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probability statistics
Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying E. Ye - Solutions
In a study that was analyzed for personnel in the Department of Biochemistry at Virginia Tech, three diets were given to groups of rats in order to study the effect of each on dietary residual zinc in the bloodstream. Five pregnant rats were randomly assigned to each diet group, and each was given
An analysis was conducted by the Laboratory for Interdisciplinary Statistical Analysis 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 the roots of the
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.
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 significantly higher yield is obtained with the catalysts than with no catalyst.
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 4 randomly selected
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:1. The intent is that the pours be identical. Thus, test that the“pour” variance component is zero. Draw conclusions.2. Show a complete
The following data show the effect of 4 operators, chosen randomly, on the output of a particular machine.1. Perform a random effects analysis of variance at the 0.05 level of significance.2. Compute an estimate of the operator variance component and the experimental error variance component.
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:1. The intent is that the pours be identical. Thus, test that the“pour” variance component is zero. Draw conclusions.2. Show a complete
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 cutoff value that is determined by the control
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 studied for their effects on the weight change of obese women. The two reduction
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, % wof ( % of the weight of a fabric), 1% wof, and 3%
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 assigned to receive each coating, and the thirty specimens were subjected to an
Scientists in the Department of Plant Pathology, Physiology, and Weed Science 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 treatments.Treatments
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 worker with a high level of arsenic can be
In a study conducted by the Department of Human Nutrition, Foods and Exercise 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 the following 3 diets in a random
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 chlorophylla, which in turn extends the growing season. To study this effect, water samples were
In a study on The Periphyton of the South River, Virginia:Mercury Concentration, Productiv-ity, and Autotropic Index Studies, conducted by the Department of Environmental and Water Resources Engineering at Virginia Tech, the total mercury concentration in periphyton total solids was measured at 6
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, that the percent of foreign additives is the same
Four kinds of fertilizer f , f , f , and f 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:i 1 2 3 4 Conduct an analysis of variance at the 0.05 level of
The following table (from A. Hald, Statistical Theory with Engineering Applications, John Wiley & Sons, New York, 1952)gives tensile strengths (in deviations from 340) for wires taken from nine cables to be used for a high-voltage network. Each cable is made from 12 wires. We want to know whether
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 growth data are taken. A control (no
Do Duncan’s test for paired comparisons for the data of Exercise 13.2 on page 538. Discuss the results.
The printout in Figure 13.5 on page 552 gives information on Duncan’s test, using PROC GLM in SAS, for the aggregate data in Example 13.1. Give conclusions regarding paired comparisons using Duncan’s test results.
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 Return, Tai Ma of Virginia Tech claims that financial leverage
It is suspected that the environmental temperature at which batteries are activated affects their life. Thirty homogeneous batteries were tested, six at each of five temperatures, and the data are shown below (activated life in seconds). Analyze and interpret the data. (From C. R. Hicks,
The following data are values of pressure (psi) in a torsion spring for several settings of the angle between the legs of the spring in a free position:Compute a one-way analysis of variance for this experiment and state your conclusion concerning the effect of angle on the pressure in the spring.
In the study An Evaluation of the Removal Method for Estimating Benthic Populations and Diversity conducted by Virginia Tech on the Jackson River, 5 different sampling procedures were used to determine the species counts. Twenty samples were selected at random, and each of the 5 sampling procedures
The study Loss of Nitrogen Through Sweat by Preadolescent Boys Consuming Three Levels of Dietary Protein was conducted by the Department of Human Nutrition, Food and Exercise at Virginia Tech to determine perspiration nitrogen loss at various dietary protein levels. Twelve preadolescent boys
Use Tukey’s test, with a 0.05 level of significance, to analyze the means of the five different brands of headache tablets in Exercise 13.6 on page 539.
An investigation was conducted to determine the source of reduction in yield of a certain chemical product. It was known that the loss in yield occurred in the mother liquor, that is, the material removed at the filtration stage. It was thought that different blends of the original material might
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 the flame-retardant finish of cotton flannelette
Consider the data of Review Exercise 13.47 on page 575.Make significance tests on the following contrasts:1. B versus A, C, and D;2. C versus A and D;3. A versus D.
Use Bartlett’s test at the 0.05 level of significance to test for homogeneity of variances in Exercise 13.2 on page 538.
For the data set in Exercise 13.7, use Bartlett’s test to check whether the variances are equal. Use α = 0.05.
Use Bartlett’s test at the 0.01 level of significance to test for homogeneity of variances in Exercise 13.5 on page 539.
Use Cochran’s test at the 0.01 level of significance to test for homogeneity of variances in Exercise 13.4 on page 539.
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 applied directly on the soil surface or mixed with
The data in the following table represent the number of hours of relief provided by five different brands of headache tablets administered to 25 subjects experiencing fevers of 38°C or more. Perform the analysis of variance and test the hypothesis at the 0.05 level of significance that the mean
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 anaerobic metabolism, and it has recently been hypothesized
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 Measurements of White-Tailed Deer and Other
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 conducted at a small supermarket for a period of 8 days on the sales of a single
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 fabricatedmetal parts and are potential hazardous waste. Independent samples from each type of solvent were tested, and their
Six different machines are being considered for use in manufacturing rubber seals. The machines are being compared with respect to tensile strength of the product. A random sample of four seals from each machine is used to determine whether the mean tensile strength varies from machine to machine.
Case Study: Consider the data set for Exercise 12.12, page 472 (hospital data), repeated here.2 1. The SAS PROC REG outputs provided in Figures 12.9 and 12.10 supply a considerable amount of information. Goals are to do outlier detection and eventually determine which model terms are to be used in
Show that in choosing the so-called best subset model from a series of candidate models, choosing the model with the smallest s is equivalent to choosing the model with the smallest.
A study was conducted to determine whether lifestyle changes could replace medication in reducing blood pressure among hypertensives. The factors considered were a healthy diet with an exercise program, the typical dosage of medication for hypertension, and no intervention. The pretreatment body
An article in the Journal of Pharmaceutical Sciences (Vol.80, 1991) presents data on the mole fraction solubility of a solute at a constant temperature. Also measured are the dispersion x and dipolar and hydrogen bonding solubility parameters x and x . A portion of the data is shown in the table
A carbon dioxide (CO ) flooding technique is used to extract crude oil. The CO floods oil pockets and displaces the crude oil. In an experiment, flow tubes are dipped into sample oil pockets containing a known amount of oil. Using three different values of flow pressure and three different values
Consider the data of Review Exercise 12.64. Suppose it is of interest to add some “interaction” terms. Namely, consider the model 1. Do we still have orthogonality? Comment.2. With the fitted model in part (a), can you find prediction intervals and confidence intervals on the mean response?Why
In exercise physiology, an objective measure of aerobic fitness is the oxygen consumption in volume per unit body weight per unit time. Thirty-one individuals were used in an experiment to model oxygen consumption against their age in years (x ), weight in kilograms (x ), time to run miles (x ),
In a chemical engineering experiment dealing with heat transfer in a shallow fluidized bed, data are collected on the following four regressor variables: fluidizing gas flow rate, lb/hr(x ); supernatant gas flow rate, lb/hr (x ); supernatant gas inlet nozzle opening, millimeters (x ); and
A small experiment was conducted to fit a multiple regression equation relating the yield y to temperature x , reaction time x , and concentration of one of the reactants x .Two levels of each variable were chosen, and measurements corresponding to the coded independent variables were recorded as
In the Department of Fisheries and Wildlife Conservation at Virginia Tech, an experiment was conducted to study the effect of stream characteristics on fish biomass. The regressor variables are as follows: average depth (of 50 cells), x ; area of in-stream cover (i.e., undercut banks, logs,
In an experiment to ascertain the effect of load, x, in lb/in , on the probability of failure of specimens of a certain fabric type, an experiment was conducted in which numbers of specimens were exposed to loads ranging from 5 lb/in. to 90 lb/in. . The numbers of “failures” were observed. The
From a set of streptonignic dose-response data, an experimenter desires to develop a relationship between the proportion of lymphoblasts sampled that contain aberrations and the dosage of streptonignic. Five dosage levels were applied to the rabbits used for the experiment. The data are as
In Exercise 12.28, page 482, we have the following data concerning wear of a bearing:1. The following model may be considered to describe the data:for i = 1, 2, …, 6. The x x is an “interaction” term. Fit this model and estimate the parameters.2. Use the models (x ), (x , x ), (x ), (x , x ,
For Exercise 12.57, test H : β = β = 0. Give P-values and comment.
The pull strength of a wire bond is an important characteristic. The following data give information on pull strength y, die height x , post height x , loop height x , wire length x , bond width on the die x , and bond width on the post x . (From Myers, Montgomery, and Anderson-Cook, 2009.)1. Fit a
In an effort to model executive compensation for the year 1979, 33 firms were selected, and data were gathered on compensation, sales, profits, and employment.p Consider the model, 1. Fit the regression with the model above.2. Is a model with a subset of the variables preferable to the full model?
A client from the Department of Mechanical Engineering approached the Laboratory for Interdisciplinary Statistical Analysis at Virginia Tech for help in analyzing an experiment dealing with gas turbine engines. The voltage output of engines was measured at various combinations of blade speed and
For the quadratic model of Exercise 12.51(b), give estimates of the variances and covariances of the estimates ofβ and β .
The following is a set of data for y, the amount of money(in thousands of dollars) contributed to the alumni association at Virginia Tech by the Class of 2005, and x, the number of years following graduation:1. Fit a regression model of the type 2. Fit a quadratic model of the type 3. Determine
For the punter data in Case Study 12.2, an additional response, “punting distance,” was also recorded. The average distance values for each of the 13 punters are given.1. Using the distance data rather than the hang times, estimate a multiple linear regression model of the type for predicting
For the data of Exercise 12.15 on page 472, use the techniques of 1 2 3 2 3 1. forward selection with a 0.05 level of significance to choose a linear regression model;2. backward elimination with a 0.05 level of significance to choose a linear regression model;3. stepwise regression with a 0.05
Consider the “hang time” punting data given in Case Study 12.2, using only the variables x and x .1. Verify the regression equation shown on page 509.2. Predict punter hang time for a punter with LLS = 180 pounds and Power = 260 foot-pounds.3. Construct a 95% confidence interval for the mean
A study was done to determine whether the gender of the credit card holder was an important factor in generating profit for a certain credit card company. The variables considered were income, the number of family members, and the gender of the card holder. The data are as follows:1. Fit a linear
A study was done to assess the cost effectiveness of driving a four-door sedan instead of a van or an SUV (sports utility vehicle). The continuous variables are odometer reading and octane of the gasoline used. The response variable is miles per gallon. The data are presented here.0 1 2 3 3 1 3 1.
For the data set given in Exericise 12.16 on page 473, can the response be explained adequately by any two regressor variables? Discuss.
In Example 12.8, a case is made for eliminating x , powder temperature, from the model since the P-value based on the F-test is 0.2156 while P-values for x and x are near zero.1. Reduce the model by eliminating x , thereby producing a full and a restricted (or reduced) model, and compare them on
Consider Example 12.3 on page 467. Compare the two competing models.1 2 3 Y 1 2 3 3 1 2 3 1 2 YUse in your comparison. Test H : β = β = β = 0. In addition, use the C.V. discussed in Exercise 12.40.
Consider Example 12.4. Figure 12.1 on page 479 displays a SAS printout of an analysis of the model containing variables x , x , and x . Focus on the confidence interval of the mean response μ at the (x , x , x ) locations representing the 13 data points. Consider an item in the printout indicated
Consider the data of Exercise 11.55 on page 457. Fit a regression model using weight and drive ratio as explanatory variables. Compare this model with the SLR (simple linear regression) model using weight alone. Use R , , and any tstatistics(or F-statistics) you may need to compare the SLR with the
Consider the data for Exercise 12.36. Compute the following:Comment.
Consider the electric power data of Exercise 12.5 on page 470. Test H : β = β = 0, making use of R(β , β |β , β ). Give a Pvalue, and draw conclusions.
Repeat Exercise 12.17 on page 481 using an F-statistic.
Test whether the regression explained by the model in Exercise 12.5 on page 470 is significant at the 0.01 level of significance.1 2 3 3
Test whether the regression explained by the model in Exercise 12.1 on page 470 is significant at the 0.01 level of significance.
Compute and interpret the coefficient of multiple determination for the variables of Exercise 12.1 on page 470.
Use the data from Exercise 12.16 on page 473.1. Estimate σ using the multiple regression of y on x , x , and x , 2. Compute a 95% prediction interval for the observed gain with the three regressors at x = 15.0, x = 220.0, and x = 6.0.
Using the data from Exercise 12.28, test the following at the 0.05 level.1. H : β = 0 versus H : β ≠ 0;2. H : β = 0 versus H : β ≠ 0.3. Do you have any reason to believe that the model in Exercise 12.28 should be changed? Why or why not?
Consider the following data from Exercise 12.13 on page 472.1. Estimate σ using multiple regression of y on x and x .2 1 2 21 2 3 4 21 2 2. Compute predicted values, a 95% confidence interval for mean wear, and a 95% prediction interval for observed wear if x = 20 and x = 1000.
Using the data of Exercise 12.5 on page 470 and the estimate of σ from Exercise 12.19, compute 95% confidence intervals for the predicted response and the mean response when x = 75, x = 24, x = 90, and x = 98.
For Exercise 12.8 on page 471, construct a 90% confidence interval for the mean compressive strength when the concentration is x = 19.5 and a quadratic model is used.
Using the data of Exercise 12.2 on page 470 and the estimate of σ from Exercise 12.17, compute 95% confidence intervals for the predicted response and the mean response when x = 900 and x = 1.00.
For the model of Exercise 12.1 on page 470, test the hypotheses that β = 2 against the alternative that β ≠ 2. Use a 22 21 2 1 4 22 11 1 1 P-value in your conclusion.
For the model of Exercise 12.2 on page 470, test the hypothesis that β = 0 at the 0.05 level of significance against the alternative that β ≠ 0.
For the model of Exercise 12.7 on page 471, test the hypothesis that β = 0 at the 0.05 level of significance against the alternative that β ≠ 0.
Referring to Exercise 12.5 on page 470, find the estimate of 1. ;2. Cov(b , b ).
Obtain estimates of the variances and the covariance of the estimators b and b of Exercise 12.2 on page 470.
For the data of Exercise 12.5 on page 470, estimate σ .
For the data of Exercise 12.1 on page 470, estimate σ .
For the data of Exercise 12.2 on page 470, estimate σ .
An engineer at a semiconductor company wants to model the relationship between the gain or hFE of a device (y) and three parameters: emitter-RS (x ), base-RS (x ), and emitter-tobase-RS (x ). The data are shown below:(Data from Myers, Montgomery, and Anderson-Cook, 2009.)1. Fit a multiple linear
The personnel department of a certain industrial firm used 12 subjects in a study to determine the relationship between job performance rating (y) and scores on four tests.The data are as follows:Estimate the regression coefficients in the model
Eleven student teachers took part in an evaluation program designed to measure teacher effectiveness and determine what factors are important. The response measure was a quantitative evaluation of the teacher. The regressor variables were scores on four standardized tests given to each teacher. The
A study was performed on a type of bearing to find the relationship of amount of wear y to x = oil viscosity and x =load. The following data were obtained. (From Response Surface Methodology, Myers, Montgomery, and Anderson-Cook, 2009.)1. Estimate the unknown parameters of the multiple linear
The following data reflect information from 17 U.S. Naval hospitals at various sites around the world. The regressors are workload variables, that is, items that result in the need for personnel in a hospital. A brief description of the variables is as follows:The goal here is to produce an
An experiment was conducted to study the size of squid eaten by sharks and tuna. The regressor variables are Y|x 0 1 22 Y |x 0 1 1 22 Y|x 0 1 2 23 3characteristics of the beaks of the squid. The data are given as follows:In the study, the regressor variables and response considered are Estimate the
The following data are given:1. Fit the cubic model μ = β + β x + β x + β x .2. Predict Y when x = 2.
1. Fit a multiple regression equation of the form μ = β + β x+ β x to the data of Example 11.8 on page 440.2. Estimate the yield of the chemical reaction for a temperature of 225°C.
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