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 varianceÏ2z . The response, cleansing efficiency, has a mean and variance that change behavior during the process. Use only significant terms in the following parts.
(a) Use the estimates in Figure 15.9 to develop the process mean efficiency 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? When B = â1?
Figure 15.9:
Solder Pot Temperature Flux Density 250 250 185 185 120 120 Low High Low High (-1) (+1) (-1) (+1) Figure 15.19: Plot showing the influence of factors on the mean response. Mean,y Mean,y
> Calculate the rank correlation coefficient for the daily rainfall and amount of particulate removed in Exercise 11.13 on page 400. Exercise 11.13: A study of the amount of rainfall and the quantity of air pollution removed produced the following data:
> With reference to the data of Exercise 11.1 on page 398, (a) calculate the rank correlation coefficient; (b) test the null hypothesis, at the 0.05 level of significance, that ρ = 0 against the alternative that ρ = 0. Compare your r
> The following table gives the recorded grades for 10 students on a midterm test and the final examination in a calculus course: (a) Calculate the rank correlation coefficient. (b) Test the null hypothesis that ρ = 0 against the alternative t
> What is the probability that at least 95% of a population will exceed the smallest value in a random sample of size n = 135?
> How large a sample is required to be 99% confident that at least 80% of the population will be less than the largest observation in the sample?
> Repeat Exercise 18.2 when 1 of the next 4 drinks overflows and the uniform prior distribution is π(p) = 10, 0.05 < p < 0.15. Exercise 18.2: Let us assume that the prior distribution for the proportion p of drinks from a vending machine that overflow is
> What is the probability that the range of a random sample of size 24 includes at least 90% of the population?
> For the situation of Review Exercise 17.8, suppose that additional data are collected as follows: Does the process appear to be in control? Explain. Exercise 17.8: Samples of size 50 are taken every hour from a process producing a certain type of item
> Samples of size 50 are taken every hour from a process producing a certain type of item that is considered either defective or not defective. Twenty samples are taken. (a) Construct a control chart for control of proportion defective. (b) Does the proces
> How large a sample is required to be 95% confident that at least 85% of the distribution of measurements is included between the sample extremes?
> Consider the situation of Example 17.2. The following data are taken on additional samples of size 5. Plot the ¯X - and S-values on the ¯X - and S-charts that were produced with the data in the preliminary sample. Does the process
> Consider an in-control process with mean μ = 25 and σ = 1.0. Suppose that subgroups of size 5 are used with control limits μ ± 3σ/ √ n, and centerline at μ. Suppose that a shift occurs in the mean, and the new mean is μ = 26.5. (a) What is the average nu
> Consider the data of Table 17.1. Suppose that additional samples of size 5 are taken and tensile strength recorded. The sampling produces the following results (in pounds per square inch). (a) Plot the data, using the ¯X - and R-charts for
> For the situation of Review Exercise 17.2, give numerical estimates of the mean and standard deviation of the diameter for the part being manufactured in the process. Exercise 17.2: Consider the following data taken on subgroups of size 5. The data cont
> Suppose for Review Exercise 17.2 that the buyer has set specifications for the part. The specifications require that the diameter fall in the range covered by 2.40000 ± 0.0100 mm. What proportion of units produced by this process will not c
> Consider the following data taken on subgroups of size 5. The data contain 20 averages and ranges on the diameter (in millimeters) of an important component part of an engine. Display ¯X- and R-charts. Does the process appear to be in control
> A cigarette manufacturer claims that the tar content of brand B cigarettes is lower than that of brand A cigarettes. To test this claim, the following determinations of tar content, in milligrams, were recorded: Use the rank-sum test with Î&plusm
> Rework Exercise 16.6 by using the signed-rank test. Exercise 16.6: Two types of instruments for measuring the amount of sulfur monoxide in the atmosphere are being compared in an air-pollution experiment. The following readings were recorded daily for a
> Rework Exercise 16.7 by using the signed-rank test. Exercise 16.7: The following figures give the systolic blood pressure of 16 joggers before and after an 8-kilometer run: Use the sign test at the 0.05 level of significance to test the null hypothesi
> The following are the numbers of prescriptions filled by two pharmacies over a 20-day period: Use the signed-rank test at the 0.01 level of significance to determine whether the two pharmacies, on average, fill the same number of prescriptions against
> Assuming that the measurements of Exercise 1.14 on page 30 were recorded successively from left to right as they were collected, use the runs test, with α = 0.05, to test the hypothesis that the data represent a random sequence. Exercise 1.14: A tire ma
> Rework Exercise 16.5 by using the signed-rank test. Exercise 16.5: It is claimed that a new diet will reduce a person’s weight by 4.5 kilograms, on average, in a period of 2 weeks. The weights of 10 women were recorded before and after
> A quality control effort is being undertaken for a process where large steel plates are manufactured and surface defects are of concern. The goal is to set up a quality control chart for the number of defects per plate. The data are given below. Set up t
> Consider X1,X2, . . . , Xn independent Poisson random variables with parameters μ1, μ2, . . . , μn. Use the properties of moment-generating functions to show that the random variable / is a Poisson random varia
> Analyze the data of Exercise 16.2 by using the signed-rank test. Exercise 16.2: The following data represent the number of hours of flight training received by 18 student pilots from a certain instructor prior to their first solo flight: Using binomia
> Analyze the data of Exercise 16.1 by using the signed-rank test. Exercise 16.1: The following data represent the time, in minutes, that a patient has to wait during 12 visits to a doctor’s office before being seen by the doctor: Use
> The following figures give the systolic blood pressure of 16 joggers before and after an 8-kilometer run: Use the sign test at the 0.05 level of significance to test the null hypothesis that jogging 8 kilometers increases the median systolic blood pres
> Two types of instruments for measuring the amount of sulfur monoxide in the atmosphere are being compared in an air-pollution experiment. The following readings were recorded daily for a period of 2 weeks: Using the normal approximation to the binomial
> It is claimed that a new diet will reduce a person’s weight by 4.5 kilograms, on average, in a period of 2 weeks. The weights of 10 women were recorded before and after a 2-week period during which they followed this diet, yielding the
> A paint supplier claims that a new additive will reduce the drying time of its acrylic paint. To test this claim, 12 panels of wood were painted, one-half of each panel with paint containing the regular additive and the other half with paint containing t
> A food inspector examined 16 jars of a certain brand of jam to determine the percent of foreign impurities. The following data were recorded: Using the normal approximation to the binomial distribution, perform a sign test at the 0.05 level of signific
> In an industrial production line, items are inspected periodically for defectives. The following is a sequence of defective items, D, and nondefective items, N, produced by this production line: D D N N N D N N D D N N N N N D D D N N D N N N N D N D Use
> The following data represent the number of hours of flight training received by 18 student pilots from a certain instructor prior to their first solo flight: Using binomial probabilities from Table A.1, perform a sign test at the 0.02 level of signific
> The weights of 5 people before they stopped smoking and 5 weeks after they stopped smoking, in kilograms, are as follows: Use the signed-rank test for paired observations to test the hypothesis, at the 0.05 level of significance, that giving up smoking
> The following data represent the time, in minutes, that a patient has to wait during 12 visits to a doctor’s office before being seen by the doctor: Use the sign test at the 0.05 level of significance to test the doctorâ€&#
> Consider Exercise 15.6. Use a +1 and −1 scaling for “high” and “low,” respectively, and do a multiple linear regression with the model Yi = β0 + Î
> Consider Exercise 15.3 once again. Three-factor interactions are often not significant, and even if they are, they are difficult to interpret. The interaction ABD appears to be important. To gain some sense of interpretation, show two AD interaction plot
> 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. Exercise 1
> It is important to study the effect 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 lev
> 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 differen
> A 22 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 oven
> A large petroleum company in the Southwest regularly conducts experiments to test additives to drilling fluids. Plastic viscosity is a rheological measure reflecting the thickness of the fluid. Various polymers are added to the fluid to increase viscosit
> Use the runs test to test, at level 0.01, whether there is a difference in the average operating time for the two calculators of Exercise 16.17 on page 670. Exercise 16.17: The following data represent the number of hours that two different types of sci
> 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 er
> A Plackett-Burman design was used to study the rheological properties of high-molecular-weight copolymers. Two levels of each of six variables were fixed in the experiment. The viscosity of the polymer is the response. The data were analyzed by the Stati
> Construct a Plackett-Burman design for 10 variables containing 24 experimental runs.
> Consider the 1/16 of the 27 factorial discussed in Section 15.9. List the additional 11 defining contrasts.
> A 25 factorial plan is used to build a regression model containing first-order coefficients 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 r
> Use the coal cleansing data of Exercise 15.2 on page 609 to fit a model of the type E(Y ) = β0 + β1x1 + β2x2 + β3x3, where the levels are x1, percent solids: 8, 12 x2, flow rate: 150, 250 gal/min x3, pH:
> Consider Case Study 15.1 involving the injection molding data. Suppose mold temperature is difficult to control and thus it can be assumed that in the process it follows a normal distribution with mean 0 and variance σ2z. Of concern is the var
> Consider the following 23 factorial with control variables x1 and x2 and noise variable z. Can x1 and x2 be chosen at levels for which Var(y) is minimized? Explain why or why not.
> 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 r
> A silver-plating process is used to coat a certain type of serving tray. When the process is in control, the thickness of the silver on the trays will vary randomly following a normal distribution with a mean of 0.02 millimeter and a standard deviation o
> In Myers, Montgomery, and Anderson-Cook (2009), an application is discussed in which an engineer is concerned with the effects on the cracking of a titanium alloy. The three factors are A, temperature; B, titanium content; and C, amount of grain refiner.
> In a metallurgy experiment, it is desired to test the effect 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 refinemen
> For Exercise 15.27, give all aliases for the six main effects. Exercise 15.27: There are six factors and only eight design points can be used. Construct a 26−3 by beginning with a 23 and use D = AB, E = −AC, and F = BC as the generators.
> Consider Exercise 15.27. Construct another 26−3 that is different from the design chosen in Exercise 15.27. Exercise 15.27: There are six factors and only eight design points can be used. Construct a 26−3 by beginning with a 23 and use D = AB, E = −AC,
> There are six factors and only eight design points can be used. Construct a 26−3 by beginning with a 23 and use D = AB, E = −AC, and F = BC as the generators.
> Consider a 25−1 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.
> 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 i
> 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 amp
> An experiment is conducted so that an engineer can gain insight into the influence 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
> 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 defining contrasts being ACD, BEF, and CEG. The data are as follows: Perform an analysis of variance on all
> A random sample of 15 adults living in a small town were selected to estimate the proportion of voters favoring a certain candidate for mayor. Each individual was also asked if he or she was a college graduate. By letting Y and N designate the responses
> (a) Using the defining contrasts ABCE and ABDF, obtain a 1/4 fraction of a 26 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-fac
> Construct a 1/4 fraction of a 26 factorial design using ABCD and BDEF as the defining contrasts. Show what effects are aliased with the six main effects.
> 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
> (a) Obtain a 1/2 fraction of a 24 factorial design using BCD as the defining 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
> List the aliases for the various effects in a 25 factorial experiment when the defining contrast is ACDE.
> Consider Figure 15.14, which represents a 22 factorial with 3 center runs. If quadratic curvature is significant, what additional design points would you select that might allow the estimation of the terms x21, x22? Explain. Figure 15.14: -1, 1 1,
> 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 t
> 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 disul
> 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 ho
> Consider a 25 experiment where the experimental runs are on 4 different machines. Use the machines as blocks, and assume that all main effects and two-factor interactions may be important. (a) Which runs would be made on each of the 4 machines? (b) Which
> In Exercise 13.6 on page 519, use the Kruskal-Wallis test at the 0.05 level of significance to determine if the organic chemical solvents differ significantly in sorption rate. Exercise 13.6: A study measured the sorption (either absorption or adsorptio
> Consider Exercise 15.11. Suppose there was some experimental difficulty 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 With only these runs, we have the signs fo
> 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 effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two level
> Consider Exercise 15.5. Compute all 15 effects and do normal probability plots of the effects. (a) Does it appear as if your assumption of negligible three- and four-factor interactions has merit? (b) Are the results of the effect plots consistent with w
> The following data are obtained from a 23 factorial experiment replicated three times. Evaluate the sums of squares for all factorial effects by the contrast method. Draw conclusions. Treatment Combination Rep 1 Rep 2 Rep 3 (1) 12 19 10 15 24 20 16
> An engineer is interested in the effects of cutting speed and tool geometry on the life in hours of a machine tool. Two cutting speeds and two different geometries are used. Three experimental tests are accomplished at each of the four combinations. The
> In Myers, Montgomery, and Anderson-Cook (2009), a scenario is discussed involving an auto bumper plating process. The response is the thickness of the material. Factors that may impact the thickness include amount of nickel (A) and pH (B). A two-factor e
> The extraction rate of a certain polymer is known to depend on the reaction temperature and the amount of catalyst used. An experiment was conducted at four levels of temperature and five levels of the catalyst, and the extraction rate was recorded in th
> An experiment was conducted to determine whether additives increase the adhesiveness of rubber products. Sixteen products were made with the new additive and another 16 without the new additive. The observed adhesiveness was as recorded below. Perform
> To determine which muscles need to be subjected to a conditioning program in order to improve one’s performance on the flat serve used in tennis, a study was conducted by the Department of Health, Physical Education and Recreation at Vi
> An experiment was designed for personnel in the Department of Animal Science at Virginia Tech to study urea and aqueous ammonia treatment of wheat straw. The purpose was to improve nutritional value for male sheep. The diet treatments were control, urea
> The following data represent the operating times in hours for three types of scientific pocket calculators before a recharge is required: Use the Kruskal-Wallis test, at the 0.01 level of significance, to test the hypothesis that the operating times fo
> Four laboratories are being used to perform chemical analysis. Samples of the same material are sent to the laboratories for analysis as part of a study to determine whether or not they give, on the average, the same results. The analytical results for t
> Three catalysts are used in a chemical process; a control (no catalyst) is also included. The following are yield data from the process: Use Dunnett’s test at the α = 0.01 level of significance to determine if a signifi
> Consider the aggregate data of Example 13.1. Perform Bartlett’s test, at level α = 0.1, to determine if there is heterogeneity of variance among the aggregates.
> To ascertain the number of tellers needed during peak hours of operation, data were collected by an urban bank. Four tellers were studied during three “busy” times: (1) weekdays between 10:00 and 11:00 A.M., (2) weekda
> The manufacturer of a certain brand of freeze-dried coffee hopes to shorten the process time without jeopardizing the integrity of the product. The process engineer wants to use 3 temperatures for the drying chamber and 4 drying times. The current drying
> In the book SN-Ratio for the Quality Evaluation, published by the Japanese Standards Association (1988), a study on how tire air pressure affects the maneuverability of an automobile was described. Three different tire air pressures were compared on thre
> Corrosion fatigue in metals has been defined as the simultaneous action of cyclic stress and chemical attack on a metal structure. A widely used technique for minimizing corrosion fatigue damage in aluminum involves the application of a protective coatin
> Exercise 14.25 on page 588 describes an experiment involving the extraction of polyethylene through use of a solvent. (a) Do a different sort of analysis on the data. Fit an appropriate regression model with a solvent categorical variable, a temperatur
> A control valve needs to be very sensitive to the input voltage, thus generating a good output voltage. An engineer turns the control bolts to change the input voltage. The book SN-Ratio for the Quality Evaluation, published by the Japanese Standards Ass
> In the experiment of Review Exercise 14.33, cake volume was also used as a response. The units are cubic inches. Test for interaction between factors and discuss main effects. Assume that both factors are fixed effects. Exercise 14.33: The Statistics C
> The following data represent the weights, in kilograms, of personal luggage carried on various flights by a member of a baseball team and a member of a basketball team. Use the rank-sum test with α = 0.05 to test the null hypothesis that
> Let us assume that the prior distribution for the proportion p of drinks from a vending machine that overflow is p 0.05 0.10 0.15 π(p) 0.3 0.5 0.2 If 2 of the next 9 drinks from this machine overflow, find (a) the posterior distribution for the proportio
> Personnel in the Materials Engineering Department at Virginia Tech conducted an experiment to study the effects of environmental factors on the stability of a certain type of copper-nickel alloy. The basic response was the fatigue life of the material. T
> A study was made to determine if humidity conditions have an effect on the force required to pull apart pieces of glued plastic. Three types of plastic were tested using 4 different levels of humidity. The results, in kilograms, are as follows: (a) Ass