Probability and Statistics for Engineers and the Scientists 9th Edition solutions

Probability and Statistics for Engineers and the Scientists 9
The following data are obtained from a \(2^3\) factorial experiment replicated three times. Evaluate the sums of squares for all factorial effects by the contrast method. Draw conclusions.

Probability and Statistics for Engineers and the Scientists 9
In an experiment conducted by the Mining Engineering Department at Virginia Tech to study a particular filtering system for coal, a coagulant was added to a solution in a tank containing coal and sludge, which was then placed in a recirculation system in order that the coal could be washed. Three fac

Probability and Statistics for Engineers and the Scientists 9
In a metallurgy experiment, it is desired to test the eect of four factors and their interactions on the concentration (percent by weight) of a particular phosphorus compound in casting material. The variables are A, percent phosphorus in the renement; B, percent remelted material; C, uxing time; an

Probability and Statistics for Engineers and the Scientists 9
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 combinations in order to supply a measure of pure experimental error. Two levels of each factor are used, result

Probability and Statistics for Engineers and the Scientists 9
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 determine whether or not there was a significant difference in the amount of aluminum obtained in an ana

Probability and Statistics for Engineers and the Scientists 9
It is important to study the eect of the 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 15% and 25%. Let the feed rate be factor B, with levels 20 lb/hr and 30 lb/hr. The experiment involve

Probability and Statistics for Engineers and the Scientists 9
Consider Exercise 15.3. It is of interest to the researcher to learn not only that AD, BC, and possibly AB are important, but also what they mean scientifically. Show two-dimensional interaction plots for all three and give an interpretation.

Probability and Statistics for Engineers and the Scientists 9
Consider Exercise 15.3 once again. Three-factor interactions are often not signicant, and even if they are, they are dicult to interpret. The interaction ABD appears to be important. To gain some sense of interpretation, show two AD interaction plots, one for B = 1 and the other for B = +1. From the

Probability and Statistics for Engineers and the Scientists 9
Consider Exercise 15.6. Use a +1 and1 scaling for high and low, respectively, and do a multiple linear regression with the model Yi = 0 + 1x1i + 2x2i + 12x1ix2i + i, with x1i = reactant concentration (1,+1) and x2i = feed rate (1,+1). (a) Compute regression coecients. (b) How do the coecients b1, b2,

Probability and Statistics for Engineers and the Scientists 9
Consider Exercise 15.5. Compute all 15 eects and do normal probability plots of the eects. (a) Does it appear as if your assumption of negligible three- and four-factor interactions has merit? (b) Are the results of the eect plots consistent with what you communicated about the importance of main eec

Probability and Statistics for Engineers and the Scientists 9
In Myers, Montgomery, and Anderson-Cook (2009), a data set is discussed in which a 23 factorial is used by an engineer to study the eects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and duplicates are r

Probability and Statistics for Engineers and the Scientists 9
Consider Exercise 15.11. Suppose there was some experimental diculty in making the runs. In fact, the total experiment had to be halted after only 4 runs. As a result, the abbreviated experiment is given by Life a 43 b 35 c 44 abc 39 With only these runs, we have the signs for contrasts given by A B

Probability and Statistics for Engineers and the Scientists 9
Consider a 25 experiment where the experimental runs are on 4 dierent machines. Use the machines as blocks, and assume that all main eects and two-factor interactions may be important. (a) Which runs would be made on each of the 4 machines? (b) Which eects are confounded with blocks?

Probability and Statistics for Engineers and the Scientists 9
An experiment is described in Myers, Montgomery, and Anderson-Cook (2009) in which optimum conditions are sought for storing bovine semen to obtain maximum survival. The variables are percent sodium citrate, percent glycerol, and equilibration time in hours. The response is percent survival of the mo

Probability and Statistics for Engineers and the Scientists 9
Oil producers are interested in nickel alloys that are strong and corrosion resistant. An experiment was conducted in which yield strengths were compared for nickel alloy tensile specimens charged in solutions of sulfuric acid saturated with carbon disulfide. Two alloys were compared: a 75% nickel co

Probability and Statistics for Engineers and the Scientists 9
Suppose a second replicate of the experiment in Exercise 15.13 could be performed. (a) Would a second replication of the blocking scheme of Exercise 15.13 be the best choice? (b) If the answer to part (a) is no, give the layout for a better choice for the second replicate. (c) What concept did you us

Probability and Statistics for Engineers and the Scientists 9
Consider Figure 15.14, which represents a 22 factorial with 3 center runs. If quadratic curvature is signicant, what additional design points would you select that might allow the estimation of the terms x2 1,x2 2? Explain.

Probability and Statistics for Engineers and the Scientists 9
List the aliases for the various eects in a 25 factorial experiment when the dening contrast is ACDE.

Probability and Statistics for Engineers and the Scientists 9
(a) Obtain a 1 2 fraction of a 24 factorial design using BCD as the dening contrast. (b) Divide the 1 2 fraction into 2 blocks of 4 units each by confounding ABC. (c) Show the analysis-of-variance table (sources of variation and degrees of freedom) for testing all unconfounded main eects, assuming th

Probability and Statistics for Engineers and the Scientists 9
Construct a 1 4 fraction of a 26 factorial design using ABCD and BDEF as the dening contrasts. Show what eects are aliased with the six main eects.

Probability and Statistics for Engineers and the Scientists 9
(a) Using the defining contrasts ABCE and ABDF, obtain a \(\frac{1}{4}\) fraction of a \(2^6\) design. (b) Show the analysis-of-variance table (sources of variation and degrees of freedom) for all appropriate tests, assuming that E and F do not interact and all three-factor and higher interaction

Probability and Statistics for Engineers and the Scientists 9
Seven factors are varied at two levels in an experiment involving only 16 trials. A 1 8 fraction of a 27 factorial experiment is used, with the dening contrasts being ACD, BEF, and CEG. The data are as follows: Treat. Treat. Comb. Response Comb. Response (1) ad abce cdef acef bcde abdf bf 31.6 28.7 3

Probability and Statistics for Engineers and the Scientists 9
An experiment is conducted so that an engineer can gain insight into the inuence of sealing temperature A, cooling bar temperature B, percent polyethylene additive C, and pressure D on the seal strength (in grams per inch) of a bread-wrapper stock. A 1 2 fraction of a 24 factorial experiment is used,

Probability and Statistics for Engineers and the Scientists 9
In an experiment conducted at the Department of Mechanical Engineering and analyzed by the Statistics Consulting Center at Virginia Tech, a sensor detects an electrical charge each time a turbine blade makes one rotation. The sensor then measures the amplitude of the electrical current. Six factors a

Probability and Statistics for Engineers and the Scientists 9
In the 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 Virginia Tech, an experimenter measured the number of breakdowns in an adhesive seal. It was postulated that con

Probability and Statistics for Engineers and the Scientists 9
Consider a 251 design with factors A, B, C, D, and E. Construct the design by beginning with a 24 and use E = ABCD as the generator. Show all aliases.

Probability and Statistics for Engineers and the Scientists 9
There are six factors and only eight design points can be used. Construct a \(2^{6?3}\) by beginning with a \(2^3\) and use D = AB, E = ?AC, and F = BC as the generators.

Probability and Statistics for Engineers and the Scientists 9
Consider Exercise 15.27. Construct another 263 that is dierent from the design chosen in Exercise 15.27.

Probability and Statistics for Engineers and the Scientists 9
For Exercise 15.27, give all aliases for the six main effects.

Probability and Statistics for Engineers and the Scientists 9
In Myers, Montgomery, and Anderson-Cook (2009), an application is discussed in which an engineer is concerned with the eects on the cracking of a titanium alloy. The three factors are A, temperature; B, titanium content; and C, amount of grain rener. The following table gives a portion of the design

Probability and Statistics for Engineers and the Scientists 9
Consider an example in which there are two control variables x1 and x2 and a single noise variable z. The goal is to determine the levels of x1 and x2 that are robust to changes in z, i.e., levels of x1 and x2 that minimize the variance produced in the response y as z moves between1 and +1. The varia

Probability and Statistics for Engineers and the Scientists 9
Consider the following 23 factorial with control variables x1 and x2 and noise variable z. Canx1 and x2 be chosen at levels for which Var(y) is minimized? Explain why or why not. z = 1 z = +1x 2 = 1 x2 = +1 x2 = 1 x2 = +1x 1 = 1x 1 = +1 4 1 6 3 8 3 10 5

Probability and Statistics for Engineers and the Scientists 9
Consider Case Study 15.1 involving the injection molding data. Suppose mold temperature is difcult to control and thus it can be assumed that in the process it follows a normal distribution with mean 0 and variance 2 z. Of concern is the variance of the shrinkage response in the process itself. In t

Probability and Statistics for Engineers and the Scientists 9
In Case Study 15.2 involving the coal cleansing data, the percent solids in the process system is known to vary uncontrollably during the process and is viewed as a noise factor with mean 0 and variance 2 z. The response, cleansing eciency, has a mean and variance that change behavior during the proc

Probability and Statistics for Engineers and the Scientists 9
Use the coal cleansing data of Exercise 15.2 on page 609 to t a model of the type E(Y )=0 + 1x1 + 2x2 + 3x3, where the levels are x1, percent solids: 8, 12 x2, ow rate: 150, 250 gal/min x3, pH: 5, 6 Center and scale the variables to design units. Also conduct a test for lack of t, and comment concer

Probability and Statistics for Engineers and the Scientists 9
A2 5 factorial plan is used to build a regression model containing rst-order coecients and model terms for all two-factor interactions. Duplicate runs are made for each factor. Outline the analysis-of-variance table, showing degrees of freedom for regression, lack of t, and pure error.

Probability and Statistics for Engineers and the Scientists 9
Consider the 1 16 of the 27 factorial discussed in Section 15.9. List the additional 11 dening contrasts.

Probability and Statistics for Engineers and the Scientists 9
Construct a Plackett-Burman design for 10 variables containing 24 experimental runs.

Probability and Statistics for Engineers and the Scientists 9
A Plackett-Burman design was used to study the rheological properties of high-molecular-weight copolymers. Two levels of each of six variables were xed in the experiment. The viscosity of the polymer is the response. The data were analyzed by the Statistics Consulting Center at Virginia Tech for pers

Probability and Statistics for Engineers and the Scientists 9
A large petroleum company in the Southwest regularly conducts experiments to test additives to drilling uids. Plastic viscosity is a rheological measure reecting the thickness of the uid. Various polymers are added to the uid to increase viscosity. The following is a data set in which two polymers a

Probability and Statistics for Engineers and the Scientists 9
A \(2^2\) factorial experiment is analyzed by the Statistics Consulting Center at Virginia Tech. The client is a member of the Department of Housing, Interior Design, and Resource Management. The client is interested in comparing cold start to preheating ovens in terms of total energy delivered to th

Probability and Statistics for Engineers and the Scientists 9
In the study The Use of Regression Analysis for Correcting Matrix Eects in the X-Ray Fluorescence Analysis of Pyrotechnic Compositions, published in the Proceedings of the Tenth Conference on the Design of Experiments in Army Research Development and Testing, ARO-D Report 65-3 (1965), an experiment w

Probability and Statistics for Engineers and the Scientists 9
Use Table 15.16 to construct a 16-run design with 8 factors that is resolution IV.

Probability and Statistics for Engineers and the Scientists 9
Verify that your design in Review Exercise 15.43 is indeed resolution IV.

Probability and Statistics for Engineers and the Scientists 9
Construct a design that contains 9 design points, is orthogonal, contains 12 total runs and 3 degrees of freedom for replication error, and allows for a lack-of-fit test for pure quadratic curvature.

Probability and Statistics for Engineers and the Scientists 9
Consider a design which is a 231 III with 2 center runs. Consider yf as the average response at the design parameter and y0 as the average response at the design center. Suppose the true regression model is E(Y )=0 + 1x1 + 2x2 + 3x3 + 11x2 1 + 22x2 2 + 33x2 3. (a) Give (and verify) E( yf y0). (b)