Statistics 402C
Final Exam, Spring 2000
Name:
INSTRUCTIONS: Read the questions carefully and completely. Answer the questions and
show work in the space provided.
1. [40 pts] A fractional factorial experiment is performed to assess the effect of six factors
on the cutting force produced during the machining of an aluminum alloy. The factors,
and their levels, are given below.
Factor
Low Level (–) High Level (+)
A: Cutting speed (sf/min)
2000
2500
B: Tool Geometry
Negative
Positive
C: Machine
American
Monarch
D: Cutting Fluid
Absent
Present
E: Feedrate (in/rev)
0.005
0.010
F: Depth of cut (in)
0.050
0.100
The experiment uses the two design generators: E=ABC and F=BCD to get the
plus/minus levels for the last two factors. The data are given below.
A
–
+
–
+
–
+
–
+
–
+
–
+
–
+
–
+
B
–
–
+
+
–
–
+
+
–
–
+
+
–
–
+
+
C
–
–
–
–
+
+
+
+
–
–
–
–
+
+
+
+
D
–
–
–
–
–
–
–
–
+
+
+
+
+
+
+
+
E=BCD F=ABD Force
–
–
21
–
+
45
+
+
119
+
–
61
+
–
59
+
+
110
–
+
65
–
–
35
+
+
122
+
–
52
–
–
31
–
+
57
–
+
57
–
–
29
+
–
70
+
+
121
(a) [5] Give the full defining relation for this experiment
(b) [2] What is the resolution of this experiment?
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(c) [8] Fill in the values of the estimated effects below.
Mean
D
A
AD
B
BD
AB
ABD
C
CD
AC
ACD
BC
BCD
ABD
ABCD
(d) [5] On the normal plot below clearly label (with names and aliases) those points
that correspond to effects that appear to be significant.
2
(e) [5] Can the experiment be projected into a smaller experiment by dropping apparently insignificant factors? If so, what factors should be dropped? How will
you estimate error variability?
(f) [5] What factors are significant at a 1% level? Give the value of the MSError
you are using. Also give the F-value and P-value for each significant factor and
interaction.
(g) [5] Suppose that for a 0.10 inch Depth of cut we wish to keep the cutting force at
68.4. What Feedrate should be used? I want the Feedrate in in/rev. Hint: You
will have to interpolate.
(h) [5] What levels of the 6 factors would you recommend to produce the lowest
cutting force? Give the predicted cutting force for your recommended levels.
3
2. [40 pts] An experiment was performed by some statistics students in Australia on the
time to swim 25 meters. There were four factors that were manipulated.
Factor
A: Wearing Shirt
B: Wearing Goggles
C: Wearing Flippers
D: Starting End
Low (−1) High (+1)
No
Yes
No
Yes
No
Yes
Shallow
Deep
One person did all the swimming with enough time between laps to aid recovery. Still,
randomization is important so that fatigue does not bias the results. Refer to the SAS
output on the next page.
(a) [5] On the normal plot below identify (by name) the estimated effects that appear
to be large.
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Number of observations in data set = 16
Dependent Variable: TIME
Source
DF
Type I SS
Mean Square
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
Error
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
7.115556
6.825156
0.166056
108.004056
1.543806
3.303306
0.007656
0.010506
0.003906
0.068906
0.033306
0.288906
0.037056
0.005256
0.003306
.
7.115556
6.825156
0.166056
108.004056
1.543806
3.303306
0.007656
0.010506
0.003906
0.068906
0.033306
0.288906
0.037056
0.005256
0.003306
.
15
127.416744
Corrected Total
F Value
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Pr > F
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Estimate
T for H0:
Parameter=0
Pr > |T|
Parameter
Std Error of
Estimate
INTERCEPT
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
19.70687500
0.66687500
-0.65312500
0.10187500
-2.59812500
-0.31062500
0.45437500
0.02187500
0.02562500
0.01562500
0.06562500
0.04562500
-0.13437500
0.04812500
0.01812500
-0.01437500
9999.99
9999.99
-9999.99
9999.99
-9999.99
-9999.99
9999.99
9999.99
9999.99
9999.99
9999.99
9999.99
-9999.99
9999.99
9999.99
-9999.99
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
(b) [10] Drop factor D: Starting End and project the experiment into a smaller design.
Give the MS Error for this smaller experiment and identify those effects that are
significant at the 1% level. Be sure to report F-values for the significant effects.
(c) [10] Use higher order interactions to come up with an alternative estimate of
Error variability. Use this alternative estimate to identify those effects that are
significant at the 1% level. Be sure to report the F values for the significant
effects.
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(d) [8] Below are interaction plots for factors A, B and C. Explain what is going
on for the significant interactions. It is not enough to say that the lines are not
parallel you must discuss what the interaction effect is.
(e) [7] Based on your analysis in (b), in order to have the fastest (lowest) 25 m swim
time, what would you recommend? Be sure to give values for each of the four
variables and predict what the swim time would be for your recommendation.
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3. [35 pts] For the following situations give the name of the design used to collect the
data. Indicate what are the factors of interest and what are nuisance variables. In
a split plot design indicate what is the whole plot and what is the sub plot factor.
Give a partial analysis of variance table indicating all sources of variability and their
associated degrees of freedom.
(a) [7] An experiment is done using 8 turtles. Four of the turtles are male and four
of the turtles are female. Each turtle has their plasma protein measured three
times; once while they were well fed (no fasting), once after ten days of fasting
and once after twenty days of fasting. The researcher wishes to know if there
are differences between the sexes. She also wants to know if there are differences
among the fasting times and if there is any interaction between sex and fasting
time.
(b) [7] A fishing magazine hires a statistician to conduct an experiment on the effects of A: reel type (Zebco and Generic), B: line type (Berkeley and Generic),
and C: sinker weight (25 g and 50 g) on the distance the line casts in meters.
Eight identical rods are purchased. Each rod is set up with a different treatment
combination of A: reel, B: line and C: sinker weight. Twenty four slips of paper,
3 slips for each of the 8 treatment combinations, are placed in a sack. Drawing
slips from the sack gives the random order that the rod is used to cast the line.
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(c) [7] In an agricultural field trial, nitrogen level (low and high), potassium level
(low and high), and water level (low and high) are investigated for their effects on
the growth of tomatoes in a green house. Sixty four tomato plants, each in its own
pot are arranged in 8 rows of 8 pots in the green house. Because of the placement
of heaters, there may be differences from the east side to the west side of the green
house. Because of the angle of the sun there may be differences from the north
end to the south end of the green house. The eight treatment combinations of
nitrogen, potassium and water are arranged so that each combination is in only
one row and only one column.
(d) [7] An article in the Journal of Quality Technology (1985), Vol. 17, pp. 198-206,
describes an experiment looked at the effect of 5 factors on the free height of leaf
springs used in an automotive application. The factors are: A=furnace temperature, B=heating time, C=transfer time, D=hold down time, and E=quench oil.
Only 16 treatment combinations were investigated with the levels of E=quench oil
set by signs associated with the ABCD interaction. Although only 16 treatment
combinations are used, each treatment combination was replicated 4 times and
the order of the 64 runs was completely randomized.
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(e) [7] An experiment is performed to see whether different operators obtain different
results in the routine analysis to determine the amount of nitrogen in soil. Soil
samples are taken from 10 different fields, each field may naturally have different
amounts of nitrogen. The soil from a field is divided into 5 portions, the portions
are randomly assigned to the five operators and each operator determines the
amount of nitrogen in the soil. This is done for each of the 10 fields.
I hope to have the finals graded and course grades posted by Friday
afternoon, May 5. You can pick up your graded final exam at my office.
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