Description
Question 1.
Hint:
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See pages L-106 and L-107 of the lecture notes for formulas and a similar example.
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Because only treatment c is replicated more than once, its variance 42.72 is automatically the MSE with 3-1=2 df.
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Do not forget to ignore “c” when you interpret the fitted effects.
Scientists conducted a half fractional factorial experiment involving factors A, B and C using the generator C=AB. Summary data are given below.
|
Treatment |
Responses |
Treatment |
Treatment |
Treatment |
|
Sample Size |
Sample Mean |
Sample |
||
|
Variance |
||||
|
c |
88.8, 94.4, 82.1 |
3 |
87.7 |
42.72 |
|
a |
69.6 |
1 |
69.6 |
NA |
|
b |
32.6 |
1 |
32.6 |
NA |
|
abc |
83.2 |
1 |
83.2 |
NA |
Notice that the treatments in the table are in (Yates) standard order if we ignore “c”. Yates algorithm produces the following values (p=3, q=1, p – q = 2 cycles):
|
Treatment |
Means |
Cycle 1 |
Cycle 2 |
Fitted effect |
|
c |
87.7 |
157.3 |
273.1 |
68.275 |
|
a |
69.6 |
115.8 |
32.5 |
8.125 |
|
b |
32.6 |
-18.1 |
-41.5 |
-10.375 |
|
abc |
83.2 |
50.6 |
68.7 |
17.175 |
Also note that treatment c was replicated 3 times. This means that we can compute r(α) which we can use to determine which fitted effects are significant. Set the significance level at α=0.05. A. Perform some calculations to show that r(0.05) = 12.84.
B. The defining relation in this experiment is I=ABC. Use this and r(0.05)=12.84 to determine which fitted effects are significant at the α =0.05 level. Just fill in the blanks in the table below to complete this exercise.
|
Fitted Effect |
Sum of Effects Estimated |
Significant? |
|
Enter YES or NO |
||
|
below. |
8.125
-10.375
17.175
C. If all interactions are negligible, which of factors A, B and C are most important?
Question 2. An experiment has 6 factors with 2 levels each. Researchers can only run 1/8 of the 26 = 64 treatments due to costs and time constraints. Let’s pick factor A, B, and C as the independent factors. Design 1 chooses the generators as D=A, E=B, E=C. Design 2 picks the generators as D=ABC, E=AB, and F=BC. Explain why design 2 is better than design 1.
Question 3. In biofiltration of wastewater, air discharged from a treatment facility is passed through a damp porous membrane that causes contaminants to dissolve in water and be transformed into harmless products. The accompanying data on x= inlet temperature (°C) and y= removal efficiency (%) was the basis for a scatter plot that appeared in the article “Treatment of Mixed Hydrogen Sulfide and Organic Vapors in a Rock Medium Biofilter”(Water Environment Research, 2001: 426–435). The scatter plot and the summary statistics are given below.
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98.0 98.5 99.0
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96.5 97.0 97.5 removal
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6 8 10 12 14 16 18
temp
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Identify the dependent and independent variables.
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From the scatter plot, do you think the two variables are linearly correlated? Why.
