Construct a

$$2^5$$

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

Use the following data:

$$\begin{matrix} x_i& \text{2} & \text{6} & \text{9} & \text{13} & \text{20}\\ y_i & \text{7} & \text{18} & \text{9} & \text{26} & \text{23}\\ \end{matrix}$$

The estimated regression equation for these data is $\hat { y } = 7.6 + .9 x .$ What percentage of the total sum of squares can be accounted for by the estimated regression equation? What is the value of the sample correlation coefficient?

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.

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

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 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:

$$\begin{matrix} \quad & \text{Temperature } (^\circ C)\\ \text{Position} & \text{800} & \text{825}\\ \text{1} & \text{570} & \text{1063}\\ \quad & \text{565} & \text{1080}\\ \quad & \text{583} & \text{1043}\\ \text{2} & \text{528} & \text{988}\\ \quad & \text{547} & \text{1026}\\ \quad & \text{521} & \text{1004}\\ \end{matrix}$$

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.

For the system ethyl ethanoate(1)/n-heptane(2) at 343.15 K, $\ln \gamma_{1}=0.95 x_{2}^{2} ; \ln \gamma_{2}=0.95 x_{1}^{2};$ $P_{1}^{\text {sat }}=79.80 \mathrm{kPa} ; \text { and } P_{2}^{\text {sat }}=40.50 \mathrm{kPa}.$ (a) Make a BUBL P calculation for T = 343.15 K, $x_1 = 0.05.$ (b) Make a DEW P calculation for T = 343.15 K, $y_1 = 0.05.$ (c) What are the azeotrope composition and pressure at T = 343.15 K?

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.

Calculate the largest permissible value of P for the beam and loading shown, knowing that the allowable normal stress is +6 ksi in tension and -18 ksi in compression.

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.

Let

$$A=\left[ \begin{array}{rr}{3} & {4} \\ {-2} & {1} \\ {3} & {4}\end{array}\right], \mathbf{b}=\left[ \begin{array}{r}{11} \\ {-9} \\ {5}\end{array}\right], \mathbf{u}=\left[ \begin{array}{r}{5} \\ {-1}\end{array}\right]$$

, and

$$v=\left[ \begin{array}{r}{5} \\ {-2}\end{array}\right]$$

. Compute Au and Av, and compare them with b. Could u possibly be a least-squares solution of Ax=b? (answer this without computing a least-squares solution.)

A paper in the Journal of the Association of Asphalt Paving Technologists (Vol. 59, 1990) describes an experiment to determine the effect of air voids on percentage retained strength of asphalt. For purposes of the experiment, air voids are controlled at three levels: low (2–4%), medium (4–6%), and high (6–8%). The data are shown in the following table.

$$\begin{matrix} \text{Air Voids } & \quad & \quad & \quad & \text{Retained Strength} (\%)\\ \text{Low} & \text{106} & \text{90} & \text{103} & \text{90} & \text{79} & \text{88} & \text{92} & \text{95}\\ \text{Medium} & \text{80} & \text{69} & \text{94} & \text{91} & \text{70} & \text{83} & \text{87} & \text{83}\\ \text{High} & \text{78} & \text{80} & \text{62} & \text{69} & \text{76} & \text{85} & \text{69} & \text{85}\\ \end{matrix}$$

(a) Do the different levels of air voids significantly affect mean retained strength? Use $\alpha = 0.01.$ (b) Find the P-value for the F-statistic in part (a).