HOMEWORK #2 due Wednesday 3-26
NOTE: Organize your homework in the following order:
title statement so that your name
will appear at the top of each output page.
title2 statement to make
it clearer what output pages belong to what problem.
Problem 1. (See also Table 4.14 p164 in the text)
Table 1. Conversion rates at different pressures over 10 days
Day Pressures (Dashes indicate missing observations)
------------------------------------------------
250 325 400 475 550
------------------------------------------------
1 16 18 -- 32 --
2 19 -- -- 46 45
3 -- 26 39 -- 61
4 -- -- 21 35 55
5 -- 19 -- 47 48
6 20 -- 33 31 --
7 13 13 34 -- --
8 21 -- 30 -- 52
9 24 10 -- -- 50
10 -- 24 31 37 --
Note that each Pressure occurs in 6 days, and each pair of Pressures
occurs together on 3 days. Answer the following questions, using SAS if
convenient:
proc
means.)
proc glm. Since BIBDs are not orthogonal
designs, the Type I and Type III tables may be different.)
Problem 2. (See Problem 19 page 232 in the text.) In order to reduce the amount of a pollutant, the waste stream of a small factory into a previously pristine mountain stream must be treated. State law requires that the average amount of this pollutant per day cannot exceed 10 pounds. Eleven (11) runs were made with various settings of three factors, Brand (of a pre-treatment chemical), Temperature, and Stirring rate. As the table in the problem indicates, the 11 runs amounted to a 2^3 design for High,Low settings for the three factors and an additional 3 runs at an average setting for all three factors. (The intermediate setting of the pre-treatment chemical was a 50-50 mix of both brands.)
proc reg in SAS with Low,High
coded as -1,+1 and the intermediate setting as 0. That will give you 11
observations for 8 parameters, so that you should be able to get P-values
for all effects. WARNING: The interaction plots will involve three levels
of each factor, High, Intermediate, and Low. Use L,M,H as the plotting
symbol and values for Low, Medium, High along the X axis that SAS will not
permute as it alphabetizes the values, such as -1 0 +1 or A B C.)
Problem 3. Consider the data in Table 2 on the amount of unburned carbon in engine exhaust.
Table 2. Unburned Carbon in 8 runs with 4 factors (see text p275)
------------------------------------------------
Run A B C D Yield
------------------------------------------------
1 -1 1 1 1 8.2
2 -1 -1 1 -1 1.7
3 -1 -1 -1 1 6.2
4 1 -1 -1 -1 3.0
5 1 -1 1 1 6.8
6 1 1 1 -1 5.0
7 -1 1 -1 -1 3.8
8 1 1 -1 1 9.3
where -1, 1 represent the Low and High levels of that factor.
proc means;
classes A B C D; Ways 1; var Yield; run; to get everything
on one page.)
Problem 4. Consider the data in Table 3:
Table 3. Data from a 2^{5-1} design with 5 factors (see text p276)
------------------------------------------------
Run A B C D E Yield
------------------------------------------------
1 -1 -1 -1 -1 1 14.8
2 1 -1 -1 -1 -1 14.5
3 -1 1 -1 -1 -1 18.1
4 1 1 -1 -1 1 19.4
5 -1 -1 1 -1 -1 18.4
6 1 -1 1 -1 1 15.7
7 -1 1 1 -1 1 27.3
8 1 1 1 -1 -1 28.2
9 -1 -1 -1 1 -1 16.0
10 1 -1 -1 1 1 15.1
11 -1 1 -1 1 1 18.9
12 1 1 -1 1 -1 22.0
13 -1 -1 1 1 1 19.8
14 1 -1 1 1 -1 18.9
15 -1 1 1 1 -1 29.9
16 1 1 1 1 1 27.4
Problem 5. (Taken from Problem 16 p278 in text) An experimenter performs a 2^{5-2} analysis with 5 factors A B C D E with confounding relations D=ABC and E=AC. After analyzing the results from this design, she decides to carry out a second 2^{5-2} design with exactly the same design matrix as the first design but with the signs changed for the main effect of C. (That is, C=Low replaced with C-High in each run and vice versa.)
Problem 6. (Taken from Problem 19 p279 in text) An experimenter wants to investigate High,Low states of five factors, temperature (T), concentration (C), pH (P), agitation rate (R), and catalyst type (K, for catalysts K1 or K2) using 8 runs. She is concerned about possible T*C and T*K interactions, but believes that any other interactions will be small. Find a 2^{5-2} design such that