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 a

$$2^2$$

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

$$SS_AB = SS$$

Blocks*

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

$$2^4$$

design in four blocks of four runs each?

Consider the experiment. Suppose that a center point with five replicates is added to the factorial runs and the responses are $2800,5600,4500,5400$, and 3600 . Compute an ANOVA with the sum of squares for curvature and conduct an $F$-test for curvature. Use $\alpha=0.05$.

In a process development study on yield, four factors were studied, each at two levels: time (A), concentration (B), pressure (C), and temperature (D). A single replicate of a $2^{4}$ design was run, and the data are shown in the table below.

$$\begin{array}{cccccc} \text { Run } & \text { Actual Run } & & & & & \text{Yield} \\ \text { Number } & \text { Order } & A & B & C & D & (\mathrm{lbs}) \\ \hline 1 & 5 & - & - & - & - & 12 \\ 2 & 9 & + & - & - & - & 18 \\ 3 & 8 & - & + & - & - & 13 \\ 4 & 13 & + & + & - & - & 16 \\ 5 & 3 & - & - & + & - & 17 \\ 6 & 7 & + & - & + & - & 15 \\ 7 & 14 & - & + & + & - & 20 \\ 8 & 1 & + & + & + & - & 15 \\ 9 & 6 & - & - & - & + & 10 \\ 10 & 11 & + & - & - & + & 25 \\ 11 & 2 & - & + & - & + & 13 \\ 12 & 15 & + & + & - & + & 24 \\ 13 & 4 & - & - & + & + & 19 \\ 14 & 16 & + & - & + & + & 21 \\ 15 & 10 & - & + & + & + & 17 \\ 16 & 12 & + & + & + & + & 23 \\ \hline \end{array}$$

$$\begin{array}{lcc} \text{Factor Levels} \\ \hline \mathrm{Low}(-) & & \text { High }(+) \\ \hline A(\mathrm{h}) & 2.5 & 3 \\ B(\%) & 14 & 18 \\ C(\mathrm{psi}) & 60 & 80 \\ D\left(^{\circ} \mathrm{C}\right) & 225 & 250 \end{array}$$

(a) Plot the effect estimates on a normal probability scale. Which factors appear to have large effects? (b) Conduct an analysis of variance using the normal probability plot in part (a) for guidance in forming an error term. What are your conclusions? (c) Analyze the residuals from this experiment. Does your analysis indicate any potential problems? (d) Can this design be collapsed into a $2^{3}$ design with two replicates? If so, sketch the design with the average and range of yield shown at each point in the cube. Interpret the results.

Construct a

$$2^5$$

design in four blocks. Select the appropriate effects to confound so that the highest possible interactions are confounded with blocks.

Consider a

$$2^2$$

factorial experiment with four center points. The data are (1) = 21, a = 125, b = 154, ab = 352, and the responses at the center point are 92, 130, 98, 152. Compute an ANOVA with the sum of squares for curvature and conduct an F-test for curvature. Use α = 0.05.

Construct a

$$2^5$$

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

Construct a $2_{\mathrm{III}}^{6-3}$ fractional factorial design. Write down the aliases, assuming that only main effects and two factor interactions are of interest.

Consider the data . Suppose that the data from the second replicate were not available. Analyze the data from replicate I only and comment on your findings.

Four factors are thought to influence the taste of a soft-drink beverage: type of sweetener $(A)$, ratio of syrup to water $(B)$, carbonation level $(C)$, and temperature $(D)$. Each factor can be run at two levels, producing a $2^4$ design. At each run in the design, samples of the beverage are given to a test panel consisting of 20 people. Each tester assigns the beverage a point score from 1 to 10 . Total score is the response variable, and the objective is to find a formulation that maximizes total score. Two replicates of this design are run, and the results are shown in the table. Analyze the data and draw conclusions. Use $\alpha=0.05$ in the statistical tests.

$$\begin{array}{ccc} a b & 166 & 168 \\ c & 175 & 178 \\ a c & 179 & 183 \\ b c & 173 & 168 \\ a b c & 179 & 182 \\ d & 164 & 159 \\ a d & 187 & 189 \\ b d & 163 & 159 \\ a b d & 185 & 191 \\ c d & 168 & 174 \\ a c d & 197 & 199 \\ b c d & 170 & 174 \\ a b c d & 194 & 198 \end{array}$$

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?

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

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.

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.