Question 1.
Hint:
• See pages L-106 and L-107 of the lecture notes for formulas and a similar example.
• Because only treatment c is replicated more than once, its variance 42.72 is automatically the MSE with 3-1=2 df.
• 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.
98.0 98.5 99.0
96.5 97.0 97.5 removal
6 8 10 12 14 16 18
temp
A. Identify the dependent and independent variables.
B. From the scatter plot, do you think the two variables are linearly correlated? Why.
