STATISTICS 402B
Spring 2016
Homework Set#3
1. Construction engineers wish to know the effect of polypropylene fibers on the compressive strength of
concrete. Fifteen concrete cubes are produced and randomly assigned to five levels of fiber content (0%,
.25%, .50%, .75%, and 1%). Data are:
Fiber content (%) Strength (ksi)
0
7.8 7.4 7.2
.25
7.9 7.5 7.3
.50
7.4 6.9 6.3
.75
6.7 6.4
Solutions from Montgomery, D. C. (2008) Design7.0
and Analysis
of Experiments, Wiley, NY
1
5.9 5.8 5.6
Chapter 4
Use JMP to analyze these data to determine if fiber content has an effect on concrete strength. Extract
numbers from the JMP output to present your own written answers to the parts below. Assume the model
Randomized Blocks, Latin Squares, and Related Designs
yij = µi + ij , i = 1, . . . , 5; j = 1, . . . , 3 where ij are iid N (0, σ 2 ).
Solutions
(a) Construct an analysis of variance table and test the hypothesis H0 : µ1 = µ2 = µ3 = µ4 = µ5 using
the p-value and α = .05. What are the best unbiased estimates of µi , i = 1, . . . , 5 and σ 2 .
Solutions from Montgomery, D. C. (2008) Design and Analysis of Experiments, Wiley, NY
4.1.engineers
The ANOVA
fromequally
a randomized
complete
outputais quantitative
shown below. factor, so that they
(b) The
selected
spaced
levelsblock
for experiment
fiber content,
can assess whether factor effects are linear, quadratic, or of higher order polynomial. Use orthogonal
Source
SS
MS
F
contrast to test whether
there is a DF
linear relationship
between the
factorPmeans and the factor levels. If
4
1010.56exists ?describe
29.84
?
your analysis indicatesTreatment
that such a relationship
that relationship
by fitting a straightline
Randomized
Blocks, Latin Squares,
and
Related
Designs
to the data with the fiber
Block content ?% as the ?x-variable.
64.765
?
?
Chapter 4
2. Problem 4.1 (Montgomery)Error
20
Solutions
169.33
?
Total
29
1503.71
4.1. The ANOVA from a randomized complete block experiment output is shown below.
(a) Fill in the blanks. You may give bounds on the P-value.
Source
Completed table is:
DF
SS
MS
F
P
4
1010.56
?
29.84
?
DF
?
SS
?
MS
64.765
F?
P
?
Treatment
Error
4
20
1010.56
169.33
252.640
?
29.84
< 0.00001
Block
Total
5
29
323.82
1503.71
64.765
Error
20
169.33
8.467
Treatment
Source
Block
(a) Fill in the blanks. You may give bounds on the P-value.
Total
29
1503.71
Completed table is:
(b) How many blocks were used in this experiment?
Source
Treatment
DF
SS
MS
F
P
4
1010.56
252.640
29.84
< 0.00001
(c) What conclusions can you draw?
Block
5
323.82
64.765
Error
20
169.33
8.467
Total
29
1503.71
4.2. How
Consider
the single-factor
randomized experiment shown in Problem 3.4. Suppose that
(b)
many blocks
were used completely
in this experiment?
this experiment had been conducted in a randomized complete block design, and that the sum of squares for
blocks
80.00.
Sixwas
blocks
wereModify
used. the ANOVA for this experiment to show the correct analysis for the randomized
complete block experiment.
(c) What conclusions can you draw?
The modified ANOVA is shown below:
1
The treatment effect is significant; the means of the five treatments are not all equal.
Solutions from Montgomery, D. C. (2008) Design and Analysis of Experiments, Wiley, NY
3. Problem 4.3 (Montgomery)
4.3. A chemist wishes to test the effect of four chemical agents on the strength of a particular type of
cloth. Because there might be variability from one bolt to another, the chemist decides to use a randomized
block design, with the bolts of cloth considered as blocks. She selects five bolts and applies all four
chemicals in random order to each bolt. The resulting tensile strengths follow. Analyze the data from this
experiment (use α = 0.05) and draw appropriate conclusions.
Chemical
1
2
3
4
1
73
73
75
73
Bolt
3
74
75
78
75
2
68
67
68
71
4
71
72
73
75
5
67
70
68
69
Design
Expert
Use
JMP
toOutput
analyze these data to determine strength means were different among the chemical agents.
Response:
Strength
ExtractANOVA
numbers
from Factorial
the JMP
output to present your own written answers to the parts below. Assume
for Selected
Model
variance
the Analysis
model ofyij
= µi table
+ ij[Partial
, i =sum
1, .of. .squares]
, 4; j = 1, . . . , 3 where ij are iid N (0, σ 2 ).
Sum of
Mean
F
Source
Squares
DF
Square
Value
Prob > F
(a)Block
Construct an
analysis 4of variance
157.00
39.25 table and test the hypothesis H0 : µ1 =
Model
12.95
4.32
2.38
0.1211
not significant
p-value
andfrom
α = .05. 3
Solutions
D. C.4.32
(2008) Design
A
12.95Montgomery,
3
2.38and Analysis
0.1211of Experiments, Wiley,
(b)Residual
Calculate the
residuals
21.80
12 and obtain
1.82 plots to check adequacy of the model.
Cor Total
191.75
19
µ2 = µ3 = µ4 using the
NY
(c) Use
LSD procedure
with α
= .05between
to compare
differences
of mean solution
strength
among the chemicals.
There
is nothe
difference
in mean bacteria
growth
solutions
1 and 2. However,
3 produces
The
"Modelyour
F-value"
of 2.38 bacteria
implies
model
is not
significant
the noise. There
is a
State
inthethe
context
ofisthe
problem.
significantly
lowerconclusion
mean
growth.
This
therelative
sameto conclusion
reached
from the Fisher LSD
12.11 % chance that a "Model F-value" this large could occur due to noise.
procedure in Problem 4.4.
4. Problem
Std. Dev.4.7 (Montgomery)
1.35
Mean
4.7.C.V.Consider
PRESS
71.75
1.88
the hardness
60.56
R-Squared
0.3727
Adj R-Squared
0.2158
Pred
R-Squared
-0.7426
testing experiment described in Section
Adeq Precision
10.558
4.1. Suppose that the experiment was
conducted as described and the following Rockwell C-scale data (coded by subtracting 40 units) obtained:
Treatment Means (Adjusted, If Necessary)
Estimated
Standard
Mean
Error
Tip
1-1
70.60
0.60
1
2-2
71.40
0.60
3-3
72.40
0.60
2
4-4
72.60
0.60
3
4
Standard
Coupon
1
2
3
9.3 9.4
9.6
9.4 9.3
9.8
9.2 9.4
9.5
9.7t for H9.6
10.0
0
4
10.0
9.9
9.7
10.2
Mean
Treatment Difference
DF
Error
Coeff=0
Prob > |t|
1 vs 2
1 experiment.
0.85
-0.94
0.3665
(a) Analyize
the-0.80
data from this
Use JMP
these1 data to
differences among the four tips. Extract numbers
1 vs 3 to analyze
-1.80
0.85determine
-2.11hardness0.0564
1 the
vs 4 JMP -2.00
1
0.85
-2.35
0.0370 to the parts below. Assume the model y =
from
output
to
present
your
own
written
answers
ij
There
is a3 difference
of the four-1.17
tips.
2 vs
-1.00between 1the means
0.85
0.2635
µi + 2vs
. . . , 4; j = 11, . . . , 40.85
where ij -1.41
are iid N (0,
σ 2 ).
ij , 4 i = 1, -1.20
0.1846
3 vs 4
-0.20
1
0.85
-0.23
0.8185
(a) Construct an analysis of variance table and test the hypothesis H0 : µ1 = µ2 = µ3 = µ4 using the
There
is no difference
p-value
and α =among
.05. the chemical types at α = 0.05 level.
(b) Calculate the residuals and obtain plots to check adequacy of the model.
(c) Use the LSD procedure with α = .05 to compare differences of mean hardness among the tips. State
your conclusion in the context of the problem.
5. An experiment was run to determine the effects of three brands of engine oil A, B, and C on the wear
of piston rings. The measure of wear was taken as the logarithm of loss of piston-ring weight (in mg.)
in a 12-hour test run. Since it is known that rings of different types were quite variable in wear, a
randomized complete block design with type of piston rings as the blocking factor. Four types of piston
rings were arbitrarily chosen to be included in the experiment and the experimenter had no interest in their
performance. The results are shown below:
4-2
2
Piston Ring
Type
1
2
3
4
Brand of Oil
A
B
C
1.782 1.568 1.570
1.306 1.223 1.240
1.982 1.919 1.874
1.149 1.029 1.068
(a) Note that 12 piston rings (3 of each type) are used in the experiment. Describe the randomization
procedure i.e., describe how the oils are assigned to the piston rings. After this is done the run order
of the test runs were also determined randomly. How is that determined?
(b) Construct an analysis of variance table and test the hypothesis H0 : µ1 = µ2 = µ3 using the p-value
and α = .05. What are the best unbiased estimates of µi , i = 1, . . . , 3 and σ 2 .
(c) Calculate the residuals and obtain plots to check adequacy of the model.
(d) Brands B and C contain a specific synthetic additive (which is not in Brand A), that is believed to
retard wear at high temperatures. Test an appropriate hypotheses and report the t-statistic and the
p-value. What is your conclusion from this test?
Note: Need to present written answers to each part, with calculation shown for parts that you are required to
do hand computation. Use the JMP output to obtain numbers for answering other parts. Attach edited JMP
output when you use the JMP output to extract numbers as part of the analysis. The JMP data files fiber.jmp,
4-3.jmp, and 4-7.jmp are available to download.
Due Wednesday, March 2nd, 2016 (turn-in at the beginning of class)
3