Question: From a mathematics class of 12 equally

> 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

> An experiment was conducted in the Department of Food Science at Virginia Tech. It was of interest to characterize the texture of certain types of fish in the herring family. The effect of sauce types used in preparing the fish was also studied. The resp

> The Statistics Consulting Center at Virginia Tech was involved in analyzing a set of data taken by personnel in the Human Nutrition and Foods Department in which it was of interest to study the effects of flour type and percent sweetener on certain physi

> A process engineer wants to determine if the power setting on the machines used to fill certain types of cereal boxes results in a significant effect on the actual weight of the product. The study consists of 3 randomly chosen types of cereal manufacture

> A manufacturer of latex house paint (brand A) would like to show that its paint is more robust to the material being painted than that of its two closest competitors. The response is the time, in years, until chipping occurs. The study involves the three

> A plant manager would like to show that the yield of a woven fabric in the plant does not depend on machine operator or time of day and is consistently high. Four randomly selected operators and 3 randomly selected hours of the day are chosen for the stu

> Three strains of rats were studied under 2 environmental conditions for their performance in a maze test. The error scores for the 48 rats were recorded. Use a 0.01 level of significance to test the hypothesis that (a) there is no difference in error s

> Consider the following analysis of variance for a random effects experiment: Test for significant variance components among all main effects and interaction effects at the 0.01 level of significance (a) by using a pooled estimate of error when appropri

> A defense contractor is interested in studying an inspection process to detect failure or fatigue of transformer parts. Three levels of inspections are used by three randomly chosen inspectors. Five lots are used for each combination in the study. The fa

> To estimate the various components of variability in a filtration process, the percent of material lost in the mother liquor is measured for 12 experimental conditions, with 3 runs on each condition. Three filters and 4 operators are selected at random f

> Assuming a random effects experiment for Exercise 14.2 on page 575, estimate the variance components for brand of orange juice concentrate, for number of days from when orange juice was blended until it was tested, and for experimental error. Exercise 1

> In the book Design of Experiments for Quality Improvement, published by the Japanese Standards Association (1989), a study is reported on the extraction of polyethylene by using a solvent and how the amount of gel (proportion) is influenced by three fact

> A scientist collects experimental data on the radius of a propellant grain, y, as a function of powder temperature, extrusion rate, and die temperature. Resources are not available to make repeated experimental trials at the eight combinations of facto

> Consider combinations of three factors in the removal of dirt from standard loads of laundry. The first factor is the brand of the detergent, X, Y , or Z. The second factor is the type of detergent, liquid or powder. The third factor is the temperature o

> Consider the data set in Exercise 14.21. (a) Construct an interaction plot for any two-factor interaction that is significant. (b) Do a normal probability plot of residuals and comment. Exercise 14.21: Electronic copiers make copies by gluing black ink

> Electronic copiers make copies by gluing black ink on paper, using static electricity. Heating and gluing the ink on the paper comprise the final stage of the copying process. The gluing power during this final process determines the quality of the copy.

> For a study of the hardness of gold dental fillings, five randomly chosen dentists were assigned combinations of three methods of condensation and two types of gold. The hardness was measured. (See Hoaglin, Mosteller, and Tukey, 1991.) Let the dentists p

> To ascertain the stability of vitamin C in reconstituted frozen orange juice concentrate stored in a refrigerator for a period of up to one week, the study Vitamin C Retention in Reconstituted Frozen Orange Juice was conducted by the Department of Human

> Corrosion fatigue in metals has been defined as the simultaneous action of cyclic stress and chemical attack on a metal structure. In the study Effect of Humidity and Several Surface Coatings on the Fatigue Life of 2024-T351 Aluminum Alloy, conducted by

> A fishing line is being manufactured by two processes. To determine if there is a difference in the mean breaking strength of the lines, 10 pieces manufactured by each process are selected and then tested for breaking strength. The results are as follows

> The method of X-ray fluorescence is an important analytical tool for determining the concentration of material in solid missile propellants. In the paper An X-ray Fluorescence Method for Analyzing Polybutadiene Acrylic Acid (PBAA) Propellants (Quarterly

> The following data are measurements from an experiment conducted using three factors A, B, and C, all fixed effects: (a) Perform tests of significance on all interactions at the α = 0.05 level. (b) Perform tests of significance on the mai

> Consider an experimental situation involving factors A, B, and C, where we assume a three-way fixed effects model of the form yijkl = μ + αi + βj + γk+(βγ)jk+ €i

> The purpose of the study The Incorporation of a Chelating Agent into a Flame Retardant Finish of a Cotton Flannelette and the Evaluation of Selected Fabric Properties, conducted at Virginia Tech, was to evaluate the use of a chelating agent as part of th

> Consider the data set in Exercise 14.12 and answer the following questions. (a) Both factors, copper and time, are quantitative in nature. As a result, a regression model may be of interest. Describe what might be an appropriate model using x1 = copper c

> In Myers, Classical and Modern Regression with Applications (Duxbury Classic Series, 2nd edition, 1990), an experiment is described in which the Environmental Protection Agency seeks to determine the effect of two water treatment methods on magnesium upt

> In an experiment conducted in the Civil Engineering Department at Virginia Tech, growth of a certain type of algae in water was observed as a function of time and the dosage of copper added to the water. The data are as follows. Response is in units of a

> A study was done to determine the impact of two factors, method of analysis and the laboratory doing the analysis, on the level of sulfur content in coal. Twenty-eight coal specimens were randomly assigned to 14 factor combinations, the structure of the

> Two factors in a manufacturing process for an integrated circuit are studied in a two-factor experiment. The purpose of the experiment is to learn their effect on the resistivity of the wafer. The factors are implant dose (2 levels) and furnace position

> An experiment was conducted to study the effects of temperature and type of oven on the life of a particular component. Four types of ovens and 3 temperature levels were used in the experiment. Twenty-four pieces were assigned randomly, two to each combi

> The following data represent the number of hours that two different types of scientific pocket calculators operate before a recharge is required. Use the rank-sum test with α = 0.01 to determine if calculator A operates longer than calcul

> Use Bartlett’s test at the 0.01 level of significance to test for homogeneity of variances in Exercise 13.5 on page 519. Exercise 13.5:

> For the data set in Exercise 13.7, use Bartlett’s test to check whether the variances are equal. Use α = 0.05. Exercise 13.7: It has been shown that the fertilizer magnesium ammonium phosphate, MgNH4PO4, is an effective s

> It has been shown that the fertilizer magnesium ammonium phosphate, MgNH4PO4, is an effective supplier of the nutrients necessary for plant growth. The compounds supplied by this fertilizer are highly soluble in water, allowing the fertilizer to be appli

> A study measured the sorption (either absorption or adsorption) rates of three different types of organic chemical solvents. These solvents are used to clean industrial fabricated-metal parts and are potential hazardous waste. Independent samples from ea

> The mitochondrial enzyme NADPH: NAD transhydrogenase of the common rat tapeworm (Hymenolepiasis diminuta) catalyzes hydrogen in the transfer from NADPH to NAD, producing NADH. This enzyme is known to serve a vital role in the tapeworm’s

> An analysis was conducted by the Statistics Consulting Center at Virginia Tech in conjunction with the Department of Forestry. A certain treatment was applied to a set of tree stumps in which the chemical Garlon was used with the purpose of regenerating

> A textile company weaves a certain fabric on a large number of looms. The managers would like the looms to be homogeneous so that their fabric is of uniform strength. It is suspected that there may be significant variation in strength among looms. Consid

> Five “pours” of metals have had 5 core samples each analyzed for the amount of a trace element. The data for the 5 randomly selected pours are as follows: (a) The intent is that the pours be identical. Thus, test th

> Immobilization of free-ranging white-tailed deer by drugs allows researchers the opportunity to closely examine the deer and gather valuable physiological information. In the study Influence of Physical Restraint and Restraint Facilitating Drugs on Blood

> The following data show the effect of 4 operators, chosen randomly, on the output of a particular machine. (a) Perform a random effects analysis of variance at the 0.05 level of significance. (b) Compute an estimate of the operator variance component

> To find out whether a new serum will arrest leukemia, nine patients, who have all reached an advanced stage of the disease, are selected. Five patients receive the treatment and four do not. The survival times, in years, from the time the experiment comm

> An experiment is conducted in which 4 treatments are to be compared in 5 blocks. The data are given below. (a) Assuming a random effects model, test the hypothesis, at the 0.05 level of significance, that there is no difference between treatment means

> Testing patient blood samples for HIV antibodies, a spectrophotometer determines the optical density of each sample. Optical density is measured as the absorbance of light at a particular wavelength. The blood sample is positive if it exceeds a certain c

> An experiment was conducted to compare three types of coating materials for copper wire. The purpose of the coating is to eliminate “flaws” in the wire. Ten different specimens of length 5 millimeters were randomly ass

> In the book Design of Experiments for the Quality Improvement, published by the Japanese Standards Association (1989), a study on the amount of dye needed to get the best color for a certain type of fabric was reported. The three amounts of dye, 1 3% wof

> In the paper “Self-Control and Therapist Control in the Behavioral Treatment of Overweight Women,” published in Behavioral Research and Therapy (Vol. 10, 1972), two reduction treatments and a control treatment were stu

> Scientists in the Department of Plant Pathology at Virginia Tech devised an experiment in which 5 different treatments were applied to 6 different locations in an apple orchard to determine if there were significant differences in growth among the treatm

> Organic arsenicals are used by forestry personnel as silvicides. The amount of arsenic that the body takes in when exposed to these silvicides is a major health problem. It is important that the amount of exposure be determined quickly so that a field wo

> In a study conducted by the Department of Health and Physical Education at Virginia Tech, 3 diets were assigned for a period of 3 days to each of 6 subjects in a randomized complete block design. The subjects, playing the role of blocks, were assigned th

> A nuclear power facility produces a vast amount of heat, which is usually discharged into aquatic systems. This heat raises the temperature of the aquatic system, resulting in a greater concentration of chlorophyll a, which in turn extends the growing se

> In an article “Shelf-Space Strategy in Retailing,” published in Proceedings: Southern Marketing Association, the effect of shelf height on the supermarket sales of canned dog food is investigated. An experiment was con

> A random sample X1, . . . , Xn comes from a uniform distribution (see Section 6.1) population U(0, θ) with unknown θ. The data are given below: 0.13, 1.06, 1.65, 1.73, 0.95, 0.56, 2.14, 0.33, 1.22, 0.20, 1.55, 1.18, 0.71, 0.01,

> In a study on The Periphyton of the South River, Virginia: Mercury Concentration, Productivity, and Autotropic Index Studies, conducted by the Department of Environmental Sciences and Engineering at Virginia Tech, the total mercury concentration in perip

> The following data represent the final grades obtained by 5 students in mathematics, English, French, and biology: Test the hypothesis that the courses are of equal difficulty. Use a P-value in your conclusions and discuss your findings. Subject S

> The following data are the percents of foreign additives measured by 5 analysts for 3 similar brands of strawberry jam, A, B, and C: Perform a randomized complete block analysis of variance to test the hypothesis, at the 0.05 level of significance, th

> Three varieties of potatoes are being compared for yield. The experiment is conducted by assigning each variety at random to one of 3 equal-size plots at each of 4 different locations. The following yields for varieties A, B, and C, in 100 kilograms per

> Four kinds of fertilizer f1, f2, f3, and f4 are used to study the yield of beans. The soil is divided into 3 blocks, each containing 4 homogeneous plots. The yields in kilograms per plot and the corresponding treatments are as follows: Conduct an anal

> The financial structure of a firm refers to the way the firm’s assets are divided into equity and debt, and the financial leverage refers to the percentage of assets financed by debt. In the paper The Effect of Financial Leverage on Ret

> In a biological experiment, four concentrations of a certain chemical are used to enhance the growth of a certain type of plant over time. Five plants are used at each concentration, and the growth in each plant is measured in centimeters. The following