Questions and Answers of Applied Statistics and Probability

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
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
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
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)
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
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
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
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
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
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
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.
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
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
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
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
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
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.
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
An article in European Food Research and Technology [€œFactorial Design Optimisation of Grape (Vitis vinifera) Seed Polyphenol Extraction€ (2009, Vol. 229(5), pp.
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
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
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
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).
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
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
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
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.
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.
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
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
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
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
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
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
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
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
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
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
An article in Applied Biochemistry and Biotechnology (€œA Statistical Approach for Optimization of Polyhydroxybutyrate Production by Bacillus sphaericus NCIM 5149 under Submerged Fermentation
An article in Analytical Biochemistry [€œApplication of Central Composite Design for DNA Hybridization Onto Magnetic Microparticles,€ (2009, Vol. 391(1), 2009, pp.
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
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
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 +
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
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
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
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
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
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
An article in Journal of Hazardous Materials [€œStatistical Factor-Screening and Optimization in Slurry Phase Bioremediation of 2,4,6-trinitrotoluene Contaminated Soil,€ (2011,
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€
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
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
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
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
Suppose that in Exercise 14-16 only a ¼ fraction of the 25design could be run. Construct the design and analyze the data that are obtained by selecting only the response for the eight runs in your
An article in Industrial and Engineering Chemistry [€œMore on Planning Experiments to Increase Research Efficiency€ (1970, pp. 60€“65)] uses a 25€“2 design to
An article in Quality Engineering [€œA Comparison of Multi-Response Optimization: Sensitivity to Parameter Selection€ (1999, Vol. 11, pp. 405€“415)] conducted a half
Montgomery (2012) described a 24€“1fractional factorial design used to study four factors in a chemical process. The factors are A = temperature, B = pressure, C = concentration, and D =
An article by L. B. Hare [€œIn the Soup: A Case Study to Identify Contributors to Filling Variability,€ Journal of Quality Technology 1988 (Vol. 20, pp. 36€“43)]
Suppose that in Exercise 14-22 it was possible to run only a 12 fraction of the 24design. Construct the design and use only the data from the eight runs you have generated to perform the
Consider the problem in Exercise 14-19. Suppose that only half of the 32 runs could be made.(a) Choose the half that you think should be run.(b) Write out the alias relationships for your design.(c)
An article in Journal of Hazardous Materials [€œBiosorption of Reactive Dye Using Acid-Treated Rice Husk: Factorial Design Analysis€ (2007, Vol. 142(1), pp. 397€“403)]
Consider the earth-moving experiment in Exercise 14-34. The experiment actually used two different operators with the production in columns 1 and 3 from operator 1 and 2, respectively. Analyze the
An article in Advanced Semiconductor Manufacturing Conference (ASMC) (May 2004, pp. 325€“29) stated that dispatching rules and rework strategies are two major operational elements that
Consider the following computer output from a single replicate of a 24experiment in two blocks with ABCD confounded.(a) Comment on the value of blocking in this experiment.(b) What effects were used
An article in Industrial and Engineering Chemistry [€œFactorial Experiments in Pilot Plant Studies€ (1951, pp. 1300€“1306)] reports on an experiment to investigate the effect of
An article in Quality Engineering [€œDesigned Experiment to Stabilize Blood Glucose Levels€ (1999€“2000, Vol. 12, pp. 83€“87)] reported on an experiment to
Consider the 26 factorial design. Set up a design to be run in four blocks of 16 runs each. Show that a design that confounds three of the four-factor interactions with blocks is the best possible
Construct a 25 design in four blocks. Select the appropriate effects to confound so that the highest possible interactions are confounded with blocks.
Consider the data from the first replicate of Exercise 14-15, assuming that four blocks are required. Confound ABD and ABC (and consequently CD) with blocks.(a) Construct a design with four blocks of
Construct a 25 design in two blocks. Select the ABCDE interaction to be confounded with blocks.
Consider the data from Exercise 14-20.(a) Construct the design that would have been used to run this experiment in two blocks of eight runs each.(b) Analyze the data and draw conclusions.Exercise
Consider the data from the first replicate of Exercise 14-15.(a) Construct a design with two blocks of eight observations each with ABCD confounded.(b) Analyze the data.Exercise 14-15Four factors are
Consider the data from the first replicate of Exercise 14-14.(a) Suppose that these observations could not all be run under the same conditions. Set up a design to run these observations in two
The book Using Designed Experiments to Shrink Health Care Costs [1997, ASQ Quality Press] presented a case study of an unreplicated 25factorial design to investigate the effect of five factors on the
An article in Journal of Construction Engineering and Management (€œAnalysis of Earth-Moving Systems Using Discrete€”Event Simulation,€ 1995, Vol. 121(4), pp. 388€“396) considered a
An article in Analytica Chimica Acta [€œDesign-of- Experiment Optimization of Exhaled Breath Condensate Analysis Using a Miniature Differential Mobility Spectrometer (DMS)€
An article in Bioresource Technology (€œInfluence of Vegetable Oils Fatty-Acid Composition on Biodiesel Optimization,€ (2011, Vol. 102(2), pp. 1059€“1065)] described an
Consider the following computer output for one replicate of a 24factorial experiment.(a) What effects are used to estimate error?(b) Calculate the entries marked with €œ?€ in the
Consider the following computer output for a 23 factorial experiment.(a) How many replicates were used in the experiment?(b) Use the appropriate equation to calculate the standard error of a
An experiment to study the effect of machining factorson ceramic strength was described atwww.itl.nist.gov/div898/handbook/. Five factors were considered at two levels each: A = Table Speed, B = Down
An article in Talanta (2005, Vol. 65, pp. 895€“899) presented a 23factorial design to find lead level by using flame atomic absorption spectrometry (FAAS). The data are in the following table.The
Consider the experiment in Exercise 14-19. Suppose that a center point with five replicates is added to the factorial runs and the responses are 45, 40, 41, 47, and 43.(a) Estimate the experimental
Consider the experiment in Exercise 14-16. Suppose that a center point with five replicates is added to the factorial runs and the responses are 2800, 5600, 4500, 5400, 3600. Compute an ANOVA with
Consider a 22 factorial experiment with four center points. The data are (1) = 21, a = 125, b = 154, ab = 352, and the responses at the center point are 92, 130, 98, 152. Compute an ANOVA with the
A two-level factorial experiment in four factors was conducted by Chrysler and described in the article €œSheet Molded Compound Process Improvement€ by P. I. Hsieh and D. E.
The following data represent a single replicate of a 25 design that is used in an experiment to study the compressive strength of concrete. The factors are mix (A), time (B), laboratory (C),
An experiment has run a single replicate of a 24design and calculated the following factor effects:(a) Construct a normal probability plot of the effects.(b) Identify a tentative model, based on the
A 24factorial design was run in a chemical process. The design factors are A = time, B = concentration, C = pressure, and D = temperature. The response variable is yield. The data follow:(a) Estimate
Consider the data from Exercise 14-14. Suppose that the data from the second replicate were not available. Analyze the data from replicate I only and comment on your findings.Exercise 14-14An
An experiment described by M. G. Natrella in the National Bureau of Standards€™ Handbook of Experimental Statistics (1963, No. 91) involves flame-testing fabrics after applying fire-retardant

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