Construct a

$$2^5$$

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

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 24 design.

$$\begin{matrix} \quad & \text{Replicate}\\ \text{Treatment Combination} & \text{I} & \text{II}\\ \text{(1)} & \text{159} & \text{163}\\ \text{a} & \text{168} & \text{175}\\ \text{b} & \text{158} & \text{163}\\ \text{ab} & \text{166} & \text{168}\\ \text{c} & \text{175} & \text{178}\\ \end{matrix}$$

$$\begin{matrix} \quad & \text{Replicate}\\ \text{Treatment Combination} & \text{I} & \text{II}\\ \text{ac} & \text{179} & \text{183}\\ \text{bc} & \text{173} & \text{168}\\ \text{abc} & \text{179} & \text{182}\\ \text{d} & \text{164} & \text{159}\\ \text{ad} & \text{187} & \text{189}\\ \text{bd} & \text{163} & \text{159}\\ \text{abd} & \text{185} & \text{191}\\ \text{cd} & \text{168} & \text{174}\\ \text{acd} & \text{197} & \text{199}\\ \text{bcd} & \text{170} & \text{174}\\ \text{abcd} & \text{194} & \text{198}\\ \end{matrix}$$

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 as shown. Analyze the data using t-ratios and draw conclusions. Use $\alpha = 0.05$ in the statistical tests.

An experiment was conducted to determine whether either firing temperature or furnace position affects the baked density of a carbon anode. The data are as follows:

Test the hypotheses in part (a) using the analysis of variance with $\alpha=0.05$. What are your conclusions?

Consider the data below, replicate I only. Suppose that a center point (with four replicates) is added to these eight runs. The tool life response at the center point is 425, 400, 437, and 418. (a) Estimate the factor effects. (b) Estimate pure error using the center points. (c) Calculate the t-ratio for curvature and test at $\alpha = 0.05.$ (d) Test for main effects and interaction effects, using $\alpha = 0.05.$ (e) Give the regression model and analyze the residuals from this experiment.An engineer is interested in the effect of cutting speed (A), metal hardness (B), and cutting angle (C) on the life of a cutting tool. Two levels of each factor are chosen, and two replicates of a $2^3$ factorial design are run. The tool life data (in hours) are shown in the following table.

$$\begin{matrix} \quad & \text{Replicate}\\ \text{Treatment Combination} & \text{I} & \text{II}\\ \text{(1)} & \text{221} & \text{311}\\ \text{a} & \text{325} & \text{435}\\ \text{b} & \text{354} & \text{348}\\ \text{ab} & \text{552} & \text{472}\\ \text{c} & \text{440} & \text{453}\\ \text{ac} & \text{406} & \text{377}\\ \text{bc} & \text{605} & \text{500}\\ \text{abc} & \text{392} & \text{419}\\ \end{matrix}$$

(a) Analyze the data from this experiment using t-ratios with $\alpha=0.05.$ (b) Find an appropriate regression model that explains tool life in terms of the variables used in the experiment. (c) Analyze the residuals from this experiment.

An article in Quality Engineering [2012, Vol. 24(1)] lescribed an experiment on a grinding wheel. The following are ome of the grinding force data (in N) from this experiment at two lifferent vibration levels.

Low $242,249,235,250,254,244,258,311,237,261,314,252$ High $302,421,419,399,317,311,350,363,392,367,301,302$

Is there evidence to support the claim that the mean grinding force increases with the vibration level?

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 the following Minitab output for a 23 factorial experiment.

(a) How many replicates were used in the experiment?

(b) Calculate the standard error of a coefficient.

(c) Calculate the entries marked with “?” in the output.

$$\begin{matrix} \mathrm{Factorial~Fit:~y~versus~A,~B,~C}~&~\quad\\ \mathrm{Estimated~Effects~and~Coefficients~for~y~(coded~units)}~&~\quad\\ \mathrm{Team}~&~\mathrm{Effects}~&~\mathrm{Coef}~&~\mathrm{SE~Coef}~&~\mathrm{T}~&~\mathrm{P}\\ \mathrm{Constant~}~&~\quad~&~\mathrm{579.33}~&~\mathrm{38.46}~&~\mathrm{15.06}~&~\mathrm{0.000}\\ \mathrm{A}~&~\mathrm{2.95}~&~\mathrm{1.47}~&~\mathrm{38.46}~&~\mathrm{0.04}~&~\mathrm{0.970}\\ \mathrm{B}~&~\mathrm{15.92}~&~\mathrm{?}~&~\mathrm{38.46}~&~\mathrm{0.21}~&~\mathrm{0.841}\\ \mathrm{C~}~&~\mathrm{-37.87}~&~\mathrm{-18.94}~&~\mathrm{38.46}~&~\mathrm{-0.49}~&~\mathrm{0.636}\\ \mathrm{A~\ast~B}~&~\mathrm{20.43}~&~\mathrm{10.21}~&~\mathrm{38.46}~&~\mathrm{?}~&~\mathrm{0.797}\\ \mathrm{A~\ast~C}~&~\mathrm{-17.11}~&~\mathrm{-8.55}~&~\mathrm{38.46}~&~\mathrm{-0.22}~&~\mathrm{0.830}\\ \mathrm{B~\ast~C}~&~\mathrm{4.41}~&~\mathrm{2.21}~&~\mathrm{38.46}~&~\mathrm{0.06}~&~\mathrm{0.956}\\ \mathrm{A~\ast~B~\ast~C}~&~\mathrm{13.35}~&~\mathrm{6.68}~&~\mathrm{?}~&~\mathrm{0.17}~&~\mathrm{0.866}\\ \mathrm{S~=~153.832~R~-~Sq~~~=~5.22\%}~&~\mathrm{R~-~Sq~(adj)~=~0.11\%}~&~\quad~&~\quad~&~\quad~&~\quad\\ \mathrm{Analysis~of~Variance~for~y~(coded~units)~}~&~\quad~&~\quad~&~\quad~&~\quad~&~\quad\\ \mathrm{Source}~&~\mathrm{DF}~&~\mathrm{Seq}~&~\mathrm{Adj~SS}~&~\mathrm{Adj~MS}~&~\mathrm{F}~&~\mathrm{P}\\ \mathrm{Main~Effects}~&~\mathrm{3}~&~\mathrm{6785}~&~\mathrm{6785}~&~\mathrm{2261.8}~&~\mathrm{?}~&~\mathrm{0.960}\\ \mathrm{2-Way~Interactions}~&~\mathrm{3}~&~\mathrm{?}~&~\mathrm{2918}~&~\mathrm{972.5}~&~\mathrm{0.04}~&~\mathrm{0.988}\\ \mathrm{3-Way~Interactions}~&~\mathrm{1}~&~\mathrm{?}~&~\mathrm{713}~&~\mathrm{713.3}~&~\mathrm{0.03}~&~\mathrm{0.866}\\ \mathrm{Residual~Error}~&~\mathrm{8}~&~\mathrm{189314}~&~\mathrm{189314}~&~\mathrm{23664.2}\\ \mathrm{Pure~Error}~&~\mathrm{8}~&~\mathrm{189314}~&~\mathrm{189314}~&~\mathrm{23664.2}\\ \mathrm{Total}~&~\mathrm{15}~&~\mathrm{199730}\\ \end{matrix}$$

For each of the following designs, answer the following questions. (a) Compute the estimates of the effects and their standard errors for this design. (b) Construct two-factor interaction plots and comment on the interaction of the factors. (c) Use the t ratio to determine the significance of each effect with $\alpha = 0.05.$ Comment on your findings. (d) Compute an approximate 95% CI for each effect. Compare your results with those in part (c) and comment. (e) Perform an analysis of variance of the appropriate regression model for this design. Include in your analysis hypothesis tests for each coefficient, as well as residual analysis. State your final conclusions about the adequacy of the model. Compare your results to part (c) and comment. An article in the IEEE Transactions on Semiconductor Manufacturing (Vol. 5, No. 3, 1992) describes an experiment to investigate the surface charge on a silicon wafer. The factors thought to influence induced surface charge are cleaning method (spin rinse dry, or SRD, and spin dry, or SD) and the position on the wafer where the charge was measured. The surface charge (×1011 q/$cm^3$) response data are as shown.

$$\begin{matrix} \quad & \text{Test Position}\\ \quad & \text{L} & \text{R}\\ \quad & \text{1.66} & \text{1.84}\\ \text{SD} & \text{1.90} & \text{1.84}\\ \quad & \text{1.92} & \text{1.62}\\ \quad & \quad & \quad\\ \quad & \text{-4.21} & \text{-7.58}\\ \text{SRD} & \text{-1.35} & \text{-2.20}\\ \quad & \text{-2.08} & \text{-5.36}\\ \end{matrix}$$

An engineer is interested in the effect of cutting speed (A), metal hardness (B), and cutting angle (C) on the life of a cutting tool. Two levels of each factor are chosen, and two replicates of a 23 factorial design are run. The tool life data (in hours) are shown in the following table. (a) Analyze the data from this experiment using t-ratios with $\alpha = 0.05.$ (b) Find an appropriate regression model that explains tool life in terms of the variables used in the experiment. (c) Analyze the residuals from this experiment.

$$\begin{matrix} \quad & \text{Replicate}\\ \text{Treatment Combination} & \text{I} & \text{II}\\ \text{(1)} & \text{221} & \text{311}\\ \text{a} & \text{325} & \text{435}\\ \text{b} & \text{354} & \text{348}\\ \text{ab} & \text{552} & \text{472}\\ \text{c} & \text{440} & \text{453}\\ \text{ac} & \text{406} & \text{377}\\ \text{bc} & \text{605} & \text{500}\\ \text{abc} & \text{392} & \text{419}\\ \end{matrix}$$

Polyisobutylene is one of the components of butyl rubber used for making inner tubes. (a) Give the structure of polyisobutylene. (b) Is this an addition polymer or a condensation polymer? (c) What conditions (cationic, anionic, free-radical) would be most appropriate for polymerization of isobutylene? Explain your answer.

Polyoxymethylene (polyformaldehyde) is the tough, self-lubricating Delrin plastic used in gear wheels. (a) Give the structure of polyformaldehyde. (b) Formaldehyde is polymerized using an acidic catalyst. Using $\mathrm{H}^{+}$ as a catalyst, propose a mechanism for the polymerization as far as the trimer. (c) Is Delrin an addition polymer or a condensation polymer?

Urylon fibers are used in premium fishing nets because the polymer is relatively stable to UV light and aqueous acid and base. Urylon has the structure shown.

(a) What functional group is contained in the Urylon structure?

(b) Is Urylon a chain-growth polymer or a step-growth polymer?

(c) Draw the products that would be formed if the polymer were completely hydrolyzed under acidic or basic conditions.

Poly(butylene terephthalate) is a hydrophobic plastic material widely used in automotive ignition systems.

(a) What type of polymer is poly(butylene terephthalate)?

(b) Is this an addition polymer or a condensation polymer?

(c) Suggest what monomers might be used to synthesize this polymer and how the polymerization might be accomplished.

Ring-opening metathesis polymerization is a promising new technique for polymerizing cyclic olefins. In its simplest form, the reaction involves a cycloalkene (preferably with some ring strain to drive the reaction) whose double bonds undergo metathesis (a trading of partners at the ends of the double bonds) to give a polymer containing both single and double bonds.