Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying Ye - Solutions

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A random sample X1,...,Xn comes from a uniform distribution (see Section 6.1) population U(0, θ) with unknown θ. The data are given below:
A random variable X follows an exponential distribution with mean 1/β. Assume the prior distribution of β is another exponential distribution with mean 2.5. Determine the Bayes estimate of β under the absolute-error loss function.
A random variable X follows a negative binomial distribution with parameters k = 5 and p [i.e., b∗(x; 5, p)]. Furthermore, we know that p follows a uniform distribution on the interval (0, 1). Find the Bayes estimate of p under the squared-error loss function.
Suppose that a sample consisting of 5, 6, 6, 7, 5, 6, 4, 9, 3, and 6 comes from a Poisson population with mean λ. Assume that the parameter λ follows a gamma distribution with parameters (3, 2). Under the squared-error loss function, find the Bayes estimate ofλ.
Suppose that the time to failure T of a certain hinge is an exponential random variable with probability density f(t) = θe−θt, t> 0.From prior experience we are led to believe that θ is a value of an exponential random variable with probability densityπ(θ)=2e−2θ, θ> 0.If we have a sample
Suppose that in Example 18.7 the electrical firm does not have enough prior information regarding the population mean length of life to be able to assume a normal distribution for μ. The firm believes, however, that μ is surely between 770 and 830 hours, and it is thought that a more realistic
The mathematics department of a large university is designing a placement test to be given to incoming freshman classes. Members of the department feel that the average grade for this test will vary from one freshman class to another. This variation of the average class grade is expressed
The daily profit from a juice vending machine placed in an office building is a value of a normal random variable with unknown mean μ and variance σ2.Of course, the mean will vary somewhat from building to building, and the distributor feels that these average daily profits can best be described
The burn time for the first stage of a rocket is a normal random variable with a standard deviation of 0.8 minute. Assume a normal prior distribution forμ with a mean of 8 minutes and a standard deviation of 0.2 minute. If 10 of these rockets are fired and the first stage has an average burn time
The developer of a new condominium complex claims that 3 out of 5 buyers will prefer a two-bedroom unit, while his banker claims that it would be more correct to say that 7 out of 10 buyers will prefer a twobedroom unit. In previous predictions of this type, the banker has been twice as reliable as
A previous study indicates that the percentage of chain smokers, p, who have lung cancer follows a beta distribution (see Section 6.8) with mean 70% and standard deviation 10%. Suppose a new data set collected shows that 81 out of 120 chain smokers have lung cancer.(a) Determine the posterior
Service calls come to a maintenance center according to a Poisson process with λ calls per minute.A data set of 20 one-minute periods yields an average of 1.8 calls. If the prior for λ follows an exponential distribution with mean 2, determine the posterior distribution of λ.
Repeat Exercise 18.2 when 1 of the next 4 drinks overflows and the uniform prior distribution isπ(p) = 10, 0.05
Let us assume that the prior distribution for the proportion p of drinks from a vending machine that overflow is p 0.05 0.10 0.15π(p) 0.3 0.5 0.2 If 2 of the next 9 drinks from this machine overflow, find(a) the posterior distribution for the proportion p;(b) the Bayes estimate of p.
Estimate the proportion of defectives being produced by the machine in Example 18.1 if the random sample of size 2 yields 2 defectives.
A quality control effort is being undertaken for a process where large steel plates are manufactured and surface defects are of concern. The goal is to set up a quality control chart for the number of defects per plate. The data are given below. Set up the appropriate control chart, using this
For the situation of Review Exercise 17.8, suppose that additional data are collected as follows:Does the process appear to be in control? Explain.
Samples of size 50 are taken every hour from a process producing a certain type of item that is considered either defective or not defective. Twenty samples are taken.(a) Construct a control chart for control of proportion defective.(b) Does the process appear to be in control? Explain.
Consider the situation of Example 17.2. The following data are taken on additional samples of size 5.Plot the X¯- and S-values on the X¯- and S-charts that were produced with the data in the preliminary sample. Does the process appear to be in control? Explain why or why not.
Consider an in-control process with mean μ = 25 and σ = 1.0. Suppose that subgroups of size 5 are used with control limits μ ± 3σ/√n, and centerline atμ. Suppose that a shift occurs in the mean, and the new mean is μ = 26.5.(a) What is the average number of samples required(following the
Consider the data of Table 17.1. Suppose that additional samples of size 5 are taken and tensile strength recorded. The sampling produces the following results (in pounds per square inch)
For the situation of Review Exercise 17.2, give numerical estimates of the mean and standard devia-tion of the diameter for the part being manufactured in the process.
Suppose for Review Exercise 17.2. that the buyer has set specifications for the part. The specifications require that the diameter fall in the range covered by 2.40000 ± 0.0100 mm. What proportion of units produced by this process will not conform to specifications?
Consider the following data taken on subgroups of size 5. The data contain 20 averages and ranges on the diameter (in millimeters) of an important component part of an engine. Display X¯- and R-charts. Does the process appear to be in control?
Consider X1, X2,...,Xn independent Poisson random variables with parameters μ1, μ2,...,μn. Use the properties of moment-generating functions to show that the random variable n i=1 Xi is a Poisson random variable with mean n i=1μi and variance n i=1μi.
Use the data from Exercise 13.14 on page 530 to see if the median amount of nitrogen lost in perspiration is different for the three levels of dietary protein.
In Review Exercise 13.45 on page 555, use the Kruskal-Wallis test, at the 0.05 level of significance, to determine if the chemical analyses performed by the four laboratories give, on average, the same results.
A study by a chemical company compared the drainage properties of two different polymers. Ten different sludges were used, and both polymers were allowed to drain in each sludge. The free drainage was measured in mL/min.(a) Use the sign test at the 0.05 level to test the null hypothesis that
In the article called “Risky Assumptions” by Paul Slovic, Baruch Fischoff, and Sarah Lichtenstein, published in Psychology Today (June 1980), the risk of dying in the United States from 30 activities and technologies is ranked by members of the League of Women Voters and also by experts who are
Two judges at a college homecoming parade rank eight floats in the following order:Float 12345678 Judge A 58436271 Judge B 75428163(a) Calculate the rank correlation coefficient.(b) Test the null hypothesis that ρ = 0 against the alternative that ρ > 0. Use α = 0.05.
A consumer panel tests nine brands of microwave ovens for overall quality. The ranks assigned by the panel and the suggested retail prices are as follows:
The following table gives the recorded grades for 10 students on a midterm test and the final examination in a calculus course:
What is the probability that at least 95% of a population will exceed the smallest value in a random sample of size n = 135?
How large a sample is required to be 99% confident that at least 80% of the population will be less than the largest observation in the sample?
What is the probability that the range of a random sample of size 24 includes at least 90% of the population?
How large a sample is required to be 95% confident that at least 85% of the distribution of measurements is included between the sample extremes?
Assuming that the measurements of Exercise 1.14 on page 30 were recorded successively from left to right as they were collected, use the runs test, withα = 0.05, to test the hypothesis that the data represent a random sequence.
In an industrial production line, items are inspected periodically for defectives. The following is a sequence of defective items, D, and nondefective items, N, produced by this production line:DDNNNDNNDDNNNN NDDDNNDNNNNDND Use the large-sample theory for the runs test, with a significance level of
Use the runs test to test, at level 0.01, whether there is a difference in the average operating time for the two calculators of Exercise 16.17 on page 670.
A silver-plating process is used to coat a certain type of serving tray. When the process is in control, the thickness of the silver on the trays will vary randomly following a normal distribution with a mean of 0.02 millimeter and a standard deviation of 0.005 millimeter. Suppose that the next 12
A random sample of 15 adults living in a small town were selected to estimate the proportion of voters favoring a certain candidate for mayor. Each individual was also asked if he or she was a college graduate. By letting Y and N designate the responses of “yes” and“no” to the education
In Exercise 13.6 on page 519, use the KruskalWallis test at the 0.05 level of significance to determine if the organic chemical solvents differ significantly in sorption rate.
The following data represent the operating times in hours for three types of scientific pocket calculators before a recharge is required:Use the Kruskal-Wallis test, at the 0.01 level of significance, to test the hypothesis that the operating times for all three calculators are equal.
The following data represent the weights, in kilograms, of personal luggage carried on various flights by a member of a baseball team and a member of a basketball team.Use the rank-sum test with α = 0.05 to test the null hypothesis that the two athletes carry the same amount of luggage on the
From a mathematics class of 12 equally capable students using programmed materials, 5 students are selected at random and given additional instruction by the teacher. The results on the final examination are as follows:Grade Additional Instruction 87 69 78 91 80 No Additional Instruction 75 88 64
A fishing line is being manufactured by two processes. To determine if there is a difference in the mean breaking strength of the lines, 10 pieces manufactured by each process are selected and then tested for breaking strength. The results are as follows:Use the rank-sum test with α = 0.1 to
The following data represent the number of hours that two different types of scientific pocket calculators operate before a recharge is required.Use the rank-sum test with α = 0.01 to determine if calculator A operates longer than calculator B on a full battery charge.
To find out whether a new serum will arrest leukemia, nine patients, who have all reached an advanced stage of the disease, are selected. Five patients receive the treatment and four do not. The survival times, in years, from the time the experiment commenced are Use the rank-sum test, at the 0.05
A cigarette manufacturer claims that the tar content of brand B cigarettes is lower than that of brand A cigarettes. To test this claim, the following determinations of tar content, in milligrams, were recorded:Brand A 1 12 9 13 11 14 Brand B 8 10 7 Use the rank-sum test with α = 0.05 to test
Rework Exercise 16.6 by using the signed-rank test.
Rework Exercise 16.7 by using the signed-rank test.
The following are the numbers of prescriptions filled by two pharmacies over a 20-day period: Use the signed-rank test at the 0.01 level of significance to determine whether the two pharmacies, on average, fill the same number of prescriptions against the alternative that pharmacy A fills more
Rework Exercise 16.5 by using the signed-rank test.
The weights of 5 people before they stopped smoking and 5 weeks after they stopped smoking, in kilograms, are as follows Use the signed-rank test for paired observations to test the hypothesis, at the 0.05 level of significance, that giving up smoking has no effect on a person’s weight against
Analyze the data of Exercise 16.2 by using the signed-rank test.
Analyze the data of Exercise 16.1 by using the signed-rank test.
The following figures give the systolic blood pressure of 16 joggers before and after an 8-kilometer run:Use the sign test at the 0.05 level of significance to test the null hypothesis that jogging 8 kilometers increases the median systolic blood pressure by 8 points against the alternative that
Two types of instruments for measuring the amount of sulfur monoxide in the atmosphere are being compared in an air-pollution experiment. The following readings were recorded daily for a period of 2 weeks:Using the normal approximation to the binomial distribution, perform a sign test to determine
It is claimed that a new diet will reduce a person’s weight by 4.5 kilograms, on average, in a period of 2 weeks. The weights of 10 women were recorded before and after a 2-week period during which they followed this diet, yielding the following data:
A paint supplier claims that a new additive will reduce the drying time of its acrylic paint. To test this claim, 12 panels of wood were painted, one-half of each panel with paint containing the regular additive and the other half with paint containing the new additive.The drying times, in hours,
A food inspector examined 16 jars of a certain brand of jam to determine the percent of foreign impurities. The following data were recorded:Using the normal approximation to the binomial distribution, perform a sign test at the 0.05 level of significance to test the null hypothesis that the median
The following data represent the number of hours of flight training received by 18 student pilots from a certain instructor prior to their first solo flight:9 12 18 14 12 14 12 10 16 11 9 11 13 11 13 15 13 14 Using binomial probabilities from Table A.1, perform a sign test at the 0.02 level of
The following data represent the time, in minutes, that a patient has to wait during 12 visits to a doctor’s office before being seen by the doctor:17 15 20 20 32 28 12 26 25 25 35 24 Use the sign test at the 0.05 level of significance to test the doctor’s claim that the median waiting time for
Consider a design which is a 23−1 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 ) = β0 + β1x1 + β2x2 + β3x3(a) Give (and verify) E(¯yf − y¯0).(b) Explain
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.
Verify that your design in Review Exercise 15.43 is indeed resolution IV.
Use Table 15.16 to construct a 16-run design with 8 factors that is resolution IV.
In the study “The Use of Regression Analysis for Correcting Matrix Effects 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
A 22 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 the
A large petroleum company in the Southwest regularly conducts experiments to test additives to drilling fluids. Plastic viscosity is a rheological measure reflecting the thickness of the fluid. Various polymers are added to the fluid to increase viscosity. The following is a data set in which two
A Plackett-Burman design was used to study the rheological properties of high-molecular-weight copolymers. Two levels of each of six variables were fixed in the experiment. The viscosity of the polymer is the response. The data were analyzed by the Statistics Consulting Center at Virginia Tech for
Construct a Plackett-Burman design for 10 variables containing 24 experimental runs.
Consider the 1 16 of the 27 factorial discussed in Section 15.9. List the additional 11 defining contrasts.
A 25 factorial plan is used to build a regression model containing first-order coefficients 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 fit, and pure error.
Use the coal cleansing data of Exercise 15.2 on page 609 to fit a model of the type E(Y ) = β0 + β1x1 + β2x2 + β3x3, where the levels are x1, percent solids: 8, 12 x2, flow rate: 150, 250 gal/min x3, pH: 5, 6 Center and scale the variables to design units. Also conduct a test for lack of fit,
In Case Study 15.2 involving the coal cleans- ing 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 varianceσ2 z . The response, cleansing efficiency, has a mean and variance that change behavior during
Consider Case Study 15.1 involving the injection molding data. Suppose mold temperature is difficult to control and thus it can be assumed that in the process it follows a normal distribution with mean 0 and variance σ2 z . Of concern is the variance of the shrinkage response in the process
Consider the following 23 factorial with control variables x1 and x2 and noise variable z. Can x1 and x2 be chosen at levels for which Var(y) is minimized?Explain why or why not.
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 between −1 and +1. The
In Myers, Montgomery, and Anderson-Cook(2009), an application is discussed in which an engineer is concerned with the effects on the cracking of a titanium alloy. The three factors are A, temperature;B, titanium content; and C, amount of grain refiner.The following table gives a portion of the
For Exercise 15.27, give all aliases for the six main effects.
Consider Exercise 15.27. Construct another 26−3 that is different from the design chosen in Exercise 15.27.
There are six factors and only eight design points can be used. Construct a 26−3 by beginning with a 23 and use D = AB, E = −AC, and F = BC as the generators.
Consider a 25−1 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
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
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
An experiment is conducted so that an engineer can gain insight into the influence 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
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 defining contrasts being ACD, BEF, and CEG. The data are as follows:Perform an analysis of variance on all seven main effects, assuming that interactions
(a) Using the defining contrasts ABCE and ABDF, obtain a 1 4 fraction of a 26 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 interactions are negligible.
Construct a 1 4 fraction of a 26 factorial design using ABCD and BDEF as the defining contrasts.Show what effects are aliased with the six main effects.
(a) Obtain a 1 2 fraction of a 24 factorial design using BCD as the defining 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 effects,
List the aliases for the various effects in a 25 factorial experiment when the defining contrast is ACDE.
Consider Figure 15.14, which represents a 22 factorial with 3 center runs. If quadratic curvature is significant, what additional design points would you select that might allow the estimation of the terms
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 use
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
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
Consider a 25 experiment where the experimental runs are on 4 different machines. Use the machines as blocks, and assume that all main effects and two-factor interactions may be important.(a) Which runs would be made on each of the 4 machines?(b) Which effects are confounded with blocks?
Consider Exercise 15.11. Suppose there was some experimental difficulty 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 With only these runs, we have the signs for contrasts given by Comment. In your comments,
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 effects 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
Consider Exercise 15.5. Compute all 15 effects and do normal probability plots of the effects.(a) Does it appear as if your assumption of negligible three- and four-factor interactions has merit?(b) Are the results of the effect plots consistent with what you communicated about the importance of

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Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying Ye - Solutions

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