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Design And Analysis Of Experiments 9th Edition Douglas C. Montgomery - Solutions
An article in the Electronic Journal of Biotechnology(“Optimization of Medium Composition for Transglutaminase Production by a Brazilian Soil Streptomyces sp,” available at The correct URL is http://www.ejbiotechnology.info/index.php/ejbiotechnology/article/viewFile/v10n4-10/110) describes the
An article in the Journal of Chromatography A (“Optimization of the Capillary Electrophoresis Separation of Ranitidine and Related Compounds,” Vol. 766, pp. 245–254)describes an experiment to optimize the production of ranitidine, a compound that is the primary active ingredient of Zantac, a
The Paper Helicopter Experiment Revisited.Reconsider the paper helicopter experiment in Problem 11.34.This experiment was actually run in two blocks. Block 1 consisted of the first 16 runs in Table P11.11 (standard order runs 1–16) and two center points (standard order runs 25 and 26).(a) Fit
Box and Liu (1999) describe an experiment flying paper helicopters where the objective is to maximize flight time. They used the central composite design shown in Table P11.11. Each run involved a single helicopter made to the following specifications: x1 = wing area (in2), −1 = 11.80 and+1 =
An article in Quality Progress (“For Starbucks, It’s in the Bag,” March 2011, pp. 18–23) describes using a central composite design to improve the packaging of one-pound coffee. The objective is to produce an airtight seal that is easy to open without damaging the top of the coffee bag.The
Table P11.9 shows a six-variable RSM design from Jones and Nachtsheim (2011). Analyze the response data from this experiment.
Myers, Montgomery, and Anderson-Cook (2009)describe a gasoline blending experiment involving three mixture components. There are no constraints on the mixture proportions, and the following 10-run design is used:Design Point x1 x2 x3 y (mi/gal)1 1 0 0 24.5, 25.1 2 0 1 0 24.8, 23.9 3 0 0 1 22.7,
An experimenter wishes to run a three-component mixture experiment. The constraints in the component proportions are as follows:0.2 ≤ x1 ≤ 0.4 0.1 ≤ x2 ≤ 0.3 0.4 ≤ x2 ≤ 0.7(a) Set up an experiment to fit a quadratic mixture model. Use n = 14 runs, with four replicates. Use the
Suppose that you want to fit a second-order response surface model in a situation where there are k = 4 factors; however, one of the factors is categorical with two levels. What model should you consider for this experiment? Suggest an appropriate design for this situation.
Suppose that you want to fit a second-order model in k = 5 factors. You cannot afford more than 25 runs. Construct both a D-optimal and on I-optimal design for this situation.Compare the prediction variance properties of the designs.Which design would you prefer?
Rework problem 11.26 assuming that the model the engineer wishes to fit is a quadratic. Obviously, only designs 2, 3, and 4 can now be considered.
A chemical engineer wishes to fit a calibration curve for a new procedure used to measure the concentration of a particular ingredient in a product manufactured in his facility.Twelve samples can be prepared, having known concentration.The engineer wants to build a model for the measured
Repeat problem 11.24 using a first-order model with the two-factor interactions.
Consider a 23 design for fitting a first-order model.(a) Evaluate the D-criterion |(X′X)−1| for this design.(b) Evaluate the A-criterion tr(X′X)−1 for this design.(c) Find the maximum scaled prediction variance for this design. Is this design G-optimal?
Suppose that you need to design an experiment to fit a quadratic model over the region −1 ≤ xi ≤ +1, i = 1,2 subject to the constraint x1 + x2 ≤ 1. If the constraint is violated, the process will not work properly. You can afford to make no more than n = 12 runs. Set up the following
Suppose that we approximate a response surface with a model of order d1, such as y = X1????1 + ????, when the true surface is described by a model of order d2 > d1; that is, E(y) =X1????1 + X1????2.(a) Show that the regression coefficients are biased, that is, E(̂????1) = ????1 + A????2, where A =
Yield during the first four cycles of a chemical process is shown in the following table. The variables are percentage of concentration (x1) at levels 30, 31, and 32 and temperature (x2) at 140, 142, and 144∘F. Analyze by EVOP methods.Conditions Cycle (1) (2) (3) (4) (5)1 60.7 59.8 60.2 64.2 57.5
How could a hexagon design be run in two orthogonal blocks?
Blocking in the central composite design. Consider a central composite design for k = 4 variables in two blocks. Can a rotatable design always be found that blocks orthogonally?
Verify that the central composite design shown in Table P11.8 blocks orthogonally:◾ T A B L E P11.8 A CCD in Three Blocks Block 1 Block 2 x1 x2 x3 x1 x2 x3 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 1 −1 1 1 −1 1 −1 −1 −1 −1 Block 3 x1 x2 x3 −1.633 0 0
The rotatable central composite design. It can be shown that a second-order design is rotatable ifΣn u=1 xa iuxb ju =0, if a or b or both are odd, and ifΣn u=1 x4 iu = 3Σn u=1 x2 iux2 ju.Show that for the central composite design these conditions lead to ???? = (nF)1∕4 for rotatability, where
Show that augmenting a 2k design with nC center points does not affect the estimates of the ????i(i = 1, 2, . . . , k)but that the estimate of the intercept ????0 is the average of all 2k + nC observations.
Verify that an orthogonal first-order design is also first-order rotatable.
A central composite design is run in a chemical vapor deposition process, resulting in the experimental data shown in Table P11.7. Four experimental units were processed simultaneously on each run of the design, and the responses are the mean and the variance of thickness, computed across the four
A manufacturer of cutting tools has developed two empirical equations for tool life in hours (y1) and for tool cost in dollars (y2). Both models are linear functions of steel hardness(x1) and manufacturing time (x2). The two equations arêy = 10 + 5x1 + 2x2̂y2 = 23 + 3x1 + 4x2 and both equations
Consider the three-variable central composite design shown in Table P11.6. Analyze the data and draw conclusions, assuming that we wish to maximize conversion (y1) with activity(y2) between 55 and 60.
An experimenter has run a Box–Behnken design and obtained the results as shown in Table P11.5, where the response variable is the viscosity of a polymer:(a) Fit the second-order model.(b) Perform the canonical analysis. What type of surface has been found?(c) What operating conditions on x1, x2,
The hexagon design in Table P11.4 is used in an experiment that has the objective of fitting a second-order model:(a) Fit the second-order model.(b) Perform the canonical analysis. What type of surface has been found?(c) What operating conditions on x1 and x2 lead to the stationary point?(d) Where
The data in Table P11.3 were collected by a chemical engineer. The response y is filtration time, x1 is temperature, and x2 is pressure. Fit a second-order model.(a) What operating conditions would you recommend if the objective is to minimize the filtration time?(b) What operating conditions would
The data shown in the Table P11.2 were collected in an experiment to optimize crystal growth as a function of three variables x1, x2, and x3. Large values of y (yield in grams)are desirable. Fit a second-order model and analyze the fitted surface. Under what set of conditions is maximum growth
The path of steepest ascent is usually computed assuming that the model is truly first order; that is, there is no interaction. However, even if there is interaction, steepest ascent ignoring the interaction still usually produces good results. To illustrate, suppose that we have fit the model̂y =
The region of experimentation for two factors are temperature (100 ≤ T ≤ 300∘F) and catalyst feed rate (10 ≤C ≤ 30 lb∕in). A first-order model in the usual ±1 coded variables has been fit to a molecular weight response, yielding the following model:̂y = 2000 + 125x1 + 40x2(a) Find the
The region of experimentation for three factors are time (40 ≤ T1 ≤ 80 min), temperature (200 ≤ T2 ≤ 300∘C), and pressure (20 ≤ P ≤ 50 psig). A first-order model in coded variables has been fit to yield data from a 23 design. The model iŝy = 30 + 5x1 + 2.5x2 + 3.5x3 Is the point T1
For the first-order model̂y = 60 + 1.5x1 − 0.8x2 + 2.0x3 find the path of steepest ascent. The variables are coded as−1 ≤ xi ≤ 1.
Verify that the following design is a simplex. Fit the first-order model and find the path of steepest ascent.x1 x2 x3 y 0√2 −1 18.5−√2 0 1 19.8 0 −√√ 2 −1 17.4 2 0 1 22.5
An industrial engineer has developed a computer simulation model of a two-item inventory system. The decision variables are the order quantity and the reorder point for each item. The response to be minimized is total inventory cost.The simulation model is used to produce the data shown in Table
A chemical plant produces oxygen by liquefying air and separating it into its component gases by fractional distillation. The purity of the oxygen is a function of the main condenser temperature and the pressure ratio between the upper and lower columns. Current operating conditions are temperature
Suppose that youwant to design an experiment for nine continuous two-level factors but you can only afford 12 runs.A co-worker suggests that you should choose 9 columns of the 11 columns from the 12-run Plackett–Burman design.(a) How many possible designs are there?(b) Are all of the designs from
Suppose that you need to conduct an experiment for four categorical factors. Factor A has five levels, factor B has four levels, and the other two factors each have three levels.You are interested in all main effects and the two-factor interactions.(a) How many runs are required?(b) How many runs
Construct a minimum-run D-optimal design for eight two-level factors. Design this experiment to estimate all main effects and as many two-factor interactions as possible. Compare this design to the 16-run no-confounding design for eight factors. Which design would you prefer? Why?
Suppose that you need to design an experiment for five factors. You are interested in the main effects of all factors, but only the AB and AC factor interactions.(a) How many runs are required for this experiment?(b) Construct a D-optimal design with the minimum number of runs. What are the
Suppose that you need to design an experiment for five factors. Three of these factors are categorical with three levels and the remaining two factors are continuous. You are interested in the main effects and the two-factor interactions of all factors, along with the pure quadratic effects of the
Suppose that you need to design an experiment for nine factors. Three of these factors are categorical with three levels and the remaining six are continuous. You are interested in the main effects and the two-factor interactions.(a) How many runs are required for this experiment?(b) Construct a
Suppose that you must design an experiment with six categorical factors. Factor A has six levels, factor B has five levels, factor C has five levels, factor D has three levels, and factors E and F have two levels. You are interested in main effects and two-factor interactions.(a) How many runs are
Suppose that you must design an experiment to investigate six continuous factors. It is thought that running all factors at two levels is adequate but that only the AB, AC, and AD two-factor interactions are of interest.(a) How many runs are required to estimate all of the effects that are of
Suppose that you must design an experiment to investigate seven continuous factors. Running all factors at two levels is thought to be appropriate but that only the two-factor interactions involving factor A are of interest.(a) How many runs are required to estimate all of the relevant effects?(b)
Suppose that you must design an experiment to investigate nine continuous factors. It is thought that running all factors at two levels is adequate but that all two-factor interactions are of interest.(a) How many runs are required to estimate all main effects and two-factor interactions?(b) Find a
Construct a minimum-run D-optimal resolution IV design for 12 factors. Find the alias relationships. What approach would you recommend for analyzing the data from this experiment?
Construct a minimum-run D-optimal resolution IV design for 10 factors. Find the alias relationships. What approach would you recommend for analyzing the data from this experiment?
An article in the Journal of Chemical Technology and Biotechnology (“A Study of Antifungal Antibiotic Production by Thermomonospora sp MTCC 3340 Using Full Factorial Design,” 2003, Vol. 78, pp. 605–610) investigated three independent variables—concentration of carbon source(glucose),
Reconsider the experiment in Problem 9.25. Suppose that you are only interested in main effects. Construct a design with N = 12 runs for this experiment.
Reconsider the experiment in Problem 9.25. Construct a design with N = 48 runs and compare it to the design you constructed in Problem 9.25.
Suppose there are four three-level categorical factor and a single two-level continuous factor. What is the minimum number of runs required to estimate all main effects and two-factor interactions? Construct this design.
Reconsider the experiment in Problem 9.23. Suppose that it was necessary to estimate all main effects and two-factor interactions, but the full factorial with 24 runs (not counting replication) was too expensive. Recommend an alternative design.
An article byW. D. Baten in the 1956 volume of Industrial Quality Control described an experiment to study the effect of three factors on the lengths of steel bars. Each bar was subjected to one of two heat treatment processes and was cut on one of four machines at one of three times during the day
In Problem 8.30, you met Harry Peterson-Nedry, a friend of the author who has a winery and vineyard in Newberg, Oregon. That problem described the application of two-level fractional factorial designs to their 1985 Pinot Noir product. In 1987, he wanted to conduct another Pinot Noir experiment. The
Starting with a 16-run 24 design, show how one three-level factor and three two-level factors can be accommodated and still allow the estimation of two-factor interactions.
Starting with a 16-run 24 design, show how two three-level factors can be incorporated in this experiment. How many two-level factors can be included if we want some information on two-factor interactions?
Outline the analysis of variance table for a 2232 factorial design. Discuss how this design may be confounded in blocks.
Construct a 4 × 23 design confounded in two blocks of 16 observations each. Outline the analysis of variance for this design.
Construct a 39−6 design and verify that it is a resolution III design.
Construct a 35−2 design with I = ABC and I = CDE.Write out the alias structure for this design. What is the resolution of this design?
Verify that the design in Problem 9.14 is a resolution IV design.
Construct a 34−1 IV design with I = ABCD. Write out the alias structure for this design.
From examining Figure 9.9, what type of design would remain if after completing the first nine runs, one of the three factors could be dropped?
Consider the data from replicate I of Problem 9.3. Suppose that only a one-third fraction of this design with I = ABC is run. Construct the design, determine the alias structure, and analyze the data.
Consider the data in Problem 9.3. If ABC is confounded in replicate I and ABC2 is confounded in replicate II, perform the analysis of variance.
Outline the analysis of variance table for the 34 design in nine blocks. Is this a practical design?
Consider the data from the first replicate of Problem 9.3. Assuming that not all 27 observations could be run on the same day, set up a design for conducting the experiment over three days with AB2C confounded with blocks. Analyze the data.
Confound a 34 design in three blocks using the AB2CD component of the four-factor interaction.
(a).Confound a 33 design in three blocks using the ABC2 component of the three-factor interaction. Compare your results with the design in Figure 9.7.(b) Confound a 33 design in three blocks using the AB2C component of the three-factor interaction. Compare your results with the design in Figure
An experiment is run in a chemical process using a 32 factorial design. The design factors are temperature and pressure, and the response variable is yield. The data that result from this experiment are as follows.Temper- Pressure, psig ature, ∘C 100 120 140 80 47.58, 48.77 64.97, 69.22 80.92,
Compute the I and J components of the two-factor interactions for Problem 9.4.
A medical researcher is studying the effect of lidocaine on the enzyme level in the heart muscle of beagle dogs. Three different commercial brands of lidocaine (A), three dosage levels(B), and three dogs (C) are used in the experiment, and two replicates of a 33 factorial design are run. The
An experiment was performed to study the effect of three different types of 32-ounce bottles (A) and three different shelf types (B)—smooth permanent shelves, end-aisle displays with grilled shelves, and beverage coolers—on the time it takes to stock ten 12-bottle cases on the shelves. Three
Compute the I and J components of the two-factor interaction in Problem 9.1.
The effects of developer strength (A) and development time (B) on the density of photographic plate film are being studied. Three strengths and three times are used, and four replicates of a 32 factorial experiment are run. The data from this experiment follow. Analyze the data using the standard
The aberration of a fractional factorial design is related to the length of the longest word in the defining relation.(a) True(b) False
In a 2k−3 design, the complete defining relation has 15 words.(a) True(b) False
In a 2k−2 design every effect has four aliases.(a) True(b) False
Consider a 24-1 fractional factorial design. If the principal fraction is run first—first block (I = ABCD)—and then later augmented with the alternate fraction—second block—the four-factor interaction effect is confounded with blocks.(a) True(b) False
It is good practice to keep the number of factor levels low and region of interest small in a screening experiment.(a) True(b) False
The design points of the 2k−p family are at the corners of a cube in a k-dimensional space and they project into a full factorial in any subset of the original k factors.(a) True(b) False
The 12-run Plackett-Burman for up to 11 factors design is a regular fraction.(a) True(b) False
For a half fraction of a two-level factorial design the maximum resolution possible is equal to the number of factors.(a) True(b) False
The resolution of a two-level fractional factorial design is the number of words in the defining relation.(a) True(b) False
Consider the fold over of the 25−2 fractional factorial constructed in Problem 8.69. Compare this design with the two one-half fractions of the 25−1. Is the fold-over design the same as either of the one-half fractions?
Consider the full 25 factorial design in Problem 6.51.Suppose that this experiment had been run as a 25−2 fractional factorial. Set up the fractional design using the principal fraction.Using the eight runs associated with this design from the original experiment, analyze the data and compare
Consider the full 25 factorial design in Problem 6.51.Suppose that this experiment had been run as a 25−1 fractional factorial. Set up the fractional design using the alternate fraction.Using the 16 runs associated with this design from the original experiment, analyze the data and compare your
Consider the full 25 factorial design in Problem 6.51.Suppose that this experiment had been run as a 25−1 fractional factorial. Set up the fractional design using the principal fraction. Using the 16 runs associated with this design from the original experiment, analyze the data and compare your
Construct a supersaturated design for h = 12 factors in N = 10 runs.8.66 How could an “optimal design” approach be used to augment a fractional factorial design to de-alias effects of potential interest?
Consider the 28−3 design in Problem 8.37. Suppose that the alias chain involving the AB interaction was large. Recommend a partial fold-over design to resolve the ambiguity about this interaction.
Construct a supersaturated design for k = 8 factors in P = 6 runs.
Reconsider the 24−1 design in Example 8.1. The significant factors are A, C, D, AC + BD, and AD + BC. Find a partial fold-over design that will allow the AC, BD, AD, and BC interactions to be estimated.
Consider a partial fold over for the 25−2 III design. Suppose that the partial fold over of this design is constructed using column A (+ signs only). Determine the alias relationships in the combined design.
Consider a partial fold over for the 27−4 III design. Suppose that the partial fold over of this design is constructed using column A (+ signs only). Determine the alias relationships in the combined design.
Consider a partial fold over for the 26−2 IV design. Suppose that the signs are reversed in column A, but the eight runs that are retained are the runs that have positive signs in column C. Determine the alias relationships in the combined design.
Consider the 27−3 IV design.(a) Suppose that a partial fold over of this design is run using column A (+ signs only). Determine the alias relationships in the combined design.(b) Rework part (a) using the negative signs to define the partial fold over. Does it make any difference which signs are
Reconsider the 27−3 IV design in Problem 8.56.(a) Suppose that a fold over of this design is run by changing the signs in column B. Determine the alias relationships in the combined design.(b) Compare the aliases from this combined design to those from the combined design from Problem 8.35. What
Consider the 27−3 IV design. Suppose that a fold over of this design is run by changing the signs in column A. Determine the alias relationships in the combined design.
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