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Design And Analysis Of Experiments 6th International Edition Douglas C. Montgomery - Solutions
Table 11.12 shows the results of Chau and Kelly's (1993) mixture process variable experiments (MPV) with printable coating material, which were described in Section 11.7.(a) Create this design and merge the response values.(b) Fit a simpli ed model of the form of Equation (11.31) with only one
The numbers in parentheses below are the responses for the design shown in Figure 11.19.(a) Fit the Schee quadratic model to the data above.(b) Verify that the design blocks orthogonally by checking to see that the type I and type III sums of squares for blocks are the same. (42) Block 1 x=1 (77)
If all three components in a mixture design have the same lower and upper constraints (L=0.20, U=0.50), use Table 11.5 to create an orthogonally blocked mixture design, for tting the Sche e quadratic model, with two blocks of four mixtures in each block.
Shumate and Montgomery (1996) developed a TiW plasma etch process for semiconductor manufacturing by studying a mixture of three gases x1: SR6, x2: He, and x3: N2. The partial pressures of the three gases were forced to add to a constant value of total pressure to form the mixture constraint.The
Consider the mixture experiment described by Anderson and McLean(1974). They presented a problem where the formula for a are was ob-tained by mixing four chemicals x1: magnesium, x2: sodium nitrate, x3:strontium nitrate, and x4: binder. Constraints on the components were(a) Given the data from the
Constraints on the mixture components are shown below.(a) Use the Xvert function in the mixexp package to nd the extreme ver-tices of the experimental region, and check to see if the ones listed in Table 11.9 form an extreme vertices design.(b) Given the data from Anik and Sukumar (1981)
Consider the problem discussed by Anik and Sukumar (1981). They studied the solubility of a butoconazole nitrate imidazole antifungal agent in a mixture of x1: polyethylene glycol 400, x2: glycerin, x3: polysorbate 60, x4:water, and x5: poloxamer
In agricultural eld tests, two or more herbicides are often mixed together in so-called tank mixes in order to nd a mixture that is more eective than individual herbicides in controlling a multitude of pest weeds. In a specic test, various mixtures of x1: a herbicide formulated to control broad
Cornell (2002) describes an experiment to make a fruit punch composed of three types of juice: x1: watermellon, x2: pineapple, and x3: orange. The general acceptance of the fruit punch was to be determined by a taste panel that would rate them on a 1 to 9 scale where 1=dislike extremely, 5=neither
Belloto et al. (1985) studied the relation between y= Soluability of pheno-barbital and mixture components x1: ethanol, x2: propylene gycol and x3:water.(a) Use the mixexp package to list the experiments required to t a linear Sche e model.(b) Use the mixexp package to list the experiments
A mixture experiment was run in three components: x1, x2, and x3.(a) If the Sche e model that best represented the data was y = 13:5x1 +3:2x2 + 9:8x3, then what is the predicted response at the pure blend x2 = 1:0? What is the predicted response at the 50-50 blend of x1 and x2?(b) If the Sche e
Use the EEw1s2 function in the daewr package to create a estimation-equivalent split-plot design.(a) Call the function with no arguments to see the list of designs available.(b) Create the design for one whole plot factor and two sub-plot factors, with seven whole plots and two sub-plots per whole
Follow Table 10.12 and use the R programming steps to create an EESPRS split-plot central composite design with one whole-plot factor and three sub-plot factors.
Fit the general quadratic model to the ceramic pipe data in Table 10.13.(a) Fit the model with the lmer function including the random whole plot e ects.(b) Fit the model with the rsm function ignoring the random whole plot e ects.(c) Verify that the regression coecients are the same for both
Find the optimal moist cake for the data in Table 10.10.(a) Do the canonical analysis and make a contour plot of the moisture read-ing, y, using the least squares coecients shown on the left side of Table 10.11.(b) Do the canonical analysis and make a contour plot of the moisture read-ing, y,
A study was conducted to nd the optimum ratio of chloride and alka-line reserves (AR) in a product that was designed to add alkalinity to livestock rations. A small composite design was planned with the factors x1 = 1~ºchloride and x2 = ln(AR), but one combination of x1 and x2 had to be dropped
Fit the general quadratic model to the data from the pastry dough experi-ment given in Table 10.9.(a) Use the rsm function to t the model including block e ects and check the adequacy of the model.(b) Use canonical analysis or ridge analysis to determine the maximum re-sponse within the cubical
Create a blocked response surface design for four factors.(a) Use the rsm package to create an orthogonally blocked central composite design. How many runs are in this design, and how many runs are in each block?(b) Use the rsm package to create an orthogonally blocked Box-Behnken design. How many
Consider the model for the chemical reaction in Section 10.7.4.(a) Use the data shown on the next page that was obtained by running 20 experiments to t model 10.13.(b) Make a contour plot of predicted concentration over the experimental region.(c) Use numerical methods to nd the time and
Consider the data in Table 10.1.(a) Fit the general quadratic model using the rsm function as shown in Section 10.6.1.(b) Use canonical analysis or ridge analysis to determine the conditions that result in the maximum workability of the cement grout within the spher-ical region −1:73 B xi B 1:73
Consider the nonlinear model y = f(x1; x2) = 1e 2x1 − 3e 4x2 .(a) Linearize the model by taking the partial derivatives @f@ i; i = 1; 4.(b) Construct a grid of candidate points in the experimental region range 1 B x1 B 7; 1 B x2 B 7.(c) Use the OptFederov function to construct a D-optimal
Consider the experimental region shown in Figure 10.9, where −1 < x1 < 1,−1 < x2 < 1, and x2 C −2x1 − 2; x2 B −2x1 + 1.(a) Construct a grid ±1 2 of candidate points in the (x1; x2) design region.(b) Construct a D-optimal subset of the candidate points for tting the general quadratic
The three factors that can be varied on the catapult are the start angle, stop angle, and pivot height.Each factor has three possible levels. A full 33 factorial would allow estima-tion of all coecients in the general quadratic model, but it would require 27 experiments with no replicates to check
Consider conducting a response surface experiment with the wooden cat-apult described in exercise 2 of Chapter
Consider performing a response surface experiment using the paper heli-copters described in exercise 1 of Chapter 2 and Sections 3.2 and 3.3.(a) Create a response surface design of your choice by varying the wing length between 3.5 inches and 7 inches and the width of the helicopters from 2.25
Following up on the experiments described in Section 6.4, AlmeidaeSilva et al. (2003) ran a response surface design to nd the optimum nutrient levels and fermentation time for producing biomass by fermentation of eu-calyptus hemicellulosic hydrolyzate. The data is shown in the table on the next
Compare characteristics of standard response surface designs for four fac-tors.(a) Create a uniform precision central composite design, a Box-Behnken design, a small composite design, and a hybrid design.(b) Compare the variance dispersion graphs or fraction of design space plots for each design,
Create a central composite design for two factors using the rsm package.(a) Create the uniform precision CCD and store the design along with ran-dom numbers (simulated response) in a data frame.(b) Use the Vdgraph package to make a variance dispersion graph or a frac-tion of design space plot of
Consider the results of the repeated measurements experiment from Us-age note: Stat-40 (1997) from the University of Texas at Austin. In this experiment the purpose was to determine how diet and exercise a ect the pulse of subjects. Eighteen subjects were randomly assigned to one of three exercises
Consider the data from the repeated measures design shown in Table 9.10.(a) Fit a multivariate model using the lm function and then use the Anova function in the car package to compute the sphericity test (as shown in Section 9.6.1) and determine if the Huynh-Feldt condition holds.(b) From the
Consider planning an experiment to compare the taste of chocolate brownie recipes. The treatments will consist of three brands of boxed brownie mixes and one from-scratch home recipe. A panel of tasters will be used to eval-uate the brownies. Each taster will taste each brownie and give it a rating
Lucas (1957) presented data from an extra-period crossover study on dairy cows. Three diet treatments were being compared and the response was the average daily production of FCM. The data is shown in Table 9.11.(a) Does every treatment follow every other treatment, including itself?(b) Analyze the
Consider the data from a study reported by Varma and Chilton (1974), that was later reanalyzed by Jones (1980).This was a simple crossover dental study comparing a test compound to a placebo with regard to their eect on dental hygiene as measured by the change in a dental hygiene index. There were
Bisgaard and Kulahci (2001) describe an experiment conducted by a man-ufacturer of boxed cake mixes. The manufacturer wanted to nd a way to make the recipe more robust so that if customers bake at a slightly dierent temperature or time from that recommended on the box, the cake would still taste
Consider a 16-run fractional factorial split-plot design with three whole-plot factors A, B, and C, and four sub-plot factors P, Q, R, and S.(a) What degree of fractionation would you use in the whole plots and why?(b) Based on your answer to (a) how many whole plots would be required?(c) How many
Using the FrF2 function, create a minimum aberration split-plot fractional factorial design with four whole-plot factors A, B, C, and D, and four sub-plot factors P, Q, R, and S utilizing a one-half fraction in the whole plots and a one-quarter fraction in the sub-plots.(a) What are the generators
Czitrom and Spagon (1997) describe a split-plot experiment conducted in a semiconductor manufacturing facility. Oxidation of silicon in a well-controlled environment is a critical step in fabrication of modern integrated circuits. A 9-nanometer thick oxide layer is grown on 200-mm silicon wafers in
Ramirez and Tobias (2007) present the data shown below from an experi-ment to compare performance, based on a quality characteristic, of a new and old lm type and three manufacturing pressures. One lm type wasselected at random and it was run through the manufacturing process ran-domly assigning
Modify the R code using the FrF2 function in Section 8.3.2 (for creating the design for the sausage-casing experiment with two whole-plot factors, two blocks of whole plots, and two split-plot factors) to create a design with two whole-plot factors, two blocks of whole plots, and three split-plot
Kuehl (2000) reports the results of an experiment conducted at a large seafood company to investigate the eect of storage temperature and type of seafood upon bacterial growth on oysters and mussels. Three storage temperatures were studied (0XC, 5XC, and 10XC). Three cold storage units were
Modify the R code in Section 8.2.1 to create(a) A randomized list for a split-plot experiment with completely random-ized whole plots where there is one whole-plot factor A with 3 levels and two replicate whole plots for each level, and one split-plot fac-tor B with three levels. Label the levels
Why wouldn't you treat the diameter of each cookie as the response in the cookie-baking experiment described in Section 8.2? What would you call the individual measurements on each cookie on each tray?
Consider the apple slice browning experiment shown in Section 7.9.(a) If the apple slicer made eight slices of each apple instead of six, nd a partially confounded 4 × 3 factorial in three blocks of eight (instead of 4 blocks of 6) by modifying the code that used the optBlock function in that
Using the optBlock in the AlgDesign package create a partially confounded 2 × 32 factorial in six blocks of size 6.
Generate random data, and use R lm function to illustrate that all main e ects and interactions can be estimated from the design you created.
Create a partially confounded 23 design in 14 blocks of size
Show why it is impossible to block a 3×4×6 into blocks of size 6 using the classical method without confounding a main e ect.
Consider blocking a 2 × 32 × 4 factorial into six blocks of 12 treatment combinations each.(a) Naming the factors A, B, C, and D, which factors must be represented by pseudo factors in order to block using the classical method?(b) What are the two sub-experiments composed of factorials with prime
For a 26−1 experiment with generator F = ABCDE that is blocked into four blocks with eight experimental units per block using the block de ning contrasts ACD and BCD,(a) Determine the de ning relation for the fractional factorial.(b) Determine the complete alias structure for the fractional
Choice based conjoint studies (see Section 6.7 for an example and de-scription of conjoint analysis) are frequently used in marketing research.In choice based conjoint studies, potential customers are asked to choose among several alternatives. The alternatives are combinations of levels of several
If the block de ning contrasts are ACD and AB (shown in Section 7.6.1), show all interactions and their aliases that will be confounded with blocks.
Consider blocking a 26−2 fractional factorial, with generators E = ABC, and F = ABD, into 4 blocks of size
Consider blocking a 25 factorial into eight blocks with four experimental units per block using the block de ning contrasts ACE, BCE, and ABCD(shown in Section 7.6.2). Show that the generalized interactions of these contrasts, which would be confounded with blocks, contain no main e ects.
Consider an incomplete block design with t=10 levels of the treatment factor.(a) If a balanced incomplete block design (BIB) was constructed using the simplest method of forming blocks as all possible combinations of 10 k , how many blocks would be required for the cases where k=3, 4, 5, or
An experimenter wanted to determine how the gear selected on a 10 speed bike would a ect the time it takes a rider to pedal up a long, gradual hill.The experimental unit will be the trial or point in time when a rider will ride up the hill. Treatment will be the gear selected on the bicycle. Since
Kamohara et al. (2001) used an 18-run orthogonal fractional factorial to investigate the eect of 8 factors (37×2) upon the Vt variance of MOSFETs(metaloxidesemiconductor eld-eect transistor). The list of factors and number of levels is shown below. The list of experiments and the response,S/N, is
Curtis et al. (1982) proposed to conduct an experiment to study the eect of four factors in the software development process upon the software life cycle cost and the number of residual errors. The factors and levels are shown in the table below:(a) A 24−1 fractional factorial design was proposed
Consider the data from the chemical reaction experiment from Box and Meyer (1993) shown in Table 6.16 on the next page. The rst ve columns dened the factor settings for the experiment, and the remaining unassigned columns represent aliased interactions.(a) Use the R lm function to calculate the
Use the pb function in the package FrF2 to create a Plackett-Burman design to study 15 factors in 20 runs.
Prince (2007) performed a 25−2 fractional factorial in the process of op-timizing a clinical assay to detect Streptococcus pyogenes with real-time PCR. Optimization of this process would allow hospitals to detect Strep infections in less than 30 minutes with 99% accuracy. The factors he stud-ied
Olsen et al. (2014) performed a 16-run 210−6 III fractional factorial in 10 factors labeled A, B, C, D, E, F, G, H, J, and K. The purpose was to identify factors in the synthesis of an Al-modied anatase catalyst support that aect the nal properties. The generators for the design were E = ABCD, F
Consider the experiment to study factors that a ect the percent removal of arsenic from drinking water using a iron coated sand lter.(a) Modify the R code in Section 6.5.1 to produce the design shown in Table 6.8.(b) Determine the de ning relation for the experiments in the rst block and show the
Create a 27−3 IV design using FrF2 with generators E = ABC, F = ABD, G = ACD.(a) Determine the de ning relation for this design.(b) Determine the confounded strings of two-factor interactions that can be estimated from this design.(c) Suppose that after the analysis of data from this experiment,
Erhardt (2007) ran a 28−4 fractional factorial experiment to determine the factors that aect ight time of paper helicopters. The standard design is shown in Figure 6.19 below with all dimensions in centimeters. The factorsand levels (in centimeters) he studied are shown in the table below. The
Use FrF2 to create a quarter fraction 25−2 design.(a) Use the design.info function to examine the aliases and determine the generators used by FrF2 to create the design.(b) Determine the de ning relation.(c) Verify that the alias structure for this design can be obtained from the de ning relation
Melo et al. (2007) used a 24−1 factional factorial design with generator D = ABC to study the factors that in uence the production of levan by aerobic fermentation using the yeast Zymomonas mobilis. Levan is a sugar polymer of the fructan group, which has been shown to have anti-tumor activity
Consider a 23−1 factional factorial design.(a) Modify the FrF2 function call in Section 6.2 to create a list of the runs in the design.(b) Determine the de ning relation.(c) Determine the complete alias pattern.(d) Examine the alias structure using FrF2.
Reanalyze the data from the golf experiment, presented in the Appendix of Chapter 4, using the lmer function. Check to see if you get the same P-values and conclusions shown in Section 4.7.
Consider the data in Table 5.20 from Smith and Beverly (1981) taken from a staggered nested design to investigate the sources of variability in impurities in raw materials received at a plant in trailer loads. Two samples of material were taken from each of nine trailer loads of pellets. Two
Consider the data (Anderson and McLean, 1974) in the following table to come from a sampling study to determine the source of variability in the average amount spent on health care per year in thousands of dollars.Towns are nested within state and households are nested within town.(a) Use the
Consider the data in Table 5.9.(a) Enter the data into R and estimate the variance components 2a, 2 b , 2 c , and 2 using the method of moments and REML.(b) What are the major source or sources of variability?(c) Is there any di erence between the method of moments or REML esti-mators of the
Plan a sampling experiment to partition the source of variability in auto gas mileage in your community between drivers, cars, and replicates.(a) Describe how you would plan the study if drivers and cars were crossed factors.(b) Describe how you would plan the study if drivers were nested within
Using the mean squares and expected mean squares from the gage R&R study presented in Section 5.5, use the macro in the appendix of this chapter to compute a 90% con dence interval on the variance component for the interaction of operator and part.
Read the data from Table 5.2 into R and(a) Calculate the ANOVA sums of squares and mean squares used in calcu-lating the con dence limits in Section 5.4.2.(b) Calculate and compare the method of moments and REML estimates of the variance components.
Carry out the sampling experiment described in Section 5.3 to partition the variability in paper helicopter ight times.(a) Read your data into R and estimate the variance components using the method of moments.(b) Using the lmer function, estimate the variance components using the REML method.(c)
Consider the situation described in Exercise 6 of Chapter 2 with t = 4 levels of the treatment factor, and = 2:0.(a) If by blocking the experimental units into blocks it was believed that the variance of the experimental error, 2, could be reduced by 50%, calculate the number of blocks that
Woodward (1970) conducted an experiment to determine the fastest path to second base in baseball. The three paths investigated were the round-out, narrow angle, and wide angle shown in the gure below. The best path is dened to be the one that minimizes the time to reach second base.He used a
le Riche and Csima (1964) evaluated four hypnotic drugs and a placebo to determine their eect on the quality of sleep in elderly patients. The treat-ment levels were labeled (A=Placebo, B=Ethchlorvynol, C=Glutethimide, D=Chloral hydrate, and E=Secobarbitol sodium). Elderly patients were given one
Lew (2007) presents the data from an experiment to determine whether cultured cells respond to two drugs. The experiment was conducted using a stable cell line plated onto Petri dishes, with each experimental run in-volving assays of responses in three Petri dishes: one treated with drug 1, one
Horiuchi et al. (2005) conducted an experiment during Japan's 2004 Upper House election. The purpose of the experiment was to test the hypothesis that voter turnout is in uenced by the amount of information available to voters. Potential voters were contacted by e-mail and asked to ll out a
The data in Table 4.6 show the yields of ve varieties of barley in a ran-domized complete block experiment carried out in Minnesota reported in The Journal of the American Society of Agronomy by Immer et al. (1934).(a) What would be the purpose of running these experiments in di erent locations and
Consider room temperature as an additional blocking factor for the cut-ower experiment described in Section 4.2.(a) What design could you use to incorporate the two blocking factors of ower type and room temperature?(b) Set up a randomization plan to conduct this experiment.(c) What is the model
Consider planning an experiment to determine how the avor duration of gum is a ected by the characteristics of the gum such as Factor A (type of gum stick or tablet), Factor B ( avor of gum), and Factor C (regular or sugar-free), see Rogness and Richardson (2003).(a) If several subjects were
Nyberg (1999) has shown that silicon nitride (SiNx) grown by Plasma En-hanced Chemical Vapor Deposition (PECVD) is a promising candidate for an antire ection coating (ARC) on commercial crystalline silicon solar cells.Silicon nitride was grown on polished (100)-oriented 4A silicon wafers using a
Consider the data in Table 3.7 with the third observation (90) replaced by an atypical value (78) for those factor settings.(a) Calculate the e ects for this data.(b) Make a normal or half-normal plot of the e ects.(c) What e ects do you judge to be signi cant?(d) Run the Gaptest function for
Kenett and Steinberg (1987) described a two-level factorial experiment con-ducted by students to study the time required to boil 1 qt of water. Factors were A= ame level (low or high), B=pan size (small or large), C=pan cover(none or glass cover), and D=salt added to water (no or yes).(a) If the
In a continuation of the experiments to investigate the eect of plant growth regulators and spear bud scales on spear elongation in asparagus, described in exercise 5 of Chapter 2, Yang-Gyu and Woolley (2006) conducted a 4×3 factorial experiment varying the concentration of the plant growth
Modify the R code on page 72 to verify that the power would be 0.80 for detecting di erences in marginal means of = 1:0 inch in the paper helicopter experiments with four levels of wing length, four levels of body width, and r = 2 replicates per cell (as shown in Section 3.5.2).
Consider performing experiments to determine the e ect of popcorn brand, power level, and time on the percentage of edible popcorn (unpopped or burnt is not edible) kernels made in a microwave oven. The object is to maximize the proportion of edible kernels. Start with 1 3 cup of kernels and do
In an experiment to maximize the Y = resolution of a peak on a gas chro-matograph, a signicant interaction between A = column temperature and C = gas ow rate was found. The table below shows the mean resolution in each combination of column temperature and gas ow rate.(a) Construct an interaction
A wooden catapult can be used to ip a foam ball. The catapult has three factors that can be adjusted: the start angle, the stop angle, and the pivot height. The distance the ball travels can be measured with a tape measure.(a) If experiments were to be conducted with the catapult by ipping the ball
A consultant was called to assist the police department of a large metropoli-tan city in evaluating its human relations course for new ocers. He planned a two-factor factorial experiment where the treatments were A|the type beat to which ocers were assigned, and B|the length of the human rela-tions
Consider an experimental situation where the investigator was interested in detecting a maximum di erence in treatment means that is twice the stan-dard deviation of the response measured on replicate experimental units assigned to the same level of the treatment factor, that is, = 2:0. If there
The eect of plant growth regulators and spear bud scales on spear elon-gation in asparagus was investigated by Yang-Gyu and Woolley (2006).Elongation rate of spears is an important factor determining nal yield of asparagus in many temperate climatic conditions. Spears were harvested from 6-year-old
In an experiment to study the eect of the amount of baking powder in a biscuit dough upon the rise heights of the biscuits, four levels of baking powder were tested and four replicate biscuits were made with each level in a random order. The results are shown in the table below.(a) What is the
In Section 2.8.2 an experiment for determining the e ect of the download site selected upon the time to download a le was discussed. In this exper-iment:(a) Describe the experimental unit.(b) Describe the treatment factor.(c) Describe the response.(d) Discuss the causes for experimental error in
Paper helicopters can be cut from one half of an 81 2 × 11 sheet of paper as shown below.These helicopters can be made quickly and inexpensively, and can be used to demonstrate experimental design concepts. An experiment can be per-formed by constructing a helicopter, dropping it from a xed
Consider the simple experiment described in Section 2.1 and used through-out this chapter to determine the e ect of rise time on the height of bread dough. Describe the steps in the planning process for this experiment sim-ilar to what was done for this historical example in Section 2.3. Describe
Kerry and Bland (1998) describe the analysis of cluster randomized studies where a group of subjects are randomized to the same treatment. For ex-ample, when women in some randomly selected districts are o ered breast cancer screening while women in other districts are not o ered the screen-ing, or
Describe a situation within your realm of experience (your work, your hobby, or school) where you might like to predict the result of some future action. Explain how an experimental design, rather than an observational study, might enhance your ability to make this prediction.
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