Ricky L Fraley MSETM
IET 421/621 (Spring 2015)
Homework 8
Total Points: 10
Due date: 04/20/2015, 10:00 PM EST
1. (6 points) An experiment was performed to improve the yield of a chemical process. Four factors were selected, and two replicates of a completely randomized experiment were run. The results are shown in the following table:
|Treatment |Replicate |Replicate |Treatment |Replicate |Replicate |
|Combination |I |II |Combination |I |II |
|(1) |91 |94 |d |99 |96 |
|a |75 |79 |ad |73 |77 |
|b |82 |86 |bd |88 |84 |
|ab |84 |81 |abd |86 |87 |
|c |78 |79 |cd |100 |91 |
|ac |82 |81 |acd |80 |76 |
|bc |89 |83 |bcd |88 |85 |
|abc |74 |71 |abcd |81 |81 |
Estimate the factor effects.
Term Effect Coef SE Coef T-Value P-Value VIF
Constant 83.781 0.489 171.28 0.000
A -9.063 -4.531 0.489 -9.26 0.000 1.00
B -1.313 -0.656 0.489 -1.34 0.198 1.00
C -2.687 -1.344 0.489 -2.75 0.014 1.00
D 3.938 1.969 0.489 4.02 0.001 1.00
A*B 4.062 2.031 0.489 4.15 0.001 1.00
A*C 0.688 0.344 0.489 0.70 0.492 1.00
A*D -2.188 -1.094 0.489 -2.24 0.040 1.00
B*C -0.562 -0.281 0.489 -0.57 0.573 1.00
B*D -0.188 -0.094 0.489 -0.19 0.850 1.00
C*D 1.687 0.844 0.489 1.72 0.104 1.00
A*B*C -5.187 -2.594 0.489 -5.30 0.000 1.00
A*B*D 4.687 2.344 0.489 4.79 0.000 1.00
A*C*D -0.937 -0.469 0.489 -0.96 0.352 1.00
B*C*D -0.937 -0.469 0.489 -0.96 0.352 1.00
A*B*C*D 2.437 1.219 0.489 2.49 0.024 1.00
Prepare an analysis of variance table, and determine which factors are important in explaining yield.
A, A B C, A B D
Write down a regression model for predicting yield, assuming that all four factors were varied over the range from -1 to +1 (in coded units).
A -4.531 A B C -2.594 ABD 2.344
Plot the residuals versus the predicted yield and on a normal probability scale. Does the residual analysis appear satisfactory? YES
Two three-factor interactions, ABC and ABD, apparently have large effects. Draw a cube plot in the factors A, B, and C with the average yields shown at each corner. Repeat using the factors A, B, and D. Do these two plots aid in data interpretation? YES
Where would you recommend that the process be run with respect to the four variables? For ABCD run them at high level.
2. (4 points) An industrial engineer employed by a beverage bottler is interested in the effects of two different types of 32-ounce bottles on the time to deliver 12-bottle cases of the product. The two bottle types are glass and plastic. Two workers are used to perform a task consisting of moving 40 cases of the product 50 feet on a standard type of hand truck and stacking the cases in a display. Four replicates of a 22 factorial design are performed, and the times observed are listed in the following table. Analyze the data and draw the appropriate conclusions. Analyze the residuals and comment on the model’s adequacy.
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Factorial Regression: Response versus Blocks, A, B, C, D
Analysis of Variance
Source DF Adj SS Adj MS F-Value P-Value
Model 17 8.3572 0.49160 1.58 0.195
Blocks 3 5.0721 1.69070 5.45 0.011
Linear 4 0.8983 0.22458 0.72 0.590
A 1 0.1596 0.15961 0.51 0.485
B 1 0.0032 0.00320 0.01 0.921
C 1 0.0800 0.08000 0.26 0.620
D 1 0.6555 0.65551 2.11 0.168
2-Way Interactions 6 1.1216 0.18693 0.60 0.724
A*B 1 0.2450 0.24500 0.79 0.389
A*C 1 0.0338 0.03380 0.11 0.746
A*D 1 0.2556 0.25561 0.82 0.379
B*C 1 0.4005 0.40051 1.29 0.275
B*D 1 0.0242 0.02420 0.08 0.784
C*D 1 0.1625 0.16245 0.52 0.481
3-Way Interactions 4 1.2652 0.31629 1.02 0.431
A*B*C 1 0.8515 0.85151 2.74 0.120
A*B*D 1 0.0050 0.00500 0.02 0.901
A*C*D 1 0.1741 0.17405 0.56 0.466
B*C*D 1 0.2346 0.23461 0.76 0.399
Error 14 4.3437 0.31026
Total 31 12.7009
None of the Factors stand out as needing to be done to accomplish the task
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Residuals are satisfactory
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