Construct a $\frac{1}{4}$ replicate of a $2^5$ factorial experiment with factors, $A,B,C,D,$ and $E$ by using the $AB$ and $DE$ interaction effects as the defining effects.

Give the alias sets for the treatment design in $1.$

Would you recommend using the treatment design in $1?$ Why or why not?

Suppose you were going to create a $\frac{1}{16}$ replicate of a $2^8$ factorial experiment.

a$)$ How many defining effects must be used?

b$)$ How many effects will be in each of the alias sets?

c$)$ How many runs will be required in this treatment design?

d$)$ Is it possible to create a resolution $V$ design for eight factors in $16$ runs? Why or why not?

e$)$ Is it possible to create a resolution IV design for eight factors in $16$ runs? If so$,$ give a set of defining effects that will create a resolution IV design.

Create a saturated resolution III design for $15$ factors in $16$ runs that is not a Plackett & Burman Design.

Using design generators given by Plackett and Burman, create a $20$ run design for eight factors. Show your work for full credit. What is the resolution of your design?

Consider a $2^{8-3}$ treatment design.

a$)$ How many factors are involved?

b$)$ How many runs are required?

c$)$ How many levels in each of the factors?

All other things being equal, why would a resolution IV design be preferred over a resolution III design?

Is it possible to construct a $2^{8-3}$ design of resolution III? Resolution IV$?$ Resolution V$?$

How would you recommend that data from a $\frac{1}{2}$ replicate of $2^7$ design be analyzed? That is$,$ would you use ANOVA or the half$-$normal plotting technique? Why?

How would you recommend that data from a $\frac{1}{16}$ replicate of a $2^7$ design be analyzed? Why?