(a) Obtain a 1 2 fraction of a 24 factorial design using

The following data are obtained from a \(2^3\) factorial experiment replicated three times. Evaluate the sums of squares for all factorial effects by the contrast method. Draw conclusions.

In an experiment conducted by the Mining Engineering Department at Virginia Tech to study a particular filtering system for coal, a coagulant was added to a solution in a tank containing coal and sludge, which was then placed in a recirculation system in order that the coal could be washed. Three factors were varied in the experimental process: Factor A: percent solids circulated initially in the overflow Factor B: flow rate of the polymer Factor C: pH of the tank The amount of solids in the underflow of the cleansing system determines how clean the coal has become. Two levels of each factor were used and two experimental runs were made for each of the 23 = 8 combinations. The response measurements in percent solids by weight in the underflow of the circulation system are as specified in the following table: Assuming that all interactions are potentially important, do a complete analysis of the data. Use P-values in your conclusion.

In a metallurgy experiment, it is desired to test the eect of four factors and their interactions on the concentration (percent by weight) of a particular phosphorus compound in casting material. The variables are A, percent phosphorus in the renement; B, percent remelted material; C, uxing time; and D, holding time. The four factors are varied in a 24 factorial experiment with two castings taken at each factor combination. The 32 castings were made in random order. The following table shows the data and an ANOVA table is given in Figure 15.8 on page 610. Discuss the eects of the factors and their interactions on the concentration of the phosphorus compound. Weight Treatment % of Phosphorus Compound Combination Rep 1 Rep 2 Total (1) 30.3 28.6 58.9 a 28.5 31.4 59.9 b 24.5 25.6 50.1 ab 25.9 27.2 53.1 c 24.8 23.4 48.2 ac 26.9 23.8 50.7 bc 24.8 27.8 52.6 abc 22.2 24.9 47.1 d 31.7 33.5 65.2 ad 24.6 26.2 50.8 bd 27.6 30.6 58.2 abd 26.3 27.8 54.1 cd 29.9 27.7 57.6 acd 26.8 24.2 51.0 bcd 26.4 24.9 51.3 abcd 26.9 29.3 56.2 Total 428.1 436.9 865.0

A preliminary experiment is conducted to study the effects of four factors and their interactions on the output of a certain machining operation. Two runs are made at each of the treatment combinations in order to supply a measure of pure experimental error. Two levels of each factor are used, resulting in the data shown next page. Make tests on all main effects and interactions at the 0.05 level of significance. Draw conclusions.

In the study An X-Ray Fluorescence Method for Analyzing Polybutadiene-Acrylic Acid (PBAA) Propellants (Quarterly Reports, RK-TR-62-1, Army Ordnance Missile Command), an experiment was conducted to determine whether or not there was a significant difference in the amount of aluminum obtained in an analysis with certain levels of certain processing variables. The data are shown below. The variables in the data are given as below. A: mixing time level 1: 2 hours level 2: 4 hours B: blade speed level 1: 36 rpm level 2: 78 rpm C: condition of nitrogen passed over propellant level 1: dry level 2: 72% relative humidity D: physical state of propellant level 1: uncured level 2: cured Assuming all three- and four-factor interactions to be negligible, analyze the data. Use a 0.05 level of significance. Write a brief report summarizing the findings.

It is important to study the eect of the concentration of the reactant and the feed rate on the viscosity of the product from a chemical process. Let the reactant concentration be factor A, at levels 15% and 25%. Let the feed rate be factor B, with levels 20 lb/hr and 30 lb/hr. The experiment involves two experimental runs at each of the four combinations (L = low and H = high). The viscosity readings are as follows.(a) Assuming a model containing two main eects and an interaction, calculate the three eects. Do you have any interpretation at this point? (b) Do an analysis of variance and test for interaction. Give conclusions. (c) Test for main eects and give nal conclusions regarding the importance of all these eects.

Consider Exercise 15.3. It is of interest to the researcher to learn not only that AD, BC, and possibly AB are important, but also what they mean scientifically. Show two-dimensional interaction plots for all three and give an interpretation.

Consider Exercise 15.3 once again. Three-factor interactions are often not signicant, and even if they are, they are dicult to interpret. The interaction ABD appears to be important. To gain some sense of interpretation, show two AD interaction plots, one for B = 1 and the other for B = +1. From the appearance of these, give an interpretation of the ABD interaction.

Consider Exercise 15.6. Use a +1 and1 scaling for high and low, respectively, and do a multiple linear regression with the model Yi = 0 + 1x1i + 2x2i + 12x1ix2i + i, with x1i = reactant concentration (1,+1) and x2i = feed rate (1,+1). (a) Compute regression coecients. (b) How do the coecients b1, b2, and b12 relate to the eects you found in Exercise 15.6(a)? (c) In your regression analysis, do t-tests on b1, b2, and b12. How do these test results relate to those in Exercise 15.6(b) and (c)?

Consider Exercise 15.5. Compute all 15 eects and do normal probability plots of the eects. (a) Does it appear as if your assumption of negligible three- and four-factor interactions has merit? (b) Are the results of the eect plots consistent with what you communicated about the importance of main eects and two-factor interactions in your summary report?

In Myers, Montgomery, and Anderson-Cook (2009), a data set is discussed in which a 23 factorial is used by an engineer to study the eects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and duplicates are run at each design point with the order of the runs being random. The data are presented here. ABC Life (1) 22, 31 a + 32, 43 b + 35, 34 ab ++ 35, 47 c + 44, 45 ac + + 40, 37 bc + + 60, 50 abc + + + 39, 41 (a) Calculate all seven eects. Which appear, based on their magnitude, to be important? (b) Do an analysis of variance and observe P-values. (c) Do your results in (a) and (b) agree?(d) The engineer felt condent that cutting speed and cutting angle should interact. If this interaction is signicant, draw an interaction plot and discuss the engineering meaning of the interaction.

Consider Exercise 15.11. Suppose there was some experimental diculty in making the runs. In fact, the total experiment had to be halted after only 4 runs. As a result, the abbreviated experiment is given by Life a 43 b 35 c 44 abc 39 With only these runs, we have the signs for contrasts given by A B C AB AC BC ABC a + ++ b + + + c ++ + abc + + + + + + + Comment. In your comments, determine whether or not the contrasts are orthogonal. Which are and which are not? Are main eects orthogonal to each other? In this abbreviated experiment (called a fractional factorial), can we study interactions independent of main eects? Is it a useful experiment if we are convinced that interactions are negligible? Explain.

Consider a 25 experiment where the experimental runs are on 4 dierent machines. Use the machines as blocks, and assume that all main eects and two-factor interactions may be important. (a) Which runs would be made on each of the 4 machines? (b) Which eects are confounded with blocks?

An experiment is described in Myers, Montgomery, and Anderson-Cook (2009) in which optimum conditions are sought for storing bovine semen to obtain maximum survival. The variables are percent sodium citrate, percent glycerol, and equilibration time in hours. The response is percent survival of the motile spermatozoa. The natural levels are found in the above reference. The data with coded levels for the factorial portion of the design and the center runs are given. (a) Fit a linear regression model to the data and determine which linear and interaction terms are significant. Assume that the x1x2x3 interaction is negligible. (b) Test for quadratic lack of t and comment. x1,% x3 Sodium x2, % Equilibration % Citrate Glycerol Time Survival 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0 57 40 19 40 54 41 21 43 63 61

Oil producers are interested in nickel alloys that are strong and corrosion resistant. An experiment was conducted in which yield strengths were compared for nickel alloy tensile specimens charged in solutions of sulfuric acid saturated with carbon disulfide. Two alloys were compared: a 75% nickel composition and a 30% nickel composition. The alloys were tested under two different charging times, 25 and 50 days. A \(2^3\) factorial was conducted with the following factors: % sulfuric acid: 4%, 6% (\(x_1\)) charging time: 25 days, 50 days (\(x_2\)) nickel composition: 30%, 75% (\(x_3\)) A specimen was prepared for each of the 8 conditions. Since the engineers were not certain of the nature of the model (i.e., whether or not quadratic terms would be needed), a third level (middle level) was incorporated, and 4 center runs were employed using 4 specimens at 5% sulfuric acid, 37.5 days, and 52.5% nickel composition. The following are the yield strengths in kilograms per square inch. The center runs gave the following strengths: 51.6, 51.4, 52.4, 52.9 (a) Test to determine which main effects and interactions should be involved in the fitted model. (b) Test for quadratic curvature. (c) If quadratic curvature is significant, how many additional design points are needed to determine which quadratic terms should be included in the model?

Suppose a second replicate of the experiment in Exercise 15.13 could be performed. (a) Would a second replication of the blocking scheme of Exercise 15.13 be the best choice? (b) If the answer to part (a) is no, give the layout for a better choice for the second replicate. (c) What concept did you use in your design selection?

Consider Figure 15.14, which represents a 22 factorial with 3 center runs. If quadratic curvature is signicant, what additional design points would you select that might allow the estimation of the terms x2 1,x2 2? Explain.

List the aliases for the various eects in a 25 factorial experiment when the dening contrast is ACDE.

(a) Obtain a 1 2 fraction of a 24 factorial design using BCD as the dening contrast. (b) Divide the 1 2 fraction into 2 blocks of 4 units each by confounding ABC. (c) Show the analysis-of-variance table (sources of variation and degrees of freedom) for testing all unconfounded main eects, assuming that all interaction eects are negligible.

Construct a 1 4 fraction of a 26 factorial design using ABCD and BDEF as the dening contrasts. Show what eects are aliased with the six main eects.

(a) Using the defining contrasts ABCE and ABDF, obtain a \(\frac{1}{4}\) fraction of a \(2^6\) design. (b) Show the analysis-of-variance table (sources of variation and degrees of freedom) for all appropriate tests, assuming that E and F do not interact and all three-factor and higher interactions are negligible.

Seven factors are varied at two levels in an experiment involving only 16 trials. A 1 8 fraction of a 27 factorial experiment is used, with the dening contrasts being ACD, BEF, and CEG. The data are as follows: Treat. Treat. Comb. Response Comb. Response (1) ad abce cdef acef bcde abdf bf 31.6 28.7 33.1 33.6 33.7 34.2 32.5 27.8 acg cdg beg adefg efg abdeg bcdfg abcfg 31.1 32.0 32.8 35.3 32.4 35.3 35.6 35.1 Perform an analysis of variance on all seven main effects, assuming that interactions are negligible. Use a 0.05 level of signicance.

An experiment is conducted so that an engineer can gain insight into the inuence of sealing temperature A, cooling bar temperature B, percent polyethylene additive C, and pressure D on the seal strength (in grams per inch) of a bread-wrapper stock. A 1 2 fraction of a 24 factorial experiment is used, with the dening contrast being ABCD. The data are presented here. Perform an analysis of variance on main eects only. Use =0 .05.

In an experiment conducted at the Department of Mechanical Engineering and analyzed by the Statistics Consulting Center at Virginia Tech, a sensor detects an electrical charge each time a turbine blade makes one rotation. The sensor then measures the amplitude of the electrical current. Six factors are rpm A, temperature B, gap between blades C, gap between blade and casing D, location of input E, and location of detection F.A1 4 fraction of a 26 factorial experiment is used, with dening contrasts being ABCE and BCDF. The data are as follows: Perform an analysis of variance on main eects and two-factor interactions, assuming that all three-factor and higher interactions are negligible. Use =0 .05.

In the study Durability of Rubber to Steel Adhesively Bonded Joints, conducted at the Department of Environmental Science and Mechanics and analyzed by the Statistics Consulting Center at Virginia Tech, an experimenter measured the number of breakdowns in an adhesive seal. It was postulated that concentration of seawater A, temperature B, pHC, voltage D, and stress E inuence the breakdown of an adhesive seal. A 1 2 fraction of a 25 factorial experiment was used, with the dening contrast being ABCDE. The data are as follows: Perform an analysis of variance on main eects and two factor interactions AD, AE, BD, BE, assuming that all three-factor and higher interactions are negligible. Use =0 .05

Consider a 251 design with factors A, B, C, D, and E. Construct the design by beginning with a 24 and use E = ABCD as the generator. Show all aliases.

There are six factors and only eight design points can be used. Construct a \(2^{6?3}\) by beginning with a \(2^3\) and use D = AB, E = ?AC, and F = BC as the generators.

Consider Exercise 15.27. Construct another 263 that is dierent from the design chosen in Exercise 15.27.

For Exercise 15.27, give all aliases for the six main effects.

In Myers, Montgomery, and Anderson-Cook (2009), an application is discussed in which an engineer is concerned with the eects on the cracking of a titanium alloy. The three factors are A, temperature; B, titanium content; and C, amount of grain rener. The following table gives a portion of the design and the response, crack length induced in the sample of the alloy. ABCResponse 1 1 1 1 1 1 1 1 1 1 1 1 0.5269 2.3380 4.0060 3.3640 (a) What is the dening relation? (b) Give aliases for all three main eects assuming that two-factor interactions may be real. (c) Assuming that interactions are negligible, which main factor is most important? (d) At what level would you suggest the factor named in (c) be for nal production, high or low? (e) At what levels would you suggest the other factors be for nal production? (f) What hazards lie in the recommendations you made in (d) and (e)? Be thorough in your answer.

Consider an example in which there are two control variables x1 and x2 and a single noise variable z. The goal is to determine the levels of x1 and x2 that are robust to changes in z, i.e., levels of x1 and x2 that minimize the variance produced in the response y as z moves between1 and +1. The variables x1 and x2 are at two levels, 1 and +1, in the experiment. The data produce the plots in Figure 15.22 above. Note that x1 and x2 interact with the noise variable z. What settings on x1 and x2 (1 or +1 for each) result in minimum variance in y? Explain.

Consider the following 23 factorial with control variables x1 and x2 and noise variable z. Canx1 and x2 be chosen at levels for which Var(y) is minimized? Explain why or why not. z = 1 z = +1x 2 = 1 x2 = +1 x2 = 1 x2 = +1x 1 = 1x 1 = +1 4 1 6 3 8 3 10 5

Consider Case Study 15.1 involving the injection molding data. Suppose mold temperature is difcult to control and thus it can be assumed that in the process it follows a normal distribution with mean 0 and variance 2 z. Of concern is the variance of the shrinkage response in the process itself. In the analysis of Figure 15.7, it is clear that mold temperature, injection velocity, and the interaction between the two are the only important factors. (a) Can the setting on velocity be used to create some type of control on the process variance in shrinkage which arises due to the inability to control temperature? Explain. (b) Using parameter estimates from Figure 15.7, give an estimate of the following models: (i)mean shrinkage across the distribution of temperature; (ii)shrinkage variance as a function of 2 z. (c) Use the estimated variance model to determine the level of velocity that minimizes the shrinkage variance. (d) Use the mean shrinkage model to determine what value of velocity minimizes mean shrinkage. (e) Are your results above consistent with your analysis from the interaction plot in Figure 15.6? Explain.

In Case Study 15.2 involving the coal cleansing data, the percent solids in the process system is known to vary uncontrollably during the process and is viewed as a noise factor with mean 0 and variance 2 z. The response, cleansing eciency, has a mean and variance that change behavior during the process. Use only signicant terms in the following parts. (a) Use the estimates in Figure 15.9 to develop the process mean eciency and variance models. (b) What factor (or factors) might be controlled at certain levels to control or otherwise minimize the process variance? (c) What conditions of factors B and C within the design region maximize the estimated mean? (d) What level of C would you suggest for minimization of process variance when B = 1? WhenB = 1?

Use the coal cleansing data of Exercise 15.2 on page 609 to t a model of the type E(Y )=0 + 1x1 + 2x2 + 3x3, where the levels are x1, percent solids: 8, 12 x2, ow rate: 150, 250 gal/min x3, pH: 5, 6 Center and scale the variables to design units. Also conduct a test for lack of t, and comment concerning the adequacy of the linear regression model.

A2 5 factorial plan is used to build a regression model containing rst-order coecients and model terms for all two-factor interactions. Duplicate runs are made for each factor. Outline the analysis-of-variance table, showing degrees of freedom for regression, lack of t, and pure error.

Consider the 1 16 of the 27 factorial discussed in Section 15.9. List the additional 11 dening contrasts.

Construct a Plackett-Burman design for 10 variables containing 24 experimental runs.

A Plackett-Burman design was used to study the rheological properties of high-molecular-weight copolymers. Two levels of each of six variables were xed in the experiment. The viscosity of the polymer is the response. The data were analyzed by the Statistics Consulting Center at Virginia Tech for personnel in the Chemical Engineering Department at the University. The variables are as follows: hard block chemistry x1, nitrogen ow rate x2, heat-up time x3, percent compression x4, scans (high and low) x5, percent strain x6. The data are presented here. Obs. x1 x2 x3 x4 x5 x6 y 1 2 3 4 5 6 7 8 9 10 11 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 194,700 588,400 7533 514,100 277,300 493,500 8969 18,340 6793 160,400 7008 3637 Build a regression equation relating viscosity to the levels of the six variables. Conduct t-tests for all main eects. Recommend factors that should be retained for future studies and those that should not. Use the residual mean square (5 degrees of freedom) as a measure of experimental error.

A large petroleum company in the Southwest regularly conducts experiments to test additives to drilling uids. Plastic viscosity is a rheological measure reecting the thickness of the uid. Various polymers are added to the uid to increase viscosity. The following is a data set in which two polymers are used at two levels each and the viscosity measured. The concentration of the polymers is indicated as low or high. Conduct an analysis of the 22 factorial experiment. Test for eects for the two polymers and interaction. Polymer 1 Polymer 2 Low High Low High 3.0 11.7 3.5 12.0 11.3 21.7 12.0 22.4

A \(2^2\) factorial experiment is analyzed by the Statistics Consulting Center at Virginia Tech. The client is a member of the Department of Housing, Interior Design, and Resource Management. The client is interested in comparing cold start to preheating ovens in terms of total energy delivered to the product. In addition, convection is being compared to regular mode. Four experimental runs are made at each of the four factor combinations. Following are the data from the experiment: Do an analysis of variance to study main effects and interaction. Draw conclusions.

In the study The Use of Regression Analysis for Correcting Matrix Eects in the X-Ray Fluorescence Analysis of Pyrotechnic Compositions, published in the Proceedings of the Tenth Conference on the Design of Experiments in Army Research Development and Testing, ARO-D Report 65-3 (1965), an experiment was conducted in which the concentrations of four components of a propellant mixture and the weights of ne and coarse particles in the slurry were each allowed to vary. Factors A, B, C, and D, each at two levels, represent the concentrations of the four components, and factors E and F, also at two levels, represent the weights of the ne and coarse particles present in the slurry. The goal of the analysis was to determine if the X-ray intensity ratios associated with component 1 of the propellant were signicantly inuenced by varying the concentrations of the various components and the weights of the particles in the mixture. A 1 8 fraction of a 26 factorial experiment was used, with the dening contrasts being ADE, BCE, and ACF. The data shown here represent the total of a pair of intensity readings. The pooled mean square error with 8 degrees of freedom is given by 0.02005. Analyze the data using a 0.05 level of signicance to determine if the concentrations of the components and the weights of the ne and coarse particles present in the slurry have a signicant inuence on the intensity ratios associated with com ponent 1. Assume that no interaction exists among the six factors. Treatment Intensity Batch Combination Ratio Total 1 abef 2.2480 2 cdef 1.8570 3 (1) 2.2428 4 ace 2.3270 5 bde 1.8830 6 abcd 1.8078 7 adf 2.1424 8 bcf 1.9122

Use Table 15.16 to construct a 16-run design with 8 factors that is resolution IV.

Verify that your design in Review Exercise 15.43 is indeed resolution IV.

Construct a design that contains 9 design points, is orthogonal, contains 12 total runs and 3 degrees of freedom for replication error, and allows for a lack-of-fit test for pure quadratic curvature.

Consider a design which is a 231 III with 2 center runs. Consider yf as the average response at the design parameter and y0 as the average response at the design center. Suppose the true regression model is E(Y )=0 + 1x1 + 2x2 + 3x3 + 11x2 1 + 22x2 2 + 33x2 3. (a) Give (and verify) E( yf y0). (b) Explain what you have learned from the result in (a).