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Ch06
Design Of Experiments (Texas Tech University)
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Ch06
Design Of Experiments (Texas Tech University)
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Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments
, Wiley, NY 6-1 Chapter 6 The 2
k
Factorial Design Solutions 6.1.
In a 2
4
factorial design, the number of degrees of freedom for the model, assuming the complete factorial model, is (a)
7 (b)
5 (c)
6 (d)
11 (e)
12 (f)
None of the above 6.2.
A 2
3
factorial is replicated twice. The number of pure error or residual degrees of freedom are (a)
4 (b)
12 (c)
15 (d)
2 (e)
8 (f)
None of the above 6.3.
A 2
3
factorial is replicated twice. The ANOVA indicates that all main effects are significant but the interactions are not significant. The interaction terms are dropped from the model. The number of residual degrees of freedom for the reduced model are (a)
12
(b)
8 (c)
16 (d)
14 (e)
10
(f)
None of the above 6.4.
A 2
3
factorial is replicated three times. The ANOVA indicates that all main effects are significant, but two of the interactions are not significant. The interaction terms are dropped from the model. The number of residual degrees of freedom for the reduced model are (a)
12 (b)
14 (c)
6 (d)
10 (e)
8
(f)
None of the above
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Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments
, Wiley, NY 6-2 6.5.
An engineer is interested in the effects of cutting speed (
A
), tool geometry (
B
), and cutting angle on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 2
3
factorial design are run. The results are as follows: Treatment
Replicate
A B C Combination
I
II
III
- - - (1)
22
31
25
+ - - a 32
43
29
- + - b 35
34
50
+ + - ab 55
47
46
- - + c 44
45
38
+ - + ac 40
37
36
- + + bc 60
50
54
+ + + abc 39
41
47
(a) Estimate the factor effects. Which effects appear to be large? From the normal probability plot of effects below, factors B
, C
, and the AC
interaction appear to be significant. (b) Use the analysis of variance to confirm your conclusions for part (a). The analysis of variance confirms the significance of factors B
, C
, and the AC
interaction. Design Expert Output Response:
Life
in hours
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Sum of
Mean
F
Source
Squares
DF
Square
Value
Prob > F Model 1612.67 7 230.38 7.64 0.0004 significant A 0.67 1 0.67 0.022 0.8837 B 770.67 1 770.67 25.55 0.0001 C 280.17 1 280.17 9.29 0.0077 AB 16.67 1 16.67 0.55 0.4681 Downloaded by Carlos Alejo (al02837270@gmail.com)
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Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments
, Wiley, NY 6-3 AC 468.17 1 468.17 15.52 0.0012 BC 48.17 1 48.17 1.60 0.2245 ABC 28.17 1 28.17 0.93 0.3483 Pure Error 482.67 16 30.17 Cor Total 2095.33 23 The Model F-value of 7.64 implies the model is significant. There is only a 0.04% chance that a "Model F-Value" this large could occur due to noise. The reduced model ANOVA is shown below. Factor A
was included to maintain hierarchy. Design Expert Output Response:
Life
in hours
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Sum of
Mean
F
Source
Squares
DF
Square
Value
Prob > F
Model 1519.67 4 379.92 12.54 < 0.0001 significant A
0.67
1
0.67
0.022
0.8836
B
770.67
1
770.67
25.44
< 0.0001
C
280.17
1
280.17
9.25
0.0067
AC
468.17
1
468.17
15.45
0.0009
Residual 575.67 19 30.30 Lack of Fit
93.00
3
31.00
1.03
0.4067
not significant
Pure Error
482.67
16
30.17
Cor Total 2095.33 23 The Model F-value of 12.54 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise. Effects B, C
and AC
are significant at 1%. (c) Write down a regression model for predicting tool life (in hours) based on the results of this experiment. 40.8333
0.1667
5.6667
3.4167
4.4167
ijk
A
B
C
A
C
y
x
x
x
x x
=
+
+
+
+
Design Expert Output Coefficient
Standard
95% CI
95% CI
Factor
Estimate
DF
Error
Low
High
VIF
Intercept 40.83 1 1.12 38.48 43.19 A-Cutting Speed 0.17 1 1.12 -2.19 2.52 1.00 B-Tool Geometry 5.67 1 1.12 3.31 8.02 1.00 C-Cutting Angle 3.42 1 1.12 1.06 5.77 1.00 AC -4.42 1 1.12 -6.77 -2.06 1.00 Final Equation in Terms of Coded Factors:
Life = +40.83 +0.17 * A +5.67 * B +3.42 * C -4.42 * A * C Final Equation in Terms of Actual Factors:
Life = +40.83333 +0.16667 * Cutting Speed +5.66667 * Tool Geometry +3.41667 * Cutting Angle -4.41667 * Cutting Speed * Cutting Angle Downloaded by Carlos Alejo (al02837270@gmail.com)
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Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments
, Wiley, NY 6-4 The equation in part (c) and in the given in the computer output form a “hierarchical” model, that is, if an interaction is included in the model, then all of the main effects referenced in the interaction are also included in the model. (d) Analyze the residuals. Are there any obvious problems? There is nothing unusual about the residual plots. (e) Based on the analysis of main effects and interaction plots, what levels of A
, B
, and C
would you recommend using? Since B
has a positive effect, set B
at the high level to increase life. The AC
interaction plot reveals that life would be maximized with C
at the high level and A
at the low level. 6.6.
Reconsider part (c) of Problem 6.5. Use the regression model to generate response surface and contour plots of the tool life response. Interpret these plots. Do they provide insight regarding the desirable operating conditions for this process? Downloaded by Carlos Alejo (al02837270@gmail.com)
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Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments
, Wiley, NY 6-5 The response surface plot and the contour plot in terms of factors A
and C
with B
at the high level are shown below. They show the curvature due to the AC
interaction. These plots make it easy to see the region of greatest tool life. 6.7.
Find the standard error of the factor effects and approximate 95 percent confidence limits for the factor effects in Problem 6.5. Do the results of this analysis agree with the conclusions from the analysis of variance? ( )
2
(
)
2
3 2
1
1
30.17
2.24
2
3 2
effect
k
SE
S
n
-
-
=
=
=
Variable
Effect
A
0.333
B
11.333
*
AB
-1.667
C
6.833
*
AC
-8.833
*
BC
-2.833
ABC
-2.167
The 95% confidence intervals for factors B, C
and AC
do not contain zero. This agrees with the analysis of variance approach. 6.8.
Plot the factor effects from Problem 6.5 on a graph relative to an appropriately scaled t
distribution. Does this graphical display adequately identify the important factors? Compare the conclusions from this plot with the results from the analysis of variance. 30.17
3.17
3
E
MS
S
n
=
=
=
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CHM151 17003 > Modules> Weeks 8 & 9 - Chapter 7 > Homework and quiz in CHEM101 - due Sunday night
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Complete the balanced neutralization equation for the reaction below:
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O CHEMICAL REACTIONS
Solving for a reactant using a chemical equation
Ammonia (NH3) chemically reacts with oxygen gas (02) to produce nitric oxide (NO) and water (H,O).
What mass of nitric oxide is produced by the reaction of 4.8 g of oxygen gas?
Round your answer to 2 significant digits.
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HW 4.4
Milk of magnesia, which is an aqueous suspension of magnesium hydroxide, is used as an
antacid in the reaction below. How many molecules of HCI would have to be present to form
17.38 g of MgCl?
7
Mg(OH)2 (s)
2 HCI (aq)2 H2O (I)MgCla (aq)
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OSHA's Hazard Communication Standard requires chemical manufacturers to evaluate hazards
relating to their products and provide a Safety Data Sheet in accordance to the Globally
Harmonized System (GHS) in which the use of pictograms (see Figure 1) is required to
communicate hazard information. Predict (with reasons) two (2) hazards related to each of
the two activities below, and assign two (2) relevant pictograms listed in Figure 1 to each of
the two activities.
Activity 1: Removing diethyl ether (last opened in 2019) using a rotary evaporator.
Activity 2: Using carbon monoxide gas as a reducing agent to reduce iron chloride.
Figure 1. Pictograms advised by the Globally Harmonized System for communication of
chemical hazards.
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2BrF3 (9) Br2 (g) + 3F2 (9)
4. AG° = AH°...
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5. AG: Pre...
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6. AG: Enthal...
Using standard thermodynamic data at 298 K, calculate the
free energy change when 2.14 moles of BrF3 (9) react at
standard conditions.
Substance AG (kJ/mol)
7. AG fro...
1req
Question
Question
Question
8. Calculat...
1req
AG°
=
9. Calculat...
1req
BrF3 (9)
-229.4
Br2(g)
3.1
F2(9)
0.0
kj
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There is 17.0 g of carbon dioxide gas stored in a cylinder with a volume of 16.0. A pressure
gauge shows the pressure to be 3.00 atm. R=0.0821 Latm/(mol-K)
In degrees Celsius, determine the temperature of the gas.
Round your answer to a whole number.
7.06 Applic x M Ihbox (1,61: x
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Calculating kinetic energy
km
Calculate the kinetic energy of a 1.7 x 10" kg airplane moving at a speed of 0.20
S
Round your answer to 2 significant digits.
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Chapter 8: Chemical Composition
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45. Calculate the percent by mass of each element in the following compounds.
a. HC103
b. UF4
с. СаН2
d. Ag,S
e. NaHSO3
f. MnO2
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Choose the correct total number of electron domains (bonding and nonbonding) about a central atom if the angle(s) between the electron domains have two different values, either about
90 or 180 degrees.
O a) The central atom has 4 electron domains.
O b) The central atom has 6 electron domains.
O c) The central atom has 3
tron domains.
O d) The central atom has 2 electron domains.
O e) The central atom has 5 electron domains.
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