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Questions and Answers of Statistics

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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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,
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
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.
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,
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
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
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
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,
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
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
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
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
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.
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
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
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
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
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.
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
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
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
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.
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
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
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
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
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
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
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)
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
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
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
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.
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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

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