Applied Statistics and Probability for Engineers, 7th edition 2017
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(c)
14.S9 An article in the Textile Research Institute Journal (1984, Vol. 54, pp. 171–179) reported the results of an experiment
that studied the effects of treating fabric with selected inorganic salts on the flammability of the material. Two
application levels of each salt were used, and a vertical burn test was used on each sample. (This finds the temperature
at which each sample ignites.) The burn test data follow.
Level
Salt
Untreated
MgCl2
NaCl
CaCO3
CaCl2
Na2CO3
1
812
752
739
733
725
751
827
728
731
728
727
761
876
764
726
720
719
755
2
945
794
741
786
756
910
881
760
744
771
781
854
919
757
727
779
814
848
(a) Test for differences between salts, application levels, and interactions. Use
= 0.01.
(b) Draw a graph of the interaction between salt and application level. What conclusions can you draw from this graph?
(c) Analyze the residuals from this experiment.
Applied Statistics and Probability for Engineers, 7th edition 2017
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(a)
Factor Type Levels Values
Level fixed 2 1 2
(b)
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(c)
14.S10 An article in the Journal of Coatings Technology (1988, Vol. 60, pp. 27–32) described a 24 factorial design used for
studying a silver automobile basecoat. The response variable is distinctness of image (DOI). The variables used in the
experiment are
A = Percentage of polyester by weight of polyester/melamine (low value = 50%, high value = 70%)
B = Percentage of cellulose acetate butyrate carboxylate (low value = 15%, high value = 30%)
C = Percentage of aluminum stearate (low value = 1%, high value = 3%)
D = Percentage of acid catalyst (low value = 0.25%, high value = 0.50%)
The responses are (1) = 63.8, a = 77.6, b = 68.8, ab = 76.5, c = 72.5, ac = 77.2, bc = 77.7, abc = 84.5, d = 60.6,
ad = 64.9, bd = 72.7, abd = 73.3, cd = 68.0, acd = 76.3, bcd = 76.0, and abcd = 75.9.
(a) Estimate the factor effects.
(b) From a normal probability plot of the effects, identify a tentative model for the data from this experiment.
(c) Using the apparently negligible factors as an estimate of error, test for significance of the factors identified in part
(b). Use
= 0.05.
(d) What model would you use to describe the process based on this experiment? Interpret the model.
(e) Analyze the residuals from the model in part (d) and comment on your findings.
Applied Statistics and Probability for Engineers, 7th edition 2017
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(a)
Estimated Effects and Coefficients for DOI
Term Effect Coef
Constant 72.894
A 5.763 2.881
B 5.563 2.781
(b)
(c) Conduct an analysis using the main factors A, B, and C and interactions among these variables to check if any are
significant.
Predictor Coef StDev T P
Constant 72.894 1.073 67.92 0.000
A 2.881 1.073 2.68 0.028
Applied Statistics and Probability for Engineers, 7th edition 2017
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(d)
The regression equation is
Predictor Coef StDev T P
Constant 72.8937 0.9365 77.84 0.000
Analysis of Variance
Source DF SS MS F P
Regression 3 412.22 137.41 9.79 0.002
(e)
14.S11 An article in Solid State Technology (1984, Vol. 29, pp. 281–284) described the use of factorial experiments in
photolithography, an important step in the process of manufacturing integrated circuits. The variables in this
experiment (all at two levels) are prebake temperature (A), prebake time (B), and exposure energy (C), and the response
variable is delta line width, the difference between the line on the mask and the printed line on the device. The data are
as follows: (1) = −2.30, a = −9.87, b = −18.20, ab = −30.20, c = −23.80, ac = −4.30,
bc = −3.80, and abc = −14.70.
Applied Statistics and Probability for Engineers, 7th edition 2017
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(a) Estimate the factor effects.
(b) Use a normal probability plot of the effect estimates to identity factors that may be important.
(c) What model would you recommend for predicting the delta line width response based on the results of this
experiment?
(d) Analyze the residuals from this experiment, and comment on model adequacy.
(a)Estimated Effects
Term Effect
A -2.74
B -6.66
(b) None of the factor effects are significant
(c)
Analysis of Variance for delta (coded units)
Source DF Seq SS Adj SS Adj MS F P
Main Effects 3 128.07 128.07 42.69 0.51 0.746
(d) The residual plots shows there are serious problems with the data.
Applied Statistics and Probability for Engineers, 7th edition 2017
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14.S12 Construct
−41
IV
2
design for the problem in Exercise 14.S10. Select the data for the eight runs that would have been
required for this design. Analyze these runs and compare your conclusions to those obtained in Exercise 14.S10 for the
full factorial.
One possible 24−1 design is I =ABCD
A B C D DOI Estimated Effects
−1 −1 −1 −1 63.8 Term Effect
1 −1 −1 1 64.9 A 3.075
−1 1 −1 1 72.7 B 7.225
1 1 −1 −1 76.5 C 5.225
−1 −1 1 1 68.0 D −3.425
1 −1 1 −1 77.2 AB −2.075
−1 1 1 −1 77.7 AC 0.625
1 1 1 1 75.9 AD −3.025
It may be useful to conduct an analysis with the main effects only to see which main effect is significant.
Term Effect Coef StDev Coef T P
Constant 72.088 1.074 67.11 0.000
A 3.075 1.538 1.074 1.43 0.248
B 7.225 3.612 1.074 3.36 0.044
C 5.225 2.612 1.074 2.43 0.093
D -3.425 -1.712 1.074 -1.59 0.209
Analysis of Variance for DOI
Source DF Seq SS Adj SS Adj MS F P
Main Effects 4 201.38 201.38 50.344 5.45 0.097
14.S13 Construct a
−52
III
2
design in eight runs. What are the alias relationships in this design?
ABCDE 25−2 8 runs
Design Generators: D = AB E = AC
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Design
StdOrder
A
B
C
D
E
1
−1
−1
−1
1
1
2
1
−1
−1
−1
−1
14.S14 Construct a
−84
IV
2
design in 16 runs. What are the alias relationships in this design?
ABCDEFGH 28−4 Resolution IV
I + ABCG + ABDH + ABEF + ACDF + ACEH + ADEG + AFGH + BCDE + BCFH + BDFG + BEGH+
CDGH + CEFG + DEFH
A + BCG + BDH + BEF + CDF + CEH + DEG + FGH
B + ACG + ADH + AEF + CDE + CFH + DFG + EGH
C + ABG + ADF + AEH + BDE + BFH + DGH + EFG
D + ABH + ACF + AEG + BCE + BFG + CGH + EFH
14.S15 An article in Rubber Chemistry and Technology (Vol. 47, 1974, pp. 825–836) described an experiment to study
the effect of several variables on the Mooney viscosity of rubber, including silica filler (parts per hundred) and
oil filler (parts per hundred). Data typical of that reported in this experiment are reported in the following
table where
−−
==
12
silica 60 oil 21
,
15 15
xx
(a) What type of experimental design has been used?
(b) Analyze the data and draw appropriate conclusions.
Coded levels
x1
x2
y
−1
−1
13.71
1
−1
14.15
−1
1
12.87
1
1
13.53
−1
−1
13.90
1
−1
14.88
−1
1
12.25
−1
1
13.35
3
−1
1
−1
−1
1
4
1
1
−1
1
−1
5
−1
−1
1
1
−1
6
1
−1
1
−1
1
7
−1
1
1
−1
−1
8
1
1
1
1
1
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14-88
(b) Factors x1 and x2 are significant. The interaction between x1 and x2 is not significant.
Term Effect Coef SE Coef T P
Constant 13.5800 0.1241 109.42 0.000
Analysis of Variance for y (coded units)
Source DF Seq SS Adj SS Adj MS F P
Main Effects 2 3.95525 3.95525 1.97763 16.05 0.012
14.S16 An article in Plant Disease [“Effect of Nitrogen and Potassium Fertilizer Rates on Severity of Xanthomonas Blight of
Syngonium podophyllum” (1989, Vol. 73(12), pp. 972–975)] showed the effect of the variable nitrogen and potassium
rates on the growth of “White Butterfly” and the mean percentage of disease. Data representative of that collected in
this experiment is provided in the following table.
Nitrogen (mg/pot/wk)
Potassium (mg/pot/wk)
30
90
120
50
61.0 61.3
45.5 42.5
59.5 58.2
150
54.5 55.9
53.5 51.9
34.0 35.9
250
42.7 40.4
36.5 37.4
32.5 33.8
(a) State the appropriate hypotheses.
(b) Use the analysis of variance to test these hypotheses with
= 0.05.
(c) Graphically analyze the residuals from this experiment.
(d) Estimate the appropriate variance components.
(a)
1. H0:
1 =
2 =
3 = 0
(b)
Analysis of Variance for Disease
Source DF SS MS F P
Nitrogen 2 924.73 462.37 311.71 0.000
All effects in the model are significant.
Applied Statistics and Probability for Engineers, 7th edition 2017
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(c) The residual plots do not show any violations of the model assumptions.
14.S17 An article in the Journal of Applied Electrochemistry (May 2008, Vol. 38(5), pp. 583–590) presented a 27–3 fractional
factorial design to perform optimization of polybenzimidazole based membrane electrode assemblies for H2/O2 fuel
cells. The design and data are shown in the following table.
Runs
A
B
C
D
E
F
G
Current Density
(CD mA cm2)
1
−1
−1
−1
−1
−1
−1
−1
160
2
+1
−1
−1
−1
+1
+1
+1
20
3
−1
+1
−1
−1
+1
+1
−1
80
4
+1
+1
−1
−1
−1
−1
+1
317
5
−1
−1
+1
−1
+1
−1
+1
19
6
+1
−1
+1
−1
−1
+1
−1
4
7
−1
+1
+1
−1
−1
+1
+1
20
8
+1
+1
+1
−1
+1
−1
−1
88
9
−1
−1
−1
+1
−1
+1
+1
1100
10
+1
−1
−1
+1
+1
−1
−1
12
11
−1
+1
−1
+1
+1
−1
+1
552
12
+1
+1
−1
+1
−1
+1
−1
880
13
−1
−1
+1
+1
+1
+1
−1
16
14
+1
−1
+1
+1
−1
−1
+1
20
15
−1
+1
+1
+1
−1
−1
−1
8
16
+1
+1
+1
+1
+1
+1
+1
15
The factors and levels are shown in the following table.
Factor
−1
+1
A
Amount of binder in the catalyst layer
0.2 mg cm2
1 mg cm2
B
Electrocatalyst loading
0.1 mg cm2
1 mg cm2
C
Amount of carbon in the gas diffusion layer
2 mg cm2
4.5 mg cm2
D
Hot compaction time
1 min
10 min
E
Compaction temperature
100°C
150°C
F
Hot compaction load
0.04 ton cm2
0.2 ton cm2
G
Amount of PTFE in the gas diffusion layer
0.1 mg cm2
1 mg cm2
Applied Statistics and Probability for Engineers, 7th edition 2017
14-90
(a) Write down the alias relationships.
(b) Estimate the main effects.
(c) Prepare a normal probability plot for the effects and interpret the results.
(d) Calculate the sum of squares for the alias set that contains the ABG interaction from the corresponding effect
estimate.
(b)
Factorial Fit: Current Density (CD mA cm2) Versus A, B, C, D, E, F, G
Applied Statistics and Probability for Engineers, 7th edition 2017
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(c) Although the effects C, D, E, F, G, AD, AG, and BG are large, these effects are not indicated as significant in the
14.S18 Consider the following results from a two-factor experiment with two levels for factor A and three levels for
factor B. Each treatment has three replicates.
A
B
Mean
StDev
1
1
21.33333
6.027714
1
2
20
7.549834
1
3
32.66667
3.511885
2
1
31
6.244998
2
2
33
6.557439
2
3
23
10
(a) Calculate the sum of squares for each factor and the interaction.
(b) Calculate the sum of squares total and error.
(c) Complete an ANOVA table with F-statistics.
(a)
A
B
Mean
StDev
Sum
1
1
21.3333
6.027714
64
1
1
2
1
31
6.244998
93
Applied Statistics and Probability for Engineers, 7th edition 2017
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Factor B
Factor A
1
2
3
yi.
(b)
10
=
−
=
2
1
StDev
()
n
i
i
xx
(c) Total trials = 6 treatments × 3 replicates = 18 trials.
The ANOVA table
Source DF SS MS F P-value
A 1 84.5 84.5 1.7624 0.209
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14.S19 The rework time required for a machine was found to depend on the speed at which the machine was run (A), the
lubricant used while working (B), and the hardness of the metal used in the machine (C). Two levels of each factor
were chosen and a single replicate of a 23 experiment was run. The rework time data (in hours) are shown in the
following table.
Treatment Combination
Time (in hours)
(1)
27
a
34
b
38
ab
59
c
44
ac
40
bc
63
abc
37
(a) These treatments cannot all be run under the same conditions. Set up a design to run these observations in two
blocks of four observations each, with ABC confounded with blocks.
(b) Analyze the data.
(a)
Std
Order
Run
Order
Center
Pt
Blocks
A
B
C
y
1
1
1
1
−1
−1
−1
27
2
2
1
1
1
1
−1
59
(b) Computer software does not detect any significant effects, but the effects for B, AC, and possibly C are large.
3
3
1
1
1
−1
1
40
4
4
1
1
−1
1
1
63
5
5
1
2
1
−1
−1
34
7
7
1
2
−1
−1
1
44
8
8
1
2
1
1
1
37
Applied Statistics and Probability for Engineers, 7th edition 2017
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Factorial Fit: Time (in hours) Versus Block, A, B, C
Estimated Effects and Coefficients for Time (in hours) (coded units)
Term Effect Coef
Constant 42.750
Block 4.500
A -0.500 -0.250
Factorial Fit: Time (in hours) Versus Block, A, B, C
Estimated Effects and Coefficients for Time (in hours) (coded units)
Term Effect Coef SE Coef T P
Constant 42.750 1.904 22.45 0.002
Block 4.500 1.904 2.36 0.142
Analysis of Variance for Time (in hours) (coded units)
Source DF Seq SS Adj SS Adj MS F P
Blocks 1 162.00 162.00 162.00 5.59 0.142
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14.S20 An article in Process Biochemistry (Dec. 1996, Vol. 31(8), pp. 773–785) presented a 27−3 fractional factorial to perform
optimization of manganese dioxide bioleaching media. The data are shown in the following table.
Runs
A
B
C
D
E
F
G
Manganese
Extraction Yield
(%)
1
−1
−1
−1
−1
−1
−1
−1
99.0
2
1
−1
−1
−1
1
−1
1
97.4
3
−1
1
−1
−1
1
1
1
97.7
4
1
1
−1
−1
−1
1
−1
90.0
5
−1
−1
1
−1
1
1
−1
100.0
6
1
−1
1
−1
−1
1
1
98.0
7
−1
1
1
−1
−1
−1
1
90.0
8
1
1
1
−1
1
−1
−1
93.5
9
−1
−1
−1
1
−1
1
1
100.0
10
1
−1
−1
1
1
1
−1
98.6
11
−1
1
−1
1
1
−1
−1
97.1
12
1
1
−1
1
−1
−1
1
92.4
13
−1
−1
1
1
1
−1
1
93.0
14
1
−1
1
1
−1
−1
−1
95.0
15
−1
1
1
1
−1
1
−1
97.0
16
1
1
1
1
1
1
1
98.0
The factors and levels are shown in the following table.
Factor
−1
+1
A
Mineral concentration (%)
10
20
B
Molasses (g/liter)
100
200
C
NH4NO3 (g/liter)
1.25
2.50
D
KH2PO4 (g/liter)
0.75
1.50
E
MgSO4 (g/liter)
0.5
1.00
F
Yeast extract (g/liter)
0.20
0.50
G
NaHCO3 (g/liter)
2.00
4.00
(a) Write down the complete defining relation and the aliases from the design.
(b) Estimate the main effects.
(c) Plot the effect estimates on normal probability paper and interpret the results
(d) Conduct a residual analysis.
Applied Statistics and Probability for Engineers, 7th edition 2017
14-96
(a) Generators are E = ABC, F = BCD, and G = ABD. Note that the generator for factor G differs from the Minitab
default.
Alias Structure (up to order 3)
I
A + B*C*E + B*D*G + C*F*G + D*E*F
B + A*C*E + A*D*G + C*D*F + E*F*G
(b)
Factorial Fit: Yield(%) Versus A, B, C, D, E, F, G
Estimated Effects and Coefficients for Yield(%) (coded units)
Term Effect Coef
Constant 96.044
A -1.362 -0.681
Estimated Effects and Coefficients for y (coded units)
Term Effect Coef
Constant 96.044
A -1.362 -0.681
B -3.162 -1.581
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(d)
===
Two-way Interaction
Two-way In
Two-way In
teraction
teraction
M67.884 9.698
7
S
SS
df
14.S21 Consider the following ANOVA table from a two-factor factorial experiment.
Two-way ANOVA: y Versus A, B
Source
DF
SS
MS
F
P
A
3
1213770
404590
?
0.341
B
2
?
17335441
58.30
0.000
Error
?
1784195
?
Total
11
37668847
(a) How many levels of each factor were used in the experiment?
(b) How many replicates were used?
(c) What assumption is made in order to obtain an estimate of error?
(d) Calculate the missing entries (denoted with “?”) in the ANOVA table.
(a) Factor A has = 3 + 1 = 4 levels. Factor B has 2 + 1 = 3 levels.
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(d) Degree of freedom of error = 11 −3 − 2 = 6