and
always will be, present and that learning is an iterative procedure (one must go
through several cycles to truly learn about something). How do statistically designed
experiments fit into the framework of statistical thinking? Be sure to mention both
aspects; variability and the iterative nature of learning.
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3. [30] An experiment is performed on a CNC milling machine. The three factors of
interest are A: Spindle Speed (1750, 2000, and 2250 rpm), B: Feed Rate (6, 12, and
18 in/sec) and Depth of Cut (0.04, 0.06, and 0.08 in). All 27 treatment combinations
are run in a completely random order.
Level
1750
2000
2250
A: Spindle Speed
Mean X1,A
63.2
–1
86.2
0
78.0
+1
X2,A
+1
–2
+1
Level
6
12
18
B: Feed Rate
Mean X1,B
54.7
–1
75.6
0
97.1
+1
X2,B
+1
–2
+1
C:
Level
0.04
0.06
0.08
Depth of Cut
Mean X1,C X2,C
71.3
–1
+1
84.0
0
–2
72.1
+1
+1
a) [13] Complete the following table of estimated slope parameters. Note MSError =
80.74.
Term
Intercept
Estimate
75.8
X1,A
7.4
X1,B
21.2
Std Error
*******
t Ratio
********
X1,C
X2,A
–5.2
1.223
X2,B
X2,C
–4.1
X1,AX1,B
–2.4
2.594
–0.93
X1,AX1,C
2.0
2.594
0.77
X1,BX1,C
–2.7
2.594
–1.04
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b) [5] According to the t-Ratios, which terms are statistically significant? Explain
briefly?
c) [4] Give the reduced model prediction equation that includes only those terms that
are statistically significant.
d) [6] A low value of the surface roughness is desirable. According to your
prediction equation in c) how would you set Spindle Speed, Feed Rate and Depth
of Cut to get the lowest surface roughness?
e) [2] What is the predicted surface roughness for your settings in d)?
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4. [30 pts] The data below are the voltage levels at which failures occurred in two types
of electrical cable insulation when specimens were subjected to an increasing voltage
stress in a laboratory experiment. The test involved 20 specimens of each type, and
the failure voltages (in kilovolts per millimeter) were:
Type 1 insulation
Type 2 insulation
32.0
46.5
39.4
57.1
35.4
46.8
45.3
57.2
36.2
47.3
49.2
57.5
39.8
47.3
49.4
59.2
41.2
47.6
51.3
61.0
43.3
49.2
52.0
62.4
45.5
50.4
53.2
63.8
46.0
50.9
53.2
64.3
46.2
52.4
54.9
67.3
46.4
56.3
55.5
67.7
a) [5] Looking at the probability plot on the next page, explain briefly why a
Weibull model is an appropriate model for the insulation failure voltage
.
b) [5] Assuming a Weibull model for Type 2 insulation, graphical estimates of the
model parameters are: βˆ = 9.1 and λˆ = 0.017 . Using these values, estimate the
chance that Type 2 insulation will withstand 50 kilovolts per millimeter.
c) [5] Using the Weibull model, estimate the median failure voltage for Type 2
insulation?
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d) [10] For Type 1 insulation, obtain graphical estimates of β and λ . Again assume
that a Weibull model is appropriate.
e) [5] For Type 1 insulation the chance that it will withstand 50 kilovolts per
millimeter is approximately 0.13 and the median voltage to failure is
approximately 44.2 kilovolts per millimeter. Which insulation should be
purchased if protection against voltage stress is needed? Why? What assumptions
are implicit in your decision?
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5. [15 pts] The three fundamental principles of a well designed experiment are control of
outside variables, randomization and replication within the experiment.
a) [4] Why is control of outside variables important?
b) [5] If an outside variable is not controlled, how can randomization help?
c) [2] What is replication within an experiment?
d) [4] What problems can occur when you do not have replication in an experiment?
Because I may not receive your final exam before grades are due at the Registrar's Office,
you may receive notification from the Registrar's Office, or see on AccessPlus, that either
no grade has been submitted or an Incomplete grade has been submitted. Please do not
worry about this. I will return graded final exams, critiques of the project and course
grades to you by May 18, 2009.
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