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Design And Analysis Of Experiments 9th Edition Douglas C. Montgomery - Solutions

Consider the 26−2 IV design.(a) Suppose that the design had been folded over by changing the signs in column B instead of column A. What changes would have resulted in the effects that can be estimated from the combined design?(b) Suppose that the design had been folded over by changing the signs
An article by L. B. Hare (“In the Soup: A Case Study to Identify Contributors to Filling Variability,” Journal of Quality Technology, Vol. 20, pp. 36–43) describes a factorial experiment used to study the filling variability of dry soup mix packages. The factors are A = number of mixing ports
Heat treating is often used to carbonize metal parts, such as gears. The thickness of the carbonized layer is a critical output variable from this process, and it is usually measured by performing a carbon analysis on the gear pitch(the top of the gear tooth). Six factors were studied in a 26−2
An experiment is run in a semiconductor factory to investigate the effect of six factors on transistor gain. The design selected is the 26−2 IV shown in Table P8.15.(a) Use a normal plot of the effects to identify the significant factors.(b) Conduct appropriate statistical tests for the model
A 16-run fractional factorial experiment in nine factors was conducted by Chrysler Motors Engineering and described in the article “Sheet Molded Compound Process Improvement,”by P. I. Hsieh and D. E. Goodwin (Fourth Symposium on Taguchi Methods, American Supplier Institute, Dearborn, MI, 1986,
A 16-run fractional factorial experiment in 10 factors on sand-casting of engine manifolds was conducted by engineers at the Essex Aluminum Plant of the Ford Motor Company and described in the article “Evaporative Cast Process 3.0 Liter Intake Manifold Poor Sandfill Study,” by D.
In an article in Quality Engineering (“An Application of Fractional Factorial Experimental Designs,” 1988, Vol. 1, pp. 19–23), M. B. Kilgo describes an experiment to determine the effect of CO2 pressure (A), CO2 temperature (B), peanut moisture (C), CO2 flow rate (D), and peanut particle size
Consider the following design:Std A B C D E y 1 −1 −1 −1 1 1 40 2 1 −1 −1 −1 1 10 3 −1 1 −1 −1 −1 30 4 1 1 −1 1 −1 20 5 −1 −1 1 −1 −1 40 6 1 −1 1 1 −1 30 7 −1 1 1 1 1 20 8 1 1 1 −1 1 30(a) What is the generator for column D?(b) What is the generator for
Consider the following design:Run A B C D E y 1 −1 −1 −1 1 −1 50 2 1 −1 −1 −1 −1 20 3 −1 1 −1 −1 1 40 4 1 1 −1 1 1 25 5 −1 −1 1 −1 1 45 6 1 −1 1 1 1 30 7 −1 1 1 1 −1 40 8 1 1 1 −1 −1 30(a) What is the generator for column D?(b) What is the generator for
Consider the following design:Run A B C D E y 1 −1 −1 −1 −1 −1 65 2 1 −1 −1 −1 1 25 3 −1 1 −1 −1 1 30 4 1 1 −1 −1 −1 89 5 −1 −1 1 −1 1 25 6 1 −1 1 −1 −1 60 7 −1 1 1 −1 −1 70 8 1 1 1 −1 1 50 9 −1 −1 −1 1 1 20 10 1 −1 −1 1 −1 70 11 −1 1
Consider the following design:Run A B C D E y 1 −1 −1 −1 −1 −1 63 2 1 −1 −1 −1 1 21 3 −1 1 −1 −1 1 36 4 1 1 −1 −1 −1 99 5 −1 −1 1 −1 1 24 6 1 −1 1 −1 −1 66 7 −1 1 1 −1 −1 71 8 1 1 1 −1 1 54 9 −1 −1 −1 1 −1 23 10 1 −1 −1 1 1 74 11 −1 1
A 26−2 factorial experiment with three replicates has been run in a pharmaceutical drug manufacturing process. The experimenter has used the following factors:Factor Natural Levels Coded Levels (x’s)A 50, 100 −1, 1 B 20, 60 −1, 1 C 10, 30 −1, 1 D 12, 18 −1, 1 E 15, 30 −1, 1 F 60, 100
An unreplicated 24−1 fractional factorial experiment has been run. The experimenter has used the following factors:Factor Natural Levels Coded Levels (x’s)A 20, 50 −1, 1 B 200, 280 −1, 1 C 50, 100 −1, 1 D 150, 200 −1, 1(a) Suppose that this design has four center runs that average 100.
An unreplicated 24−1 fractional factorial experiment with four center points has been run. The experimenter has used the following factors:Factor Natural Levels Coded Levels (x’s)A - time 10, 50 (minutes) −1, 1 B - temperature 200, 300 (deg C) −1, 1 C - concentration 70, 90 (percent) −1,
An unreplicated 25−1 fractional factorial experiment with four center points has been run in a chemical process.The response variable is molecular weight. The experimenter has used the following factors:Factor Natural Levels Coded Levels (x’s)A - time 20, 40 (minutes) −1, 1 B - temperature
An article in the International Journal of Research in Marketing (“Experimental design on the front lines of marketing: Testing new ideas to increase direct mail sales,”2006, Vol. 23, pp. 309–319) describes the use of a 20-run Plackett–Burman design to investigate the effects of 19 factors
An article in Soldering & Surface Mount Technology(“Characterization of a Solder Paste Printing Process and Its Optimization,” 1999, Vol. 11, No. 3, pp. 23–26) describes the use of a 28−3 fractional factorial experiment to study the effect of eight factors on two responses; percentage
An article in Thin Solid Films (504, “A Study of Si/SiGe Selective Epitaxial Growth by Experimental Design Approach,” 2006, Vol. 504, pp. 95–100) describes the use of a fractional factorial design to investigate the sensitivity of low-temperature (740–760∘C) Si/SiGe selective epitaxial
An article in the Journal of Chromatography A(“Simultaneous Supercritical Fluid Derivatization and Extraction of Formaldehyde by the Hantzsch Reaction,” 2000, Vol. 896, pp. 51–59) describes an experiment where the Hantzsch reaction is used to produce the chemical derivatization of
Consider the 24 factorial experiment in Problem 6.46.Suppose that the experimenters could only afford eight runs.Set up the 24−1 fractional factorial design with I = ABCD and select the responses for the runs from the full factorial data in Problem 6.46.(a) Analyze the data for all of the
Consider the 24 factorial experiment for surfactin production in Problem 6.44. Suppose that the experimenters could only afford eight runs. Set up the 24−1 fractional factorial design with I = ABCD and select the responses for the runs from the full factorial data in Problem 6.44.(a) Analyze the
Consider the 25 factorial in Problem 6.43. Suppose that the experimenters could only afford 16 runs. Set up the 25−1 fractional factorial design with I = ABCDE and select the responses for the runs from the full factorial data in Problem 6.43.(a) Analyze the data and draw conclusions.(b) Compare
Consider the isatin yield data from the experiment described in Problem 6.42. The original experiment was a 24 full factorial. Suppose that the original experimenters could only afford eight runs. Set up the 24−1 fractional factorial design with I = ABCD and select the responses for the runs from
Harry Peterson-Nedry (a friend of the author) owns a vineyard and winery in Newberg, Oregon. He grows several varieties of grapes and produces wine. Harry has used factorial designs for process and product development in the winemaking segment of the business. This problem describes the experiment
A spin coater is used to apply photoresist to a bare silicon wafer. This operation usually occurs early in the semiconductor manufacturing process, and the average coating thickness and the variability in the coating thickness have an important impact on downstream manufacturing steps. Six
A 16-run experiment was performed in a semiconductor manufacturing plant to study the effects of six factors on the curvature or camber of the substrate devices produced. The six variables and their levels are shown in Table P8.2.(a) What type of design did the experimenters use?(b) What are the
Carbon anodes used in a smelting process are baked in a ring furnace. An experiment is run in the furnace to determine which factors influence the weight of packing material that is stuck to the anodes after baking. Six variables are of interest, each at two levels: A = pitch∕fines ratio (0.45,
Nonregular fractions of the 2k [John (1971)].Consider a 24 design. We must estimate the four main effects and the six two-factor interactions, but the full 24 factorial cannot be run. The largest possible block size contains 12 runs.These 12 runs can be obtained from the four one-quarter replicates
Construct a 27−2 design. Show how the design may be run in four blocks of eight observations each. Are any main effects or two-factor interactions confounded with blocks?
Construct a 25−1 design. Show how the design may be run in two blocks of eight observations each. Are any main effects or two-factor interactions confounded with blocks?
An industrial engineer is conducting an experiment using a Monte Carlo simulation model of an inventory system.The independent variables in her model are the order quantity(A), the reorder point (B), the setup cost (C), the backorder cost (D), and the carrying cost rate (E). The response variable
Fold over a 25−2 III design to produce a six-factor design.Verify that the resulting design is a 26−2 IV design. Compare this design to the 26−2 IV design in Table 8.10.
Fold over the 27−4 III design in Table 8.19 to produce an eight-factor design. Verify that the resulting design is a 28−4 IV design. Is this a minimal design?
Consider the 26−3 III design in Problem 8.19. Determine the effects that may be estimated if a single factor fold over of this design is run with the signs for factor A reversed.
Construct a 26−3 III design. Determine the effects that may be estimated if a full fold over of this design is performed.
Construct a 25−2 III design. Determine the effects that may be estimated if a full fold over of this design is performed.
Project the 24−1 IV design in Example 8.1 into two replicates of a 22 design in the factors A and B. Analyze the data and draw conclusions.
Repeat Problem 8.15 using I = −ABCD. Does the use of the alternate fraction change your interpretation of the data?
Analyze the data in Problem 6.32 as if it came from a 24−1 IV design with I = ABCD. Project the design into a full factorial in the subset of the original four factors that appear to be significant.
Consider the 25 design in Problem 6.30. Suppose that only a one-half fraction could be run. Furthermore, two days were required to take the 16 observations, and it was necessary to confound the 25−1 design in two blocks. Construct the design and analyze the data.
Construct a 27−2 design by choosing two four-factor interactions as the independent generators. Write down the complete alias structure for this design. Outline the analysis of variance table. What is the resolution of this design?
Consider the leaf spring experiment in Problem 8.10.Suppose that factor E (quench oil temperature) is very difficult to control during manufacturing. Where would you set factors A, B, C, and D to reduce variability in the free height as much as possible regardless of the quench oil temperature used?
An article in Industrial and Engineering Chemistry(“More on Planning Experiments to Increase Research Efficiency,”1970, pp. 60–65) uses a 25−2 design to investigate the effect of A = condensation temperature, B = amount of material 1, C = solvent volume, D = condensation time, and E =
An article by J. J. Pignatiello Jr. and J. S. Ramberg in the Journal of Quality Technology (Vol. 17, 1985, pp. 198–206) describes the use of a replicated fractional factorial to investigate the effect of five factors on the free height of leaf springs used in an automotive application. The
R. D. Snee (“Experimenting with a Large Number of Variables,” in Experiments in Industry: Design, Analysis and Interpretation of Results, by R. D. Snee, L. B. Hare, and J. B.Trout, Editors, ASQC, 1985) describes an experiment in which a 25−1 design with I = ABCDE was used to investigate the
Continuation of Problem 8.6. Reconsider the 24−1 fractional factorial design with I = ABCD from Problem 8.6.Set a partial fold over of this fraction to isolate the AB interaction.Select the appropriate set of responses from the full factorial data in Example 6.6 and analyze the resulting data.
Continuation of Problem 8.6. In Problem 6.6, we found that all four main effects and the two-factor AB interaction were significant. Show that if the alternate fraction(I = −ABCD) is added to the 24−1 design in Problem 8.6 that the analysis of the results from the combined design produce
In Example 6.10, a 24 factorial design was used to improve the response rate to a credit card mail marketing offer. Suppose that the researchers had used the 24−1 fractional factorial design with I = ABCD instead. Set up the design and select the responses for the runs from the full factorial
Continuation of Problem 8.4. Suppose you have made the eight runs in the 25−2 design in Problem 8.4. What additional runs would be required to identify the factor effects that are of interest? What are the alias relationships in the combined design?
Problem 6.30 describes a process improvement study in the manufacturing process of an integrated circuit. Suppose that only eight runs could be made in this process. Set up an appropriate 25−2 design and find the alias structure. Use the appropriate observations from Problem 6.28 as the
Consider the plasma etch experiment described in Example 6.1. Suppose that only a one-half fraction of the design could be run. Set up the design and analyze the data.
Suppose that in Problem 6.19, only a one-half fraction of the 24 design could be run. Construct the design and perform the analysis, using the data from replicate I.
Suppose that in the chemical process development experiment described in Problem 6.11, it was only possible to run a one-half fraction of the 24 design. Construct the design and perform the statistical analysis, using the data from replicate I.
Suppose that you are designing an experiment for four factors and that due to material properties it is necessary to conduct the experiment in blocks. Material availability restricts you to the use of two blocks but each batch of material is large enough for up to 10 runs. You can afford to make
Suppose that you are designing an experiment for four factors and that due to material properties it is necessary to conduct the experiment in blocks. Material availability restricts you to the use of two blocks; however, each batch of material is only sufficient for six runs. So the standard 24
Consider the full 25 factorial design in Problem 6.51.Suppose that this experiment had been run in two blocks with ABCDE confounded with the blocks. Set up the blocked design and perform the analysis. Compare your results with the results obtained for the completely randomized design in Problem
The information on the interaction confounded with the block can always be separated from the block effect.(a) True(b) False
Consider the 25 factorial design in two blocks.If ABCDE is confounded with blocks, then which of the following runs is in the same block as run acde?(a) a (b) acd (c) bcd(d) be (e) abe (f) None of the above
When constructing the 27 design confounded in eight blocks, three independent effects are chosen to generate the blocks, and there are a total of eight interactions confounded with blocks.(a) True(b) False
Suppose that a 22 design has been conducted. There are four replicates and the experiment has been conducted in four blocks. The error sum of squares is 500 and the block sum of squares is 250. If the experiment had been conducted as a completely randomized design, the estimate of the error
Construct a 23 design with ABC confounded in the first two replicates and BC confounded in the third. Outline the analysis of variance and comment on the information obtained.
Suppose that in Problem 6.11 ABCD was confounded in replicate I and ABC was confounded in replicate II. Perform the statistical analysis of this design.
Repeat the analysis of Problem 6.5 assuming that ABC was confounded with blocks in each replicate.
Suppose that in Problem 6.5 we had confounded ABC in replicate I, AB in replicate II, and BC in replicate III.Calculate the factor effect estimates. Construct the analysis of variance table.
Consider the data in Example 7.2. Suppose that all the observations in block 2 are increased by 20. Analyze the data that would result. Estimate the block effect. Can you explain its magnitude? Do blocks now appear to be an important factor? Are any other effect estimates impacted by the change you
Consider the 22 design in two blocks with AB confounded. Prove algebraically that SSAB = SSBlocks.
Consider the 26 design in eight blocks of eight runs each with ABCD, ACE, and ABEF as the independent effects chosen to be confounded with blocks. Generate the design.Find the other effects confounded with blocks.
Design an experiment for confounding a 26 factorial in four blocks. Suggest an appropriate confounding scheme, different from the one shown in Table 7.9.
The design in Problem 6.46 is a 23 factorial replicated twice. Suppose that each replicate was a block. Analyze all of the responses from this blocked design. Are the results comparable to those from Problem 6.46? Is the block effect large?
The design in Problem 6.44 is a 24 factorial. Set up this experiment in two blocks with ABCD confounded. Analyze the data from this design. Is the block effect large?
Repeat Problem 7.16 using a design in two blocks.
The experiment in Problem 6.43 is a 25 factorial.Suppose that this design had been run in four blocks of eight runs each.(a) Recommend a blocking scheme and set up the design.(b) Analyze the data from this blocked design. Is blocking important?
Consider the isatin yield experiment in Problem 6.42.Set up the 24 experiment in this problem in two blocks with ABCD confounded. Analyze the data from this design. Is the block effect large?
Consider the direct mail experiment in Problem 6.28.Suppose that each group of customers is in a different part of the country. Suggest an appropriate analysis for the experiment.
Using the data from the 24 design in Problem 6.26, construct and analyze a design in two blocks with ABCD confounded with blocks.
Consider the putting experiment in Problem 6.25. Analyze the data considering each replicate as a block.
Consider the fill height deviation experiment in Problem 6.24. Suppose that only four runs could be made on each shift. Set up a design with ABC confounded in replicate I and AC confounded in replicate II. Analyze the data and comment on your findings.
Consider the fill height deviation experiment in Problem 6.24. Suppose that each replicate was run on a separate day. Analyze the data assuming that days are blocks.
Consider the data from the 25 design in Problem 6.30.Suppose that it was necessary to run this design in four blocks with ACDE and BCD (and consequently ABE) confounded.Analyze the data from this design.
Repeat Problem 7.7 assuming that four blocks are necessary. Suggest a reasonable confounding scheme.
Using the data from the 25 design in Problem 6.30, construct and analyze a design in two blocks with ABCDE confounded with blocks.
Repeat Problem 7.5 assuming that four blocks are required. Confound ABD and ABC (and consequently CD)with blocks.
Consider the data from the first replicate of Problem 6.11. Construct a design with two blocks of eight observations each with ABCD confounded. Analyze the data.
Consider the data from the first replicate of Problem 6.5. Suppose that these observations could not all be run using the same bar stock. Set up a design to run these observations in two blocks of four observations each with ABC confounded.Analyze the data.
Consider the alloy cracking experiment described in Problem 6.19. Suppose that only 16 runs could be made on a single day, so each replicate was treated as a block. Analyze the experiment and draw conclusions.
Consider the experiment described in Problem 6.9.Analyze this experiment assuming that each one of the four replicates represents a block.
Consider the experiment described in Problem 6.5.Analyze this experiment assuming that each replicate represents a block of a single production shift.
The display below summarizes the results of analyzing a 24 factorial design.Term Intercept Effect Estimate Sum of Squares %Contribution A 6.25 3.25945 B 5.25 110.25 57.4967 C 3.5 49 25.5541 D 0.75 1.1734 AB 0.75 2.25 1.1734 AC –0.5 1 0.521512 AD 0.75 2.25 1.1734 BC 1.5 9 BD 0.25 0.25 0.130378 CD
Suppose that you want to replicate 2 of the 8 runs in a 23 factorial design. How many ways are there to choose the 2 runs to replicate? Suppose that you decide to replicate the run with all three factors at the high level and the run with all three factors at the low level.(a) Is the resulting
If a D-optimal design algorithm is used to create a 12-run design for fitting a first-order model in three variables with all three two-factor interactions, the algorithm will construct a 23 factorial with four center runs.(a) True(b) False
A 2k factorial design is a D-optimal design for fitting a first-order model.(a) True(b) False
The mean square for pure error in a replicated factorial design can get smaller if nonsignificant terms are added to a model.(a) True(b) False
Adding center runs to a 2k design affects the estimate of the intercept term but not the estimates of any other factor effects.(a) True(b) False
In a replicated 23 design (16 runs), the estimate of the model intercept is equal to one-half of the total of all 16 runs.(a) True(b) False
In an unreplicated design, the degrees of freedom associated with the “pure error” component of error are zero.(a) True(b) False
A half-normal plot of factor effects plots the expected normal percentile versus the effect estimate.(a) True(b) False
Consider the 23 shown below:When running a designed experiment, it is sometimes difficult to reach and hold the precise factor levels required by the design. Small discrepancies are not important, but large ones are potentially of more concern. To illustrate, the experiment presented in Table P6.16
An article in Quality and Reliability Engineering International(2010, Vol. 26, pp. 223–233) presents a 25 factorial design. The experiment is shown in Table P6.15.(a) Analyze the data from this experiment. Identify the significant factors and interactions.(b) Analyze the residuals from this
Suppose that a full 24 factorial uses the following factor levels:Factor Low (−) High(+)A: Acid strength (%) 85 95 B: Reaction time (min) 15 35 C: Amount of acid (mL) 35 45 D: Reaction temperature (∘C) 60 80 The fitted model from this experiment is ̂y = 24 + 16x1 −34x2 + 12x3 + 6x4 −
Suppose that you want to run a 23 factorial design. The variance of an individual observation is expected to be about 4. Suppose that you want the length of a 95 percent confidence interval on any effect to be less than or equal to 1.5. How many replicates of the design do you need to run?

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