Suppose that only a $1 / 4$ fraction of the $2^5$ design could be run. Construct the design and analyze the data that are obtained by selecting only the response for the eight runs in your design.

Construct a sign table for the principal fraction for a $2^4$ design. Then indicate all the alias pairs.

An article in Industrial and Engineering Chemistry ["More on Planning Experiments to Increase Research Efficiency" (1970, pp. 60-65)] uses a $2^{5-2}$ design to investigate the effect on process yield of $A=$ condensation temperature, $B=$ amount of material 1,C= solvent volume, $D=$ condensation time, and $E$ $=$ amount of material 2. The results obtained are as follows:

Prepare an analysis of variance table. Verify that the $A B$ and $A D$ interactions are available to use as error.

An article in Industrial and Engineering Chemistry ["More on Planning Experiments to Increase Research Efficiency" (1970, pp. 60-65)] uses a $2^{5-2}$ design to investigate the effect on process yield of $A=$ condensation temperature, $B=$ amount of material 1,C= solvent volume, $D=$ condensation time, and $E$ $=$ amount of material 2. The results obtained are as follows:

Plot the residuals versus the fitted values. Also construct a normal probability plot of the residuals. Comment on the results.

Construct a $\frac{1}{4}$ fraction of a $2^6$ factorial design using ABCD and BDEF as the defining contrasts. Show what effects are aliased with the six main effects.

An article in the Journal of Marketing Research presented a $2^{7-4}$ fractional factorial design to conduct marketing research:

Estimate the main effects.

Consider a

$$2^3$$

design. Suppose that the largest number of runs that can be made in one block is four, but you can afford to perform a total of 32 observations. a. Suggest a blocking scheme will provides some information on all interactions. b. Show an outline (source of variability, degrees of freedom only) for the analysis of variance for this design.

Consider the $2^{6-2}$ design in table below. Suppose that after analyzing the original data, we find that factors C and E can be dropped. What type of $2^{k}$ design is left in the remaining variables?

$$\scriptstyle\begin{array}{rrrrrrr} \text { Run } & A & B & C & D & E=A B C & F=B C D & (\times 10) \\ \hline 1 & - & - & - & - & - & - & 6 \\ 2 & + & - & - & - & + & - & 10 \\ 3 & - & + & - & - & + & + & 32 \\ 4 & + & + & - & - & - & + & 60 \\ 5 & - & - & + & - & + & + & 4 \\ 6 & + & - & + & - & - & + & 15 \\ 7 & - & + & + & - & - & - & 26 \\ 8 & + & + & + & - & + & - & 60 \\ 9 & - & - & - & + & - & + & 8 \\ 10 & + & - & - & + & + & + & 12 \\ 11 & - & + & - & + & + & - & 34 \\ 12 & + & + & - & + & - & - & 60 \\ 13 & - & - & + & + & + & - & 16 \\ 14 & + & - & + & + & - & - & 5 \\ 15 & - & + & + & + & - & + & 37 \\ 16 & + & + & + & + & + & + & 52 \\ \hline \end{array}$$

Consider a

$$2^2$$

design in two blocks with AB confounded with blocks. Prove algebraically that

$$SS_AB = SS$$

Blocks*

Consider the following computer output from a single replicate of a

$$2^4$$

experiment in two blocks with ABCD confounded. a. Comment on the value of blocking in this experiment. b. What effects were used to generate the residual error in the ANOVA? c. Calculate the entries marked with "?" in the output.

Construct a

$$2^5$$

design in two blocks. Select the ABCDE interaction to be confounded with blocks.

Consider the

$$2^6$$

factorial design. Set up a design to be run in four blocks of 16 runs each. Show that a design that confounds three of the four-factor interactions with blocks is the best possible blocking arrangement.

What blocking scheme would you recommend if it were necessary to run a

$$2^4$$

design in four blocks of four runs each?

Montgomery (2001) describes a $2^{4-1}$ fractional factorial design used to study four factors in a chemical process. The factors are A = temperature, B = pressure, C = concentration, and D = stirring rate, and the response is filtration rate. The design and the data are as follows:

$$\scriptstyle\begin{array}{cccccc} & & & & &\text{Treatment} & \text{Filtration} \\ \text { Run } & A & B & C & D=A B C & \text { Combination } & \text { Rate } \\ \hline 1 & - & - & - & - & (1) & 45 \\ 2 & + & - & - & + & a d & 100 \\ 3 & - & + & - & + &b d & 45 \\ 4 & + & + & - & - & a b & 65 \\ 5 & - & - & + &+ & c d & 75 \\ 6 & + & - & + & - & a c & 60 \\ 7 & - & + & + & - &b c & 80 \\ 8 & + & + & + & - &a b c d & 96 \\ \hline \end{array}$$

(a) Write down the alias relationships. (b) Estimate the factor effects. Which factor effects appear large? (c) Project this design into a full factorial in the three apparently important factors and provide a practical interpretation of the results.