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mathematics
applied statistics and probability
Applied Statistics And Probability For Engineers 6th Edition Douglas C. Montgomery, George C. Runger - Solutions
An article in Electronic Journal of Applied Statistical Analysis [“Survival Analysis of Acute Myocardial Infarction Patients Using Non-Parametric and Parametric Approaches” (2009, Vol. 2(1), pp. 22–36)] described the use of a Weibull distribution to model the survival time of acute myocardial
An article in Electric Power Systems Research [“On the Self-Scheduling of a Power Producer in Uncertain Trading Environments” (2008, Vol. 78(3), pp. 311–317)] considered a self-scheduling approach for a power producer. In addition to price and forced outages, another uncertainty was due to
An article in Journal of Theoretical Biology [“Computer Model of Growth Cone Behavior and Neuronal Morphogenesis” (1995, Vol. 174(4), pp. 381–389)] developed a model for neuronal morphogenesis in which neuronal growth cones have a significant function in the development of the nervous system.
Among homeowners in a metropolitan area, 25% recycle paper each week. A waste management company services 10,000 homeowners (assumed independent). Approximate the following probabilities:(a) More than 2600 recycle paper in a week(b) Between 2400 and 2600 recycle paper in a week(c) Number of
Provide approximate sketches for beta probability density functions with the following parameters. Comment on any symmetries and show any peaks in the probability density functions in the sketches.(a) α = β < 1(b) α = β = 1.(c) α = β > 1.
Consider the regional right ventricle transverse wall motion in patients with pulmonary hypertension (PH). The right ventricle ejection fraction (EF) is approximately normally distributed with standard deviation of 12 for PH subjects, and with mean and standard deviation of 56 and 8, respectively,
An article in IEEE Journal on Selected Areas in Communications [“Impulse Response Modeling of Indoor Radio Propagation Channels” (1993, Vol. 11(7), pp. 967–978)] indicated that the successful design of indoor communication systems requires characterization of radio propagation. The
Suppose that the construction of a solar power station is initiated. The project’s completion time has not been set due to uncertainties in financial resources. The proportion of completion within one year has a beta distribution with parameters α =1 and β = 5. Determine the following:(a) Mean
The waiting time for service at a hospital emergency department follows an exponential distribution with a mean of three hours. Determine the following:(a) Waiting time is greater than four hours(b) Waiting time is greater than six hours given that you have already waited two hours(c) Value x (in
The size of silver particles in a photographic emulsion is known to have a log normal distribution with a mean of 0.001 mm and a standard deviation of 0.002 mm.(a) Determine the parameter values for the lognormal distribution.(b) What is the probability of a particle size greater than 0.005 mm?
Suppose that the construction of a solar power station is initiated. The project’s completion time has not been set due to uncertainties in financial resources. The completion time for the first phase is modeled with a beta distribution and the minimum, most likely (mode), and maximum completion
An allele is an alternate form of a gene, and the proportion of alleles in a population is of interest in genetics. An article in BMC Genetics [“Calculating Expected DNA Remnants From Ancient Founding Events in Human Population Genetics” (2008, Vol. 9:66)] used a beta distribution with mean 0.3
The maximum time to complete a task in a project is 2.5 days. Suppose that the completion time as a proportion of this maximum is a beta random variable with α = 2 and β = 3. What is the probability that the task requires more than two days to complete?
The length of stay at a hospital emergency department is the sum of the waiting and service times. Let X denote the proportion of time spent waiting and assume a beta distribution with α = 10 and β =1. Determine the following:(a) P(X > 0.9)(b) P(X < 0.5)(c) Mean and variance
A European standard value for a low-emission window glazing uses 0.59 as the proportion of solar energy that enters a room. Suppose that the distribution of the proportion of solar energy that enters a room is a beta random variable.(a) Calculate the mode, mean, and variance of the distribution for
Suppose that X has a beta distribution with parameters α = 1 and β = 4.2. Determine the following:(a) P(X < 0.25)(b) P(0.5 < X)(c) Mean and variance
Suppose that x has a beta distribution with parameters α = 2.5 and β = 1. Determine the following:(a) P(X < 0.25)(b) P(0.25 < X < 0.75)(c) Mean and variance
Suppose that X has a beta distribution with parameters α = 2.5 and β = 2.5. Sketch an approximate graph of the probability density function. Is the density symmetric?
Consider the lifetime of a laser in Example 4-26. Determine the following in parts (a) and (b):(a) Probability the lifetime is less than 1000 hours(b) Probability the lifetime is less than 11,000 hours given that it is more than 10,000 hours(c) Compare the answers to parts (a) and (b) and comment
An article in Chemosphere [“Statistical Evaluations Reflecting the Skewness in the Distribution of TCDD Levels in Human Adipose Tissue” (1987, Vol.16(8), pp. 2135- 2140)] concluded that the levels of 2,3,7,8-TCDD (colorless persistent environmental contaminants with no distinguishable odor at
An article in Applied Mathematics and Computation [“Confidence Intervals for Steady State Availability of a System with Exponential Operating Time and Lognormal Repair Time” (2003, Vol.137(2), pp. 499-509)] considered the long-run availability of a system with an assumed lognormal distribution
An article in Journal of Hydrology [“Use of a Lognormal Distribution Model for Estimating Soil Water Retention Curves from Particle-Size Distribution Data” (2006, Vol. 323(1), pp. 325–334)] considered a lognormal distribution model to estimate water retention curves for a range of soil
Suppose that the length of stay (in hours) at a hospital emergency department is modeled with a lognormal random variable X with θ = 1.5 and ω = 0.4. Determine the following in parts (a) and (b):(a) Mean and variance(b) P(X <8)(c) Comment on the difference between the probability P(X < 0)
An article in Health and Population: Perspectives and Issues (2000, Vol. 23, pp. 28–36) used the lognormal distribution to model blood pressure in humans. The mean systolic blood pressure (SBP) in males age 17 was 120.87 mm Hg. If the co-efficient of variation (100% × Standard deviation/mean) is
An article in IEEE Transactions on Dielectrics and Electrical Insulation [“Statistical Analysis of the AC Breakdown Voltages of Ester Based Transformer Oils” (2008, Vol. 15(4))] used Weibull distributions to model the breakdown voltage of insulators. The breakdown voltage is the minimum voltage
An article in Financial Markets Institutions and Instruments [“Pricing Reinsurance Contracts on FDIC Losses” (2008, Vol. 17(3)] modeled average annual losses (in billions of dollars) of the Federal Deposit Insurance Corporation (FDIC) with a Weibull distribution with parameters δ = 1.9317 and
An article in Electronic Journal of Applied Statistical Analysis [“Survival Analysis of Dialysis Patients Under Parametric and Non-Parametric Approaches” (2012, Vol. 5(2), pp. 271–288)] modeled the survival time of dialysis patients with chronic kidney disease with a Weibull distribution. The
An article in the Journal of the Indian Geophysical Union titled “Weibull and Gamma Distributions for Wave Parameter Predictions” (2005, Vol. 9, pp. 55–64) described the use of the Weibull distribution to model ocean wave heights. Assume that the mean wave height at the observation station is
An article in Mathematical Biosciences [“Influence of Delayed Viral Production on Viral Dynamics in HIV-1 Infected Patients” (1998, Vol.152(2), pp. 143–163)] considered the time delay between the initial infection by immunodeficiency virus type 1 (HIV-1) and the formation of productively
An article in Sensors and Actuators A: Physical [“Characterization and Simulation of Avalanche PhotoDiodes for Next-Generation Colliders” (2011, Vol.172(1), pp.181– 188)] considered an avalanche photodiode (APD) to detect charged particles in a photo. The number of arrivals in each detection
The total service time of a multi step manufacturing operation has a gamma distribution with mean 18 minutes and standard deviation 6.(a) Determine the parameters λ and r of the distribution.(b) Assume that each step has the same distribution for service time. What distribution for each step and
Patients arrive at a hospital emergency department according to a Poisson process with a mean of 6.5 per hour.(a) What is the mean time until the 10th arrival?(b) What is the probability that more than 20 minutes is required for the third arrival?
Given the probability density function f(x) = 0.013 x2e−0.01x / Γ(3), determine the mean and variance of the distribution.
An article in Vaccine [“Modeling the Effects of Influenza Vaccination of Health Care Workers in Hospital Departments” (2009, Vol.27(44), pp. 6261–6267)] considered the immunization of healthcare workers to reduce the hazard rate of influenza virus infection for patients in regular hospital
Requests for service in a queuing model follow a Poisson distribution with a mean of five per unit time.(a) What is the probability that the time until the first request is less than 4 minutes?(b) What is the probability that the time between the second and third requests is greater than 7.5 time
An article in Journal of National Cancer Institute [“Breast Cancer Screening Policies in Developing Countries: A Cost-Effectiveness Analysis for India” (2008, Vol.100(18), pp. 1290–1300)] presented a screening analysis model of breast cancer based on data from India. In this analysis, the
The length of stay at a specific emergency department in a hospital in Phoenix, Arizona had a mean of 4.6 hours. Assume that the length of stay is exponentially distributed.(a) What is the standard deviation of the length of stay?(b) What is the probability of a length of stay of more than 10
Web crawlers need to estimate the frequency of changes to Web sites to maintain a current index for Web searches. Assume that the changes to a Web site follow a Poisson process with a mean of 3.5 days.(a) What is the probability that the next change occurs in less than 2.0 days?(b) What is the
The number of (large) inclusions in cast iron follows a Poisson distribution with a mean of 2.5 per cubic millimeter. Approximate the following probabilities:(a) Determine the mean and standard deviation of the number of inclusions in a cubic centimeter (cc).(b) Approximate the probability that
Cabs pass your workplace according to a Poisson process with a mean of five cabs per hour.(a) Determine the mean and standard deviation of the number of cabs per 10-hour day.(b) Approximate the probability that more than 65 cabs pass within a 10-hour day.(c) Approximate the probability that between
Among homeowners in a metropolitan area, 75% recycle plastic bottles each week. A waste management company services 1500 homeowners (assumed independent). Approximate the following probabilities:(a) At least 1150 recycle plastic bottles in a week(b) Between 1075 and 1175 recycle plastic bottles in
Consider the code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (five bars and four spaces). The code for a character starts and ends with a bar (either narrow or wide) and a
A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit the site. Determine the probability that the ad color is red and the font size is not the smallest
Consider the well failure data given below. Suppose that two failed wells are selected randomly (without replacement) for a follow-up review.(a) What is the probability that both are from the gneiss geological formation group?(b) What is the probability that both are from the same geological
Consider the hospital emergency room data given below. Suppose that three visits that resulted in LWBS are selected randomly (without replacement) for a follow-up interview.(a) What is the probability that all three are selected from hospital 2?(b) What is the probability that all three are from
Consider the hospital emergency room data given below. Let A denote the event that a visit is to hospital 4 and let B denote the event that a visit results in LWBS (at any hospital). Determine the following probabilities.(a) P(A ∩ B)(b) P(A ∪ B)(c) P(A'∪ B')(d) Use the total probability
Consider the endothermic reactions given below. Let A denote the event that a reaction's final temperature is 271 K or less. Let B denote the event that the heat absorbed is above target.Determine the following
A computer system uses passwords that are exactly seven characters and each character is one of the 26 letters (a–z) or 10 integers (0–9). You maintain a password for this computer system. Let A denote the subset of passwords that begin with a vowel
Consider the square root of the sum of squares for curvature and divide by the square root of mean square error. Explain why the statistic that results has a t distribution and why it can be used to conduct a t test for curvature that is equivalent to the F test in the ANOVA.
Set up a 4III7–4 design using D = AB, E = AC, F = BC, and G = ABC as the design generators. Ignore all interactions above two factors.(a) Verify that each main effect is aliased with three two-factor interactions.(b) Suppose that a second 4III7–4 design with generators D = –AB, E = –AC, F =
Construct a 3IV7–3 design. Show how this design can be confounded in two blocks of eight runs each without losing information on any of the two-factor interactions.
Construct a 2IV7–2 design. Show how this design may be confounded in four blocks of eight runs each. Are any two-factor interactions confounded with blocks?
Construct a 25–1 design. Suppose that it is necessary to run this design in two blocks of eight runs each. Show how this can be done by confounding a two-factor interaction (and its aliased three-factor interaction) with blocks.
Consider a 23 design. Suppose that the largest number of runs that can be made in one block is four, but you can afford to perform a total of 32 observations.(a) Suggest a blocking scheme will provides some information on all interactions.(b) Show an outline (source of variability, degrees of
Consider a 22 design in two blocks with AB confounded with blocks. Prove algebraically that SSAB = SSBlocks.
What blocking scheme would you recommend if it were necessary to run a 24 design in four blocks of four runs each?
Consider an unreplicated 2kfactorial, and suppose that one of the treatment combinations is missing. One logical approach to this problem is to estimate the missing value with a number that makes the highest order interaction estimate zero. Apply this technique to the data in Example 14-5, assuming
An article in Journal of Applied Microbiology ["Use of Response Surface Methodology to Optimize Protease Synthesis by a Novel Strain of Bacillus sp. Isolated from Portuguese Sheep Wool" (2012, Vol. 113(1), pp. 36-43)] described a fractional factorial design with three center points to study six
Graphically study the assumption of no interactions between blocks and treatments in the previous exercise.
Consider the data in Exercise 14-97. Use only the first eight rows of data from the 23factorial design and assume that the experiment was conducted in blocks based on seed type.(a) Analyze the factorial effects and comment on which effects are important.(b) Develop a regression model to predict the
An article in European Food Research and Technology [€œFactorial Design Optimisation of Grape (Vitis vinifera) Seed Polyphenol Extraction€ (2009, Vol. 229(5), pp. 731€“742)] used a central composite design to study the effects of basic factors (time, ethanol, and pH)
An article in Process Biochemistry (Dec. 1996, Vol. 31(8), pp. 77-785) presented a 27€“3fractional factorial to perform optimization of manganese dioxide bioleaching media. The data are shown in the following table.The factors and levels are shown in the following table.(a) Write down the
Consider the following ANOVA table from a two-factor factorial experiment.(a) How many levels of each factor were used in the experiment?(b) How many replicates were used?(c) What assumption is made in order to obtain an estimate of error?(d) Calculate the missing entries (denoted with
Consider the following results from a two-factor experiment with two levels for factor A and three levels for factor B. Each treatment has three replicates.(a) Calculate the sum of squares for each factor and the interaction.(b) Calculate the sum of squares total and error.(c) Complete an ANOVA
The rework time required for a machine was found to depend on the speed at which the machine was run (A), the lubricant used while working (B), and the hardness of the metal used in the machine (C). Two levels of each factor were chosen and a single replicate of a 23experiment was run. The rework
An article in Biotechnology Progress (December 2002, Vol. 18(6), pp. 1170€“1175) presented a 27€“3fractional factorial to evaluate factors promoting astaxanthin production. The data are shown in the following table.The factors and levels are shown in the following table.(a) Write down the
An article in the Journal of Applied Electrochemistry (May 2008, Vol. 38(5), pp. 583€“590) presented a 27€“3 fractional factorial design to perform optimization of polybenzimidazole-based membrane electrode assemblies for H2/O2fuel cells. The design and data are shown in the following
An article in Biotechnology Progress (2001, Vol. 17, pp. 366€“368) reported on an experiment to investigate and optimize the operating conditions of the nisin extraction in aqueous two-phase systems (ATPS). A 22full factorial design with center points was used to verify the most significant
An article in Plant Disease [€œEffect of Nitrogen and Potassium Fertilizer Rates on Severity of Xanthomonas Blight of Syngonium Podophyllum€ (1989, Vol. 73(12), pp. 972€“975)] showed the effect of the variable nitrogen and potassium rates on the growth of
An article in Tropical Science [€œProximate Composition of the Seeds of Acacia Nilotica var Adansonti (Bagaruwa) and Extraction of Its Protein€ (1992, Vol. 32(3), pp. 263€“268)] reported on research extracting and concentrating the proteins of the bagaruwa seeds in
An article in the Journal of Quality Technology (1985, Vol. 17, pp. 198€“206) described the use of a replicated fractional factorial to investigate the effect of five factors on the free height of leaf springs used in an automotive application. The factors are A = furnace
Construct a 2III5–2 design in eight runs. What are the alias relationships in this design?
Construct a 2IV8–4 design in 16 runs. What are the alias relationships in this design?
Construct a 2IV4€“1design for the problem in Exercise 14-81. Select the data for the eight runs that would have been required for this design. Analyze these data and compare your conclusions to those obtained in Exercise 14-81 for the full factorial.Exercise 14-81An article in the Journal
Construct a 2IV4-1 design for the problem in Exercise 14-80. Select the data for the eight runs that would have been required for this design. Analyze these runs and compare your conclusions to those obtained in Exercise 14-80 for the full factorial.Exercise 14-80An article in the Journal of
An article in the Journal of Manufacturing Systems (1991, vol. 10, pp. 32€“40) described an experiment to investigate the effect of four factors, P = waterjet pressure, F = abrasive flow rate, G = abrasive grain size, and V = jet traverse speed, on the surface roughness of a waterjet
An article in the Journal of Coatings Technology (1988, Vol. 60, pp. 27–32) described a 24 factorial design used for studying a silver automobile basecoat. The response variable is distinctness of image (DOI). The variables used in the experiment are A = Percentage of polyester by weight of
An article in Solid State Technology (1984, Vol. 29, pp. 281–284) described the use of factorial experiments in photolithography, an important step in the process of manufacturing integrated circuits. The variables in this experiment (all at two levels) are prebake temperature (A), prebake time
An article in the IEEE Transactions on Components, Hybrids, and Manufacturing Technology (1992, Vol. 15) described an experiment for aligning optical chips onto circuit boards. The method involves placing solder bumps onto the bottom of the chip. The experiment used three solder bump sizes and
An article in the Textile Research Institute Journal (1984, Vol. 54, pp. 171€“179) reported the results of an experiment that studied the effects of treating fabric with selected inorganic salts on the flammability of the material. Two application levels of each salt were used, and a
Heat-treating metal parts is a widely used manufacturing process. An article in the Journal of Metals (1989, Vol. 41) described an experiment to investigate flatness distortion from heat-treating for three types of gears and two heat-treating times. The data follow:(a) Is there any evidence that
An article in Process Engineering (1992, No. 71, pp. 46€“47) presented a two-factor factorial experiment to investigate the effect of pH and catalyst concentration on product viscosity (cSt). The data are as follows:(a) Test for main effects and interactions using α = 0.05.
An article in Applied Biochemistry and Biotechnology (€œA Statistical Approach for Optimization of Polyhydroxybutyrate Production by Bacillus sphaericus NCIM 5149 under Submerged Fermentation Using Central Composite Design€ (2010, Vol. 162(4), pp. 996€“1007)] described an experiment to
An article in Analytical Biochemistry [€œApplication of Central Composite Design for DNA Hybridization Onto Magnetic Microparticles,€ (2009, Vol. 391(1), 2009, pp. 17€“23)] considered the effects of probe and target concentration and particle number in immobilization
An article in the Journal of Materials Processing Technology (1997, Vol. 67, pp. 55€“61) used response surface methodology to generate surface roughness prediction models for turning EN 24T steel (290 BHN). The data are shown in the following table.The factors and levels for the experiment are
Two responses y1 and y2 are related to two inputs x1 and x2 by the models y1 = 5 (x1 – 2)2 + (x2 − 2)3 and y2 = x2 – x1 + 3. Suppose that the objectives are y1 ≤ 9 and y2 ≥ 6.(a) Is there a feasible set of operating conditions for x1 and x2? If so, plot the feasible region in the
Suppose that a response y1 is a function of two inputs x1 and x2 with y1 = x22 – 4x21 –x1x2 + 4.(a) Draw the contours of this response function.(b) Consider another response y2 = (x1 – 2)2 + (x2 − 3)2.(c) Add the contours for y2 and discuss how feasible it is to minimize both y1 and y2 with
Consider the first-order model y = 12 + 1.2x1 − 2.1x2 + 1.6x3 − 0.6x4 where −1 ≤ xi ≤ 1.(a) Find the direction of steepest ascent.(b) Assume that the current design is centered at the point (0, 0, 0, 0). Determine the point that is three units from the current center point in the
In their book Empirical Model Building and Response Surfaces (John Wiley, 1987), Box and Draper described an experiment with three factors. The data in the following table are a variation of the original experiment from their book. Suppose that these data were collected in a semiconductor
An article in Tappi (1960, Vol. 43, pp. 38-44) describes an experiment that investigated the ash value of paper pulp (a measure of inorganic impurities). Two variables, temperature T in degrees Celsius and time t in hours, were studied, and some of the results are shown in the following table. The
A manufacturer of cutting tools has developed two empirical equations for tool life (y1) and tool cost (y2). Both models are functions of tool hardness (x1) and manufacturing time (x2). The equations areŷ1 = 10 + 5x1 + 2x2ŷ2 = 23 + 3x1 + 4x2and both are valid over the range –1.5 ≤ xi ≤
Consider the first-order model ŷ = 50 + 1.5x1 − 0.8x2 where –1 ≤ xi ≤ 1. Find the direction of steepest ascent.
An article in Quality Engineering [€œMean and Variance Modeling with Qualitative Responses: A Case Study€ (1998€“1999, Vol. 11, pp. 141€“148)] studied how three active ingredients of a particular food affect the overall taste of the product. The measure of
An article in Rubber Age (1961, Vol. 89, pp. 453€“ 458) describes an experiment on the manufacture of a product in which two factors were varied. The factors are reaction time (hr) and temperature (°C). These factors are coded as x1= (time €“ 12) / 8 and x2= (temperature
An article in Journal of Hazardous Materials [€œStatistical Factor-Screening and Optimization in Slurry Phase Bioremediation of 2,4,6-trinitrotoluene Contaminated Soil,€ (2011, Vol. 188(1), pp. 1€“9)] described an experiment to optimize the removal of TNT
An article in Bioresource Technology [€œMedium Optimization for Phenazine-1-carboxylic Acid Production by a gacA qscR Double Mutant of Pseudomonas sp. M18 Using Response Surface Methodology€ (Vol. 101(11), 2010, pp. 4089-4095)] described an experiment to optimize culture medium factors to
An article in the Journal of Marketing Research (1973, Vol. 10(3), pp. 270€“276) presented a 27€“4fractional factorial design to conduct marketing research:The factors and levels are shown in the following table.(a) Write down the alias relationships.(b) Estimate the main effects.(c)
An article in the Journal of Radioanalytical and Nuclear Chemistry (2008, Vol. 276(2), pp. 323€“328) presented a 28€“4 fractional factorial design to identify sources of Pu contamination in the radioactivity material analysis of dried shellfish at the National Institute of
Consider the 26€“2design in Table 14-29.(a) Suppose that after analyzing the original data, we find that factors C and E can be dropped. What type of 2k design is left in the remaining variables?(b) Suppose that after the original data analysis, we find that factors D and F can be
For each of the following designs, write down the aliases, assuming that only main effects and two factor interactions are of interest.(a) 2III6–3 (b) 2IV8–4
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