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mathematics
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Probability & Statistics For Engineers & Scientists 7th Edition Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying - Solutions
In Exercise 13.43, if we are interested in testing for the significance of the operator variance component, do we have large enough samples to ensure with a probability as large as 0.95 a significant variance component if the true σ2α is 1.5σ2? If not, how many runs are necessary for
If one assumes a fixed effects model in Exercise 13.45 and uses an ? = 0.05 level test, how many blocks are needed in order that we accept the hypothesis of equality of treatment means with probability 0.1 when, in fact,
Verify the values given for A and Φ2 in Table 13.19 for the randomized complete block design.
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 cutofr value that is determined by the control
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 that the "pour" variance component is zero. Draw conclusions.(b) Show a complete ANOVA
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. Consider the following data for the 4 randomly
An analysis was conducted by the Statistics Consulting Center at Virginia Polytechnic Institute and State University in conjunction with the Department of Forestry. A certain treatment was applied to a set of tree stumps. The chemical Garlon was used with the purpose of regenerating the roots of
Consider the aggregate data of Example 13.1. Perform Bartlett's test to determine if there is heterogeneity of variance among the aggregates,
In 1983 the Department of Dairy Science at the Virginia Polytechnic Institute and State University conducted an experiment to study the effect of feed rations, differing by source of protein, on the average daily milk production of cows. There were 5 rations used in the experiment. A 5 x 5 Latin
Three catalysts are used in a chemical process with a control (no catalyst) being included. The following are yield data from the process. Use Dunnett's test at ? = 0.01 level of significance to determine if a significantly higher yield is obtained with the catalysts than with no catalyst.
Four laboratories are being used to perform chemical analysis. Samples of the same material are sent to the laboratories for analysis as part of the study to determine whether or not they give, on the average, the same results. The analytical results for the four laboratories are as follows:(a) Use
Use Bartlett's test at the 0.01 level of significance to test for homogeneity of variances in Exercise 13.9.
Use Cochran's test at the 0.01 level of significance to test for homogeneity of variances in Exercise 13.6.
Use Bartlett's test at the 0.05 level of significance to test for homogeneity of variances in Exercise 13.8.
An experiment was designed for personnel in the Department of Animal Science at Virginia Polytechnic Institute and State University with the purpose of studying urea and aqueous ammonia treatment of wheat straw. The purpose was to improve nutrition value for male sheep. The diet treatments are:
In a data set that was analyzed for personnel in the Department of Biochemistry at Virginia Polytechnic Institute and State University, three diets were given to a group of rats in order to study the effect of each on dietary residual zinc in the bloodstream. Five pregnant rats were randomly
A study is conducted to compare gas mileage for 3 competing brands of gasoline. Four different automobile models of varying size are randomly selected. The data, in miles per gallon, follow. The order of testing is random for each model.(a) Discuss the need for the use of more than a single model
An experiment was conducted to compare three types of paint to determine if there is evidence of differences in their wearing qualities. They were exposed to abrasive action and the time in hours was observed until abrasion was noticed. Six specimens were used for each type of paint. The data are
Four different locations in the northeast are used for collecting ozone measurements in parts per million. Amounts of ozone were collected in 5 samples at each location.(a) Is then: sufficient information here to suggest that there are differences in the mean ozone levels across locations? Be
An experiment was conducted to study the effect of temperature and type of oven on the life of a particular component being tested. Four types of ovens and 3 temperature levels were used in the experiment. Twenty-four pieces were assigned randomly, 2 to each combination of treatments, and the
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 Nutrition and Foods at the Virginia
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 as follows. Use a 0.01 level of significance to test the hypothesis that(a) There is no difference in error scores for different environments;(b)
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 coating. In a study conducted by the Department of
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 the Virginia Polytechnic Institute and State University. Five
An experiment was conducted to increase the adhesiveness of rubber products. Sixteen products were made with the new additive and another 16 without the new additive. The observed adhesiveness is recorded below. Perform an analysis of variance to test for significant main and interactioneffects.
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 the following table. Perform an analysis of
In Myers and Montgomery (2002) a scenario is discussed in which an auto bumper plating process is described. 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 experiment is designed. The plan is a completely
An engineer is interested in the effect 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 data are as follows.(a) Show an analysis-of
Two factors in a manufacturing process for an integrated circuit were 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 (3 levels). Experimentation is costly so
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 28 factor combinations, the structure of the experimental units represented by
In an experiment conducted in the civil engineering department at Virginia Tech, a 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 algae.(a) Do an analysis of variance and
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 uptake. Magnesium levels in grams per cubic
Consider the data set in Exercise 14.12 and answer the following questions.(a) Both factors of 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 x = copper content and x2 = time.Fit the model to the
The following data are taken in a study involving measurements. An experiment was conducted using 3 three: factors A, B, and C, all fixed effects:(a) Perform tests of significance on all interactions at ? = 0.05 level.(b) Perform tests of significance on the main effects at the Q as 0.05 level.(c)
Consider an experimental situation involving factors A, B, and C, where we assume a three way fixed effects model of the form IJijkl = ft. + ? i + 0j + ? k + (? ?) j k + ? i j k l. All other interactions arc considered to be nonexistent or negligible. The data are presented here. (a) Perform a test
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 the Department of Mechanical Engineering at
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 Report, RK-TR-62-1, Army Ordinance Missile
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. It is postulated that temperature, surface
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?s (1991)] The data are presented here. Let the dentists play the role
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 of the water, hot or warm. The experiment was
A scientist collects experimental data on the radius of a propellant grain, y, as a function of powder temperature, extrusion rate, and die temperature. The three factor experiment is as follows, trials at the eight combinations of factors. It is felt as if extrusion rate does not interact, with
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 factors, the type of solvent, extraction
Consider the data set in Exercise 14.19.(a) Construct an interaction plot for any two factor interaction that is significant.(b) Do a normal probability plot of residuals and comment.
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. 3 runs on each condition. Three filters and 4 operators are selected at random to use in the experiment, resulting in the
Assuming a model II experiment for Exercise 14.2, 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.
Consider the following analysis of variance for a model II 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 appropriate;(b) By not pooling sums of squares of
Are 2 observations for each treatment combination in Exercise 14.16 sufficient if the power of our test for detecting differences among the levels of factor C at the 0.05 level of significance is to be at least 0.8 when γ1 = —0.2, γ2 = —0.4, and γ3 = —0.2? Use the same pooled estimate of
Using the estimates of the variance components in Exercise 14.25 evaluate the power when we test the variance component due to filters to be zero.
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 factor levels are given in the data. The
A manufacturer of latex house paint (brand A) would like to show that their paint is more robust to the material being painted than their two closest competitors. The response is the time, in years, until chipping occurs. The study involves the three brands of paint and three randomly chosen
A plant manager would like to show that the yield of a woven fabric in his plant does not depend on machine operator or time of day and is consistently high. Four randomly selected operators and three randomly selected hours of the day are chosen for the study. The yield is measured in yards
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 manufactured by the company and 3 fixed power settings.
The Statistics Consulting Center at Virginia Polytechnic Institute and State University 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 physical
An experiment was conducted in the Department of Food Science at Virginia Polytechnic Institute and State University. 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 response in
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 a retested use 4 different levels of humidity. The results, in kilograms, are as follows:(a) Assuming a model I experiment, perform an analysis of
Personnel in the Materials Engineering Department at Virginia Polytechnic Institute and State University 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. The
In the experiment of Review Exercise 14.34, 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 factorsare
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. In the book SN-Ratio for the Quality Evaluation published by the Japanese Standards Association (1988), a study 011 how these
Exercise 14.23 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 temperature term, a time term, and a temperature by time interaction,
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 conducted. Three different tire air pressures were compared on three different driving surfaces. The three air
The manufacturer of a certain brand of freeze dried coffee hopes to shorten the process time without jeopardizing the integrity of the product. He wants to use 3 temperatures for the drying chamber and 4 drying times. The current drying time is 3 hours at a temperature of -15?C. The flavor response
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) Weekday afternoons between 2:00 and 3:00 P.M., and(3) Saturday mornings between 11:00 A.M.
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. Drawconclusions.
In an experiment conducted by the Mining Engineering Department at the Virginia Polytechnic Institute and State University to study a particular filtering system for coal, a coagulant was added to a solution in a tank containing e: oal and sludge, which was then placed in a recirculation system in
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 refinement; B, percent remelted material; C, fluxing
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 error. Two levels of each factor are used,
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 is a significant difference in the amount of aluminum achieved in the
It is important to study the effect of 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 the levels being 20 lb/hr and 30 lb/hr. The
Consider Exercise 15.3. It is of interest to the researcher to learn not only that AD, BC, and possibly AB are important. But there is also interest in what they mean scientifically. Show two-dimensional interaction plots for all three and give interpretation.
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 plots, one for B = —1 and the other for B = +1.
Consider Exercise 15.6. Use a "+1" and "—1" scaling for '-high" and "low,'- respectively, and do a multiple linear regression with the model Y1 = β0 + β1x1i + β2x2i + β12x1ix2i + ε i, with x u = reactant concentration (—1, +1) and x2i = feed rate (-1, +1).(a) Compute regression
Consider Exercise 15.5. Compute all 15 effects and do normal probability plots of the effects.(a) Docs 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 what you communicated about, the importance of
In Myers and Montgomery (2002), 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 levels of each factor are chosen, and duplicates were run at, each
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 for contrasts given by Comment. In your comments,
In a 23 factorial experiment with 3 replications, show the block arrangement and indicate by means of tin analysis-of-variance table the effects to be tested and their degrees of freedom, when the AB interaction is confounded with blocks.
The following experiment was rnn to study main effects and all interactions. Four factors are used at two levels each. The experiment is replicated and two blocks are necessary in each replication. The data are presented here.(a) What effect is confounded with blocks in the first replication of the
Divide the treatment combinations of a 24 factorial experiment into four blocks by confounding ABC and ABD. What additional effect is also confounded with blocks?
An experiment is conducted to determine the breaking strength of a certain alloy containing five metals, A, B, C, D, and E. Two different percentages of each metal are used in forming the 25 = 32 different alloys. Since only eight alloys can be tested on a given day, the experiment is conducted
By confounding ABC in two replicates and AB in the third, show the block arrangement and the analysis-of-variance table for a 23 factorial experiment with three replicates. What is the relative information on the confounded effects?
The following coded data represent the strength of a certain type of bread-wrapper stock produced under 16 different conditions, the latter representing two levels of each of four process variables. An operator effect was introduced into the model, since it was necessary to obtain half the
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 effects are confounded with blocks?
An experiment is revealed in Myers and Montgomery (2002) in whom optimum conditions are sought storing bovine semen to obtain maximum survival. The variables are percent sodium citrate, percent glycerol, glycerol, and equilibration time in hours. The response is percent survival of the motile
Oil producers are interested in high strength nickel alloys that are strong and corrosion resistant. An experiment conducted in which yield strengths were compared for nickel alloy tensile specimens charged in sulfuric acid solution saturated with carbon disulfide. Two alloys were combined; a 75%
Suppose a second replicate of the experiment in Exercise 15.19 could be performed.(a) Would a second replication of the blocking scheme of Exercise 15.19 be the best choice?(b) If the answer to part (a) is no, give the layout for a better choice for the second replicate.(c) What concept did you use
Consider Figure, 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.
List the aliases for the: various effects in a 25 factorial experiment when the defining contrast is ACDE.
(a) Obtain a i2 fraction of a 24 factorial design using BCD as the defining contrast.(b) Divide the ½ fraction into 2 blocks of A units each by confounding ABC.(c) Show the analysis-of-variance table (sources of variation and degrees of freedom) for testing all unconfounded main effects, assuming
Construct a 1/4 fraction a 2° factorial elesign using ABCD and BDEF as the defining contrasts. Show what effects are: aliased with the six main effects.
(a) Using the defining contrasts ABCE and ABDF obtain a 1/4 fraction of a 2° elesign.(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-factor and higher interactions are negligible.
Seven factors are' varied at, two levels in an experiment involving only 16 trials. A 1/8 fraction of a 2 factorial experiment is used with the defining contrasts being ACD, BEF, and CEG. The data are as follows. Perform an analysis of variance em all seven main effects, assuming that interactions
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 1/2 fraction of a 2'1 factorial experiment
In an experiment conducted at the Department of Mechanical Engineering and analyzed by the Statistics Consulting Center at the Virginia Polytechnic Institute and State University, a sensor detects an electrical charge each time a turbine blade makes one; rotation. The sensor then measures the
In a 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 the Virginia Polytechnic Institute: and Stale University, an experimenter measures the number of breakdowns in an
Consider a 25-1 elesign with factors A, B, C, D, and E. Construct the design by beginning with a 2'1 and use E = ABCD as the generator. Show all aliases.
There are six factors and only 8 elesign points can be used. Construct 26-3 by beginning with a 2d and use D = AB, E = AC and F = BC ns the generators.
Consider Exercise 15.33. Construct another 26-3 that is different from the design chosen in Exercise 15.33.
For Exercise 15.33 give all aliases for the six main effects.
In Myers and Montgomery (2002), an application was discussed in which an engineer is concerned with the effects on the cracking of a fit antum alloy. The three factors are A, temperature, B, titanium content, and C, amount of grain refiner. The following gives a portion of the design and the
Use: the coal cleansing data of Exercise 15.2 to fit a model of the type E (Y) = β0 + ftjpi + β2x2 +β3x3 where the levels are x1: percent solids: 8; 12, x2: flow rate: 150: 250 gal/min, x3: pH: 5; 6 Center and scale the variables to design units. Also conduct a test for lack of fit,
A 2° 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 regression, lack of fit and pure error.
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