Applied Statistics and Probability for Engineers 6th Edition solutions

Applied Statistics and Probability for Engineers 6
Set up a \(2_{I I I}^{7-4}\) design using \(D=A B, E=A C, F=B C, \text { and } G=A B C\) 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 \(2_{I I I}^{7-4}\) design with g

Applied Statistics and Probability for Engineers 6
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 (\({ }^{\circ} \mathrm{C}\)). These factors are coded as \(x_{1}\) = (time 12) / 8 and \(x_{2}\) = (tempe

Applied Statistics and Probability for Engineers 6
An article in Quality Engineering [Mean and Variance Modeling with Qualitative Responses: A Case Study (19981999, Vol. 11, pp. 141148)] studied how three active ingredients of a particular food affect the overall taste of the product. The measure of the overall taste is the overall mean liking score

Applied Statistics and Probability for Engineers 6
Consider the first-order model \(\hat{y}=50+1.5 x_{1}-0.8 x_{2}\) where \(-1 \leq x_{i} \leq 1\). Find the direction of steepest ascent. Equation Transcription: Text Transcription: \hat y=50+1.5 x_1-0.8 x_2 -1 \leq x_i \leq 1

Applied Statistics and Probability for Engineers 6
A manufacturer of cutting tools has developed two empirical equations for tool life \(\left(y_{1}\right)\) and tool cost \(\left(y_{2}\right)\). Both models are functions of tool hardness \(\left(x_{1}\right)\) and manufacturing time \(\left(x_{2}\right)\). The equations are \(\hat{y}_{1}=10+5 x_{1

Applied Statistics and Probability for Engineers 6
An article in Tappi (1960, Vol. 43, pp. 3844) 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 cod

Applied Statistics and Probability for Engineers 6
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 manufacturi

Applied Statistics and Probability for Engineers 6
Consider the first-order model \(y=12+1.2 x_{1}-2.1 x_{2}+1.6 x_{3}-0.6 x_{4}\) where \(-1 \leq x_{i} \leq 1\). (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

Applied Statistics and Probability for Engineers 6
An article in the Journal of Materials Processing Technology (1997, Vol. 67, pp. 5561) used response surface meth-odology to generate surface roughness prediction models for turning \(\text { EN } 24 T\) steel (290 BHN). The data are shown in the following table. The factors and levels for the

Applied Statistics and Probability for Engineers 6
An article in Process Engineering (1992, No. 71, pp. 4647) presented a two-factor factorial experiment to investigate the effect of pH and catalyst concentration on prod-uct viscosity (cSt). The data are as follows: (a) Test for main effects and interactions using \(\alpha=0.05\). What are your

Applied Statistics and Probability for Engineers 6
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 th

Applied Statistics and Probability for Engineers 6
An article in the Textile Research Institute Journal (1984, Vol. 54, pp. 171179) 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 vertical burn test

Applied Statistics and Probability for Engineers 6
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 three

Applied Statistics and Probability for Engineers 6
Problem 79SE An article in Solid State Technology (1984, Vol. 29, pp. 281284) 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), prebak

Applied Statistics and Probability for Engineers 6
Problem 80SE An article in the Journal of Coatings Technology (1988, Vol. 60, pp. 2732) 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

Applied Statistics and Probability for Engineers 6
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 cutter. A \(2^4\)

Applied Statistics and Probability for Engineers 6
Construct a \(2_{I V}^{4-1}\) 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. Equation Transcription:

Applied Statistics and Probability for Engineers 6
Construct a \(2_{\mathrm{IV}}^{4-1}\) design 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.

Applied Statistics and Probability for Engineers 6
Construct a \(2_{I V}^{8-4}\) ? design in 16 runs. What are the alias relationships in this design? Equation Transcription: Text Transcription: 2_I V^8-4

Applied Statistics and Probability for Engineers 6
Construct a \(2_{I I I}^{5-2}\) ? design in eight runs. What are the alias relationships in this design? Equation Transcription: Text Transcription: 2_I I I^5-2

Applied Statistics and Probability for Engineers 6
An article in the Journal of Quality Technology (1985, Vol. 17, pp. 198206) 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 temperature, \(B=\) heat

Applied Statistics and Probability for Engineers 6
An article in Rubber Chemistry and Technology (Vol. 47, 1974, pp. 825836) described an experiment to study the effect of several variables on the Mooney viscosity of rubber, including silica filler (parts per hundred) and oil filler (parts per hundred). Data typical of that reported in this experi

Applied Statistics and Probability for Engineers 6
An article in Tropical Science [Proximate Composition of the Seeds of Acacia Nilotica var Adansonti (Bagaruwa) and Extraction of Its Protein (1992, Vol. 32(3), pp. 263268)] reported on research extracting and concentrating the proteins of the bagaruwa seeds in livestock feeding in Nigeria to e

Applied Statistics and Probability for Engineers 6
An article in Plant Disease [Effect of Nitrogen and Potassium Fertilizer Rates on Severity of Xanthomonas Blight of Syngonium Podophyllum (1989, Vol. 73(12), pp. 972975)] showed the effect of the variable nitrogen and potassium rates on the growth of White Butterfly and the mean percentage of disea

Applied Statistics and Probability for Engineers 6
An article in Biotechnology Progress (2001, Vol. 17, pp. 366368) reported on an experiment to investigate and opti-mize the operating conditions of the nisin extraction in aqueous two-phase systems (ATPS). A \(2^{2}\) full factorial design with center points was used to verify the most significant fa

Applied Statistics and Probability for Engineers 6
An article in the Journal of Applied Electrochemistry (May 2008, Vol. 38(5), pp. 583590) presented a \(2^{7-3}\) fractional factorial design to perform optimization of polybenzimidazole-based membrane electrode assemblies for \(\mathrm{H}_{2} / \mathrm{O}_{2}\) fuel cells. The design and data are s

Applied Statistics and Probability for Engineers 6
An article in Biotechnology Progress (December 2002, Vol. 18(6), pp. 11701175) presented a \(2^{7-3}\) fractional 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 do

Applied Statistics and Probability for Engineers 6
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 \(2^{3}\) experiment was run. The re

Applied Statistics and Probability for Engineers 6
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) Calcu

Applied Statistics and Probability for Engineers 6
Consider the following ANOVA table from a two-fac-tor 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 m

Applied Statistics and Probability for Engineers 6
An article in Process Biochemistry (Dec. 1996, Vol. 31(8), pp. 773785) presented a \(2^{7-3}\) fractional 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

Applied Statistics and Probability for Engineers 6
An article in European Food Research and Technology [Factorial Design Optimisation of Grape (Vitis vinifera) Seed Polyphenol Extraction (2009, Vol. 229(5), pp. 731742)] used a central composite design to study the effects of basic factors (time, ethanol, and pH) on the extractability of polyphenolic

Applied Statistics and Probability for Engineers 6
Consider the data in Exercise 14-97. Use only the first eight rows of data from the \(2^{3}\) factorial 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 pre

Applied Statistics and Probability for Engineers 6
Graphically study the assumption of no interactions between blocks and treatments in the previous exercise.

Applied Statistics and Probability for Engineers 6
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 factors

Applied Statistics and Probability for Engineers 6
Consider an unreplicated \(2^{k}\) factorial, 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, as

Applied Statistics and Probability for Engineers 6
Problem 102MEE What blocking scheme would you recommend if it were necessary to run a 24 design in four blocks of four runs each?

Applied Statistics and Probability for Engineers 6
Consider a 22 design in two blocks with AB confounded with blocks. Prove algebraically that SSAB = SSBlocks.

Applied Statistics and Probability for Engineers 6
Problem 104MEE 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, d

Applied Statistics and Probability for Engineers 6
Problem 105MEE Construct a 251 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.

Applied Statistics and Probability for Engineers 6
Construct a \(2_{\mathrm{IV}}^{7-2}\) design. Show how this design may be confounded in four blocks of eight runs each. Are any two-factor interactions confounded with blocks?

Applied Statistics and Probability for Engineers 6
Construct a \(2_{I V}^{7-2}\) 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. Equation Transcription: Text Transcription: 2_I V^7-2

Applied Statistics and Probability for Engineers 6
Problem 109MEE 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.