Description
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
Sample Size
Treatment
Sample Mean
Treatment
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.
A. Identify the dependent and independent variables.
B. From the scatter plot, do you think the two variables are linearly correlated? Why
