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.

The article “An Application of Fractional Factorial Designs" describes a $2^{5-1}$ (half-replicate of a $2^{5}$ design) involving the use of carbon dioxide $\left(\mathrm{CO}_{2}\right)$ at high pressure to extract oil from peanuts. The outcomes were the solubility of the peanut oil in the $CO_2$ (in mg oil/liter $CO_2),$ and the yield of peanut oil (in percent). The five factors were A: $CO_2$ pressure, B: $CO_2$ temperature, C: peanut moisture, D: $CO_2$ flow rate, and E: peanut particle size. The results are presented in the following table.

$$\begin{matrix} \text{Treatment} & \text{Solubility} & \text{Yield} & \text{Treatment} & \text{Solubility} & \text{Yield}\\ \text{e} & \text{29.2} & \text{63} & \text{d} & \text{22.4} & \text{23}\\ \text{a} & \text{23.0} & \text{21} & \text{ade} & \text{37.2} & \text{74}\\ \text{b} & \text{37.0} & \text{36} & \text{bde} & \text{31.3} & \text{80}\\ \text{abe} & \text{139.7} & \text{99} & \text{abd} & \text{48.6} & \text{33}\\ \text{c} & \text{23.3} & \text{24} & \text{cde} & \text{22.9} & \text{63}\\ \text{ace} & \text{38.3} & \text{66} & \text{acd} & \text{36.2} & \text{21}\\ \text{bce} & \text{42.6} & \text{71} & \text{bcd} & \text{33.6} & \text{44}\\ \text{abc} & \text{141.4} & \text{54} & \text{abcde} & \text{172.6} & \text{96}\\ \end{matrix}$$

a. Assuming third- and higher-order interactions to be negligible, compute estimates of the main effects and interactions for the solubility outcome. b. Plot the estimates on a normal probability plot. Does the plot show that some of the factors influence the solubility? If so, which ones? c. Assuming third- and higher-order interactions to be negligible, compute estimates of the main effects and interactions for the yield outcome. d. Plot the estimates on a normal probability plot. Does the plot show that some of the factors influence the yield? If so, which ones?

Construct a $2_{\mathrm{IV}}^{4-1}$ design for the problem. Select the data for the eight runs that would have been required for this design. Analyze these runs and compare your conclusions to those obtained for the full factorial.

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:

Write down the complete defining relation and the aliases from the design.

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

$$2^5$$

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

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

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 elevator starts at the basement with 8 people (not including the elevator operator) and discharges them all by the time it reaches the top floor, number 6. In how many ways could the operator have perceived the people leaving the elevator if all people look alike to him? What if the 8 people consisted of 5 men and 3 women and the operator could tell a man from a woman?

The previous balance of Sam's savings account was $1345.67. He makes the following transactions: deposits,$228.48, $74.36; withdrawals,$435, $110. What is his new balance?

Let X and Y represent two dimensions of an injection molded part. Suppose X and Y have a bivariate normal distribution with $\sigma_{X}=0.04$, $\sigma_{Y}=0.08$, $\mu_{X}=3.00$, $\mu_{Y}=7.70,$ and $\rho_{Y}=0$. Determine P(2.95 < X < 3.05, 7.60 < Y < 7.80).

(a) Give the (R, S) designations for each chirality center in compound A and for compound B. (b) Write the Fischer projection formula for a compound C that is the diastereomer of A and B. (c) Would C be optically active?