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

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18.15 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:0.13, 1.06, 1.65, 1.73, 0.95, 0.56, 2.14, 0.33, 1.22, 0.20, 1.55, 1.18, 0.71, 0.01, 0.42, 1.03, 0.43, 1.02, 0.83, 0.88 Suppose the prior
18.14 A random variable X follows an exponential distribution with mean 1/β. Assume that the prior distribution of β is another exponential distribution with mean 4, determine the Bayes estimate of β under the squared-error loss function.
18.13 A random variable X follows a negative binomial distribution with parameters k = 8 and p[or, b∗(x; 8, p)]. Furthermore, we know that p follows a uniform distribution at the interval (0, 1). Find the Bayes estimate of p under the squared-error loss function.
18.12 Suppose that a sample consisting of the values 7, 6, 6, 8, 7, 6, 5, 9, 4, and 6 comes from a Poisson population with mean θ. Assume that θ follows a gamma distribution with parameters (4, 3). Under the squared-error loss function, find the Bayes estimate of θ.
18.11 Suppose that the time to failure T of a certain hinge is an exponential random variable with probability density f(t) = θe−θ t, 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
18.10 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
18.9 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
18.8 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
18.7 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
18.6 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
18.5 A previous study indicates that the percentage(p) of frequent users of a certain brand of chewing tobacco who have mouth cancer follows a beta distribution(see Section 6.8), with a mean of 80% and standard deviation of 8%. Suppose a new data set is collected and shows that 70 out of 120 users
18.4 A washing machine maintenance center receives service calls according to a Poisson process, with λ calls per minute. A data set of 30 one-minute periods yields an average of 1.2 calls. If the prior for λ follows an exponential distribution with mean 1, determine the posterior distribution of
18.3 Repeat Exercise 18.2 when 1 of the next 4 drinks overflows and the uniform prior distribution isπ(p) = 10, 0.05 < p < 0.15.
18.2 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.
18.1 Estimate the proportion of defectives being produced by the machine in Example 18.1 if the random sample of size 2 yields 2 defectives.
17.10 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
17.9 For the situation of Review Exercise 17.8, suppose that additional data are collected as follows:Sample Number of Defective Items 1 3 2 4 3 2 4 2 5 3 6 1 7 3 8 5 9 7 10 7 Does the process appear to be in control? Explain.
17.8 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.Number
17.7 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.Sample X¯ Si 1
17.6 Consider an in-control process with μ = 36 andσ = 1.0. Suppose that subgroups of size 6 are used with control limits μ ± 3σ/√n, with the centerline atμ. If a shift occurs in the mean, and the new mean isμ = 37.5.(a) What is the average number of samples required(following the shift)
17.5 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).Sample ¯X R 1 1511 22 2 1508 14 3 1522 11 4 1488 18 5 1519 6 6 1524 11 7 1519 8 8 1504 7 9 1500 8 10 1519
17.4 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.
17.3 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?
17.2 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?Sample ¯X R 1 2.3972 0.0052 2 2.4191 0.0117 3
17.1 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 variablen i=1 Xi is a Poisson random variable with meann i=1μi and variancen i=1μi .
16.41 Use the data from Exercise 13.16 on page 550 to see if the median amount of nitrogen lost in perspiration is different for the three levels of dietary protein.
16.40 In Review Exercise 13.47 on page 575, 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.
16.39 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.Sludge Type Polymer A Polymer B 123456789 10 12.7 14.6 18.6 17.5 11.8
16.38 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
16.37 Two judges at a college homecoming parade rank eight floats in the following order:Float 1 2 3 4 5 6 7 8 Judge A 5 8 4 3 6 2 7 1 Judge B 7 5 4 2 8 1 6 3(a) Calculate the rank correlation coefficient.(b) Test the null hypothesis that ρ = 0 against the alternative that ρ > 0. Use α = 0.05.
16.36 A consumer panel tests nine brands of television for overall quality. The ranks assigned by the panel and the suggested retail prices are as follows:Panel Suggested Manufacturer Rating Price AB CD EF GH I692851743$480 395 575 550 510 545 400 465 420 Is there a significant relationship between
16.35 With reference to the weights and chest sizes of infants in Exercise 11.47 on page 456,(a) calculate the rank correlation coefficient;(b) test the hypothesis, at the 0.025 level of significance, that ρ = 0 against the alternative that ρ > 0.
16.34 Calculate the rank correlation coefficient for the daily rainfall and amount of particulate removed in Exercise 11.13 on page 420.
16.33 With reference to the data of Exercise 11.1 on page 418,(a) calculate the rank correlation coefficient;(b) test the null hypothesis, at the 0.05 level of significance, that ρ = 0 against the alternative thatρ = 0. Compare your results with those obtained in Exercise 11.44 on page 455.
16.32 The following table gives the recorded grades for 10 students on a midterm test and the final examination in a calculus course:Midterm Final Student Test Examination L.S.A.W.P.B.R.W.K.J.R.L.J.K.L.D.L.P.B.L.P.D.W.M.M.N.M.R.H.S.84 98 91 72 86 93 80 092 87 73 63 87 66 78 78 91 088 77(a)
16.31 What is the probability that at least 85% of a population will exceed the smallest value in a random sample of size n = 29?
16.30 How large a sample is required to be 95% confident that at least 99% of the population will be less than the largest observation in the sample?
16.29 What is the probability that the range of a random sample of size 30 includes at least 85% of the population?
16.28 How large a sample is required to be 90%confident that at least 80% of the distribution of a certain measurement is included between the sample extremes?
16.27 Assuming that the measurements of Exercise 1.14 on page 50 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.
16.26 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:D D N N N D N N D D N N N N N D D D N N D N N N N D N D Use the large-sample theory for the runs
16.25 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 690.
16.24 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
16.23 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
16.22 In Exercise 13.2 on page 538, use the Kruskal-Wallis test at the 0.05 level of significance to determine if the organic chemical solvents differ significantly in sorption rate.
16.21 The following data represent the working times, in hours, of three brands of mobile phones before a recharge is required.Brand A 17.2 19.8 16.1 16.4 17.6 Brand B 18.4 18.2 20.2 19.6 18.4 17.6 16.8 Brand C 21.4 22.8 18.6 21.8 20.6 22.6 Use the Kruskal–Wallis test, at the 0.01 level of
16.20 The following data represent the average weights, in kilograms, of the luggage carried by passengers traveling to Asia and Europe.Europe Asia 17.3 16.1 14.1 14.8 13.8 16.9 19.3 13.1 16.1 18.9 16.9 19.3 18.2 12.2 Europe Asia 19.8 13.8 15.2 19.1 12.5 16.9 14.8 15.2 19.1 17.6 18.9 15.7 Europe
16.19 From a science class of 12 equally capable students preparing for an entrance examination,5 students are selected at random and given additional coaching by an expert. Their grades are noted:Grades of students who receive additional coaching 85 75 72 81 90 Grades of students who do not
16.18 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:Process 1 10.4 9.8 11.5 10.0 9.9 9.6
16.17 The following data represent the number of hours that two different types of scientific pocket calculators operate before a recharge is required.Calculator A 5.5 5.6 6.3 4.6 5.3 5.0 6.2 5.8 5.1 Calculator B 3.8 4.8 4.3 4.2 4.0 4.9 4.5 5.2 4.5 Use the rank-sum test with α = 0.01 to determine
16.16 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 Treatment 2.1 5.3 1.4 4.6 0.9
16.15 A bulb manufacturer claims that brand A bulbs last longer than brand B bulbs. To test this claim, the life of both brands of bulbs, per hour, were recorded:Brand A 810 850 760 690 820 830 Brand B 765 720 700 Use the rank-sum test with α = 0.05 to test whether the claim is valid.
16.14 Rework Exercise 16.6 by using the signed-rank test.
16.13 Rework Exercise 16.7 by using the signed-rank test.
16.12 Following are the numbers of typographical errors made by two typists over a 20-day period:Day Typist A Typist B 123456789 10 11 12 13 14 15 16 17 18 19 20 28 35 33 32 41 32 42 32 34 31 29 31 37 35 39 32 25 38 32 34 26 29 30 27 33 35 34 31 28 34 26 25 31 33 33 28 13 37 30 28 Use the
16.11 Rework Exercise 16.5 by using the signed-rank test.
16.10 Consider the results of a clinical trial to test the effectiveness of a sleeping drug in which the sleep patterns of 5 patients were observed. Each patient was given the drug for one night and a placebo for another, and their sleep times, in hours, were recorded.Patients 1 2 3 4 5 Placebo 4 8
16.9 Analyze the data of Exercise 16.2 by using the signed-rank test.
16.8 Analyze the data of Exercise 16.1 by using the signed-rank test.
16.7 The following figures give the systolic blood pressure of 16 joggers before and after an 8-kilometer run:Jogger Before After 123456789 10 11 12 13 14 15 16 158 149 160 155 164 138 163 159 165 145 150 161 132 155 146 159 164 158 163 160 172 147 167 169 173 147 156 164 133 161 154 170 Use the
16.6 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:Sulfur Monoxide Day Instrument A Instrument B 123456789 10 11 12 13 14 0.96 0.82 0.75
16.5 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:Woman Weight Before Weight After
16.4 The aggressiveness scores of 12 sets of identical twins are given below. Use the sign test at the 0.05 level of significance to test the null hypothesis that the Twin I is not less aggressive than Twin II.Twin Aggressiveness Aggressiveness set score of Twin I score of Twin II 123456789 10 11
16.3 A food inspector examined 20 cans of a tuna to determine the percentage of foreign impurities in them.The following data were recorded:1.1 1.3 2.1 1.3 1.4 1.7 1.0 1.4 1.2 1.4 1.7 2.6 1.3 2.1 1.8 1.6 2.5 1.1 1.4 1.2 Using the normal approximation to binomial distribution, perform a sign test at
16.2 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
16.1 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
15.46 Consider a design which is a 23−1 I I I 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+ β11x21+ β22x22+ β33x23.(a) Give (and
15.45 Verify that your design in Review Exercise 15.43 is indeed resolution IV.
15.44 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.
15.43 Use Table 15.16 to construct a 16-run design with 8 factors that is resolution IV.
15.42 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),
15.41 A 22 factorial experiment is analyzed by the Laboratory for Interdisciplinary Statistical Analysis at Virginia Tech. The client is a member of the Department of Apparel, Housing, and Resource Management.The client is interested in comparing cold start to preheating ovens in terms of total
15.40 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 Laboratory for Interdisciplinary Statistical
15.39 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
15.38 Construct a Plackett-Burman design for 10 variables containing 24 experimental runs.
15.37 Consider the 1 16 of the 27 factorial discussed in Section 15.9. List the additional 11 defining contrasts.
15.36 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.z = −1 z = +1 x2 = −1 x2 = +1 x2 = −1 x2 = +1 x1 = −1 x1 = +1 41 63 83 10 5
15.35 Use the coal cleansing data of Exercise 15.2 on page 629 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
15.34 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 varianceσ2 z . The response, cleansing efficiency, has a mean and variance that change behavior
15.33 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
15.32 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.
15.31 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
15.30 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
15.29 For Exercise 15.27, give all aliases for the six main effects.
15.28 Consider Exercise 15.27. Construct another 26−3 that is different from the design chosen in Exercise 15.27.
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.
15.26 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 aliases.
15.25 In the study Durability of Rubber to Steel Adhesively Bonded Joints, conducted at the Department of Environmental Science and Mechanics and analyzed by the Laboratory for Interdisciplinary Statistical Analysis at Virginia Tech, an experimenter measured the number of breakdowns in an adhesive
15.24 In an experiment conducted at the Department of Mechanical Engineering and analyzed by the Laboratory for Interdisciplinary Statistical Analysis 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
15.23 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 12 fraction of a 24 factorial experiment
15.22 Seven factors are varied at two levels in an experiment involving only 16 trials. A 18 fraction of a 27 factorial experiment is used, with the defining 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
15.21 (a) Using the defining contrasts ABCE and ABDF, obtain a 14 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
15.20 List the aliases for the various effects in a 25 factorial experiment when the defining contrast is ACDE.Table 15.15: Analysis of Variance for the Data of a Half-Replicate of a 25 Factorial Experiment Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f Main effect:A B
15.19 (a) Obtain a 12 fraction of a 24 factorial design using BCD as the defining contrast.(b) Divide the 12 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,
15.18 Construct a 14 fraction of a 26 factorial design using ABCD and BDEF as the defining contrasts.Show what effects are aliased with the six main effects.
15.17 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 x21, x22? Explain. x2 -1,1 -1,-1 (0, 0) X1 1,1 1,-1
15.16 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
15.15 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%
15.14 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
15.13 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?

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

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