Exercise 4

1. (a) Why do we “block” experimental designs?

(b) Write a 2³ factorial design.

effects are not confounded with blocks.

2. To reduce the amount of a certain pollutant, a waste stream of a small plastic molding factory has to be treated before it is discharged. State laws stipulate that the daily average of this pollutant cannot exceed 10 pounds. The following 11 experiments were performed to determine the best way to treat this waste stream. Analyze these data, and list the conclusions you reach.

Order in which experiments were performed	Chemical brand	Temperature (deg C)	Stirring	Pollutant (lb/day)
5	A	72	None	5
6	B	72	None	30
1	A	100	None	6
9	B	100	None	33
11	A	72	Fast	4
4	B	72	Fast	3
2	A	100	Fast	5
7	B	100	Fast	4
3	AB	86	Intermediate	7
8	AB	86	Intermediate	4
10	AB	86	Intermediate	3

Where AB denotes a 50-50 mixture of both brands.

3. Write generators for a 2^8-4 design with Resolution = IV and for a 2^6-2 design with Resolution = IV. Does a 2^6-2 design with Resolution = V exist?

4. (a) Construct a 2^6-2 fractional factorial design with as high a resolution as possible.

(b) What are the generators of your design?

(d) What is confounded with the main effect of variable 3 in your design?

(e) What is confounded with the two-factor interaction 12 in your design?

5. (a) Write a design of resolution III in seven variables and eight runs.

(b) From (a), obtain a design for eight variables in 16 runs of resolution IV, arranged in eight blocks of two.

(d) What are the generators for your design?

(e) If no blocking is to be used, how could this design be employed to study the main effects and interactions of three principal variables and the main effects of eight minor variables not expected to interact with the principal variables and with each other?

6. Show how to construct a 2^7-1 resolution VII design. Show how the design may be divided into eight blocks of eight runs each so that no main effect or two-factor interaction is confounded with any block effect.