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statistics for engineers and scientists

Introduction To Probability And Statistics For Engineers And Scientists 4th Edition Sheldon M. Ross - Solutions

Give a method for generating a random variable having distribution function F (x) = xn, 0< x < 1
Give a method for generating a random variable having density function f (x) = ex /(e − 1), 0< x < 1
Show that the discrete inverse transform algorithm for generating a geometric random variable with parameter p reduces to the following:1. Generate a random number U 2. Set X = Int(log(1−U)log(1−p) ) + 1 Give a second algorithm for generating a geometric random variable with parameter p that
Write an algorithm, similar to what was done in the text to generate a binomial random variable, that uses the discrete inverse transform algorithm to generate a Poisson random variable with mean λ.
Do Problem 16 in Chapter 12 by using a permutation test.Use the normal approximation to approximate the p-value.
Do Problem 13 in Chapter 12 by using a permutation test.Use the normal approximation to approximate the p-value.
A group of 16 mice were exposed to 300 rads of radiation at the age of 5 weeks.The group was then randomly divided into two subgroups. Mice in the first subgroup lived in a normal laboratory environment, whereas those from the second subgroup were raised in a special germ-free environment. The
A baseball player has the reputation of starting slowly at the beginning of a season but then continually improving as the season progresses. Do the following data, which indicate the number of hits he has in consecutive five-game strings of the season, strongly validate the player’s
The following are a student’s weekly exam scores. Do they prove that the student improved (as far as exam score) as the semester progressed?68, 64, 72, 80, 72, 84, 76, 86, 94, 92
Let X1, . . . , X8 be independent and identically distributed random variables with mean μ. LetEstimate p if the values of the Xi are 5, 2, 8, 6, 24, 6, 9, 4. 8 P = P /8 < i=1
Suppose that X1, . . . , Xn is a sample from a distribution whose variance σ2 is unknown. Suppose we are planning to estimate σ2 by the sample variance S2 = ni=1(Xi − ¯X)2/(n −1), and we want to use the bootstrap technique to estimate Var(S2).(a) If n = 2 and X1 = 1 and X2 = 3, what is the
Suppose that we are to observe the independent and identically distributed vectors(X1, Y1), (X2, Y2), . . . , (Xn, Yn), and that we want to use these data to estimateθ ≡ E[X1]/E[Y1].(a) Give an estimator of θ.(b) Explain how you could estimate the mean square error of this estimator.
Another method of generating a random permutation, different from the one given in Example 15.2b, is to successively generate a random permutation of the numbers 1, 2, . . . , n starting with n = 1, then n = 2, and so on. (Of course, the random permutation when n = 1 is 1.) Once we have a random
If x0 = 5, and xn = 3 xn−1 mod 5 find x1, x2, . . . , x10.
To generate a Bernoulli random variable X such that P(X = 1) = p = 1 − P(X = 0)generate a random number U, and set 1, x = { 1); 0, if Up if U> p.
To determine if the weekly sales of DVD players is on a downward trend, the manager of a large electronics store has been tracking such sales for the past 12 weeks, with the following sales figures from week 1 to week 12 (the current week) resulting:22, 24, 20, 18, 16, 14, 15, 15, 13, 17, 12, 14
The following are the PSATmath scores of a random sample of 16 students from a certain school district.522, 474, 644, 708, 466, 534, 422, 480, 502, 655, 418, 464, 600, 412, 530, 564 Use them to estimate(a) the average score of all students in the district;(b) the probability that the estimator of
If U is uniformly distributed on (0, 1), show that −logU has an exponential distribution with mean 1. Now use Equation 14.3.7 and the results of the previous problems to establish Equation 14.5.7.
Let X(i) denote ith smallest of a sample of size n from a continuous distribution function F . Also, let U(i) denote the ith smallest from a sample of size n from a uniform (0, 1) distribution.(a) Argue that the density function of U(i) is given by[Hint : In order for the ith smallest of n uniform
If X is a continuous random variable having distribution function F , show that(a) F (X ) is uniformly distributed on (0, 1);(b) 1 − F (X ) is uniformly distributed on (0, 1).
If U is uniformly distributed on (0, 1) — that is, U is a random number —show that [−(1/α) logU]1/β is a Weibull random variable with parameters(α, β).The next three problems are concerned with verifying Equations 14.5.5 and 14.5.7.
Show that if X is a Weibull random variable with parameters (α, β), then αX β is an exponential random variable with mean 1.
If the following are the sample data from a Weibull population having unknown parameters α and β, determine the least square estimates of these quantities, using either of the methods presented.Data: 15.4, 16.8, 6.2, 10.6, 21.4, 18.2, 1.6, 12.5, 19.4, 17
Show that if X is aWeibull random variable with parameters (α, β), then [[(({ + 1) +1) ( 2 + 1 ) ]], - 2 8/12- Var(X)=a
If X is aWeibull random variable with parameters (α, β), show thatand make the change of variables E[X] ar(1+1/8) == where I'(y) is the gamma function defined by (y) = e-xx-1dx Hint: Write E[X] = tat-1 exp{-at) dt
Suppose that the life distributions of two types of transistors are both exponential.To test the equality of means of these two distributions, n1 type 1 transistors are simultaneously put on a life test that is scheduled to end when there have been a total of r1 failures. Similarly, n2 type 2
The following data represent failure times, in minutes, for two types of electrical insulation subject to a certain voltage stress.Test the hypothesis that the two sets of data come from the same exponential distribution. Type I 212, 88.5, 122.3, 116.4, 125, 132, 66 Type II 34.6, 54, 162, 49, 78,
What is the Bayes estimate of λ = 1/θ in Problem 18 if the prior distribution onλ is exponential with mean 1/30?
In Problem 17, suppose that prior to the testing phase and based on past experience one felt that the value of λ = 1/θ could be thought of as the outcome of a gamma random variable with parameters 1, 100. What is the Bayes estimate of λ?
Suppose that the remission time, in weeks, of leukemia patients that have undergone a certain type of chemotherapy treatment is an exponential random variable having an unknown mean θ. A group of 20 such patients is being monitored and, at present, their remission times are (in weeks) 1.2, 1.8∗,
A testing laboratory has facilities to simultaneously life test 5 components. The lab tested a sample of 10 components from a common exponential distribution by initially putting 5 on test and then replacing any failed component by one still waiting to be tested. The test was designed to end either
Verify that the maximum likelihood estimate corresponding to Equation 14.3.9 is given by Equation 14.3.10.
From a sample of items having an exponential life distribution with unknown mean θ, items are tested in sequence. The testing continues until either the rth failure occurs or after a time T elapses.(a) Determine the likelihood function.(b) Verify that the maximum likelihood estimator of θ is
Using the fact that a Poisson process results when the times between successive events are independent and identically distributed exponential random variables, show that P{X ≥ n} = Fχ2 2n(x)when X is a Poisson random variable with mean x/2 and Fχ2 2n is the chi-square distribution function
A one-at-a-time sequential life testing scheme is scheduled to run for 300 hours.A total of 16 items fail within that time. Assuming an exponential life distribution with unknown mean θ (measured in hours):(a) Determine the maximum likelihood estimate of θ. (b) Test at the 5 percent level of
Vacuum tubes produced at a certain plant are assumed to have an underlying exponential life distribution having an unknown mean θ. To estimate θ it has been decided to put a certain number n of tubes on test and to stop the test at the 10th failure. If the plant officials want the mean length of
Suppose that 20 items are to be put on test that is to be terminated when the 10th failure occurs. If the lifetime distribution is exponential with mean 10 hours, compute the following quantities.(a) The mean length of the testing period.(b) The variance of the testing period.
Suppose 30 items are put on test that is scheduled to stop when the 8th failure occurs. If the failure times are, in hours, .35, .73, .99, 1.40, 1.45, 1.83, 2.20, 2.72, test, at the 5 percent level of significance, the hypothesis that the mean life is equal to 10 hours. Assume that the underlying
Consider a test of H0 : θ = θ0 versus H1 : θ = θ0 for the model of Section 14.3.1. Suppose that the observed value of 2τ/θ0 is v. Show that the hypothesis should be rejected at significance level α whenever α is less than the p-value given by p-value = 2 min(P{χ2 2r < v}, 1 − P{χ2 2r <
When 30 transistorswere simultaneously put on a life test that was to be terminated when the 10th failure occurred, the observed failure times were (in hours) 4.1, 7.3, 13.2, 18.8, 24.5, 30.8, 38.1, 45.5, 53, 62.2. Assume an exponential life distribution.(a) What is the maximum likelihood estimate
For the model of Section 14.3.1, explain how the following figure can be used to show thatwhere j=1
Show that the uniform distribution on (a,b) is an IFR distribution.
A continuous life distribution is said to be an IFR (increasing failure rate) distribution if its failure rate function λ(t ) is nondecreasing in t .(a) Show that the gamma distribution with density f (t ) = λ2te−λt , t > 0 is IFR.(b) Show, more generally, that the gamma distribution with
Suppose the life distribution of an item has failure rate function λ(t ) = t 3, 0
The lung cancer rate of a t -year-old male smoker, λ(t ), is such thatAssuming that a 40-year-old male smoker survives all other hazards, what is the probability that he survives to (a) age 50, (b) age 60, without contracting lung cancer? In the foregoingwe are assuming that he remains a smoker
If X and Y are independent random variables having failure rate functions λx (t )and λy(t ), show that the failure rate function of Z = min(X , Y ) isλz (t ) = λx (t ) + λy(t )
A random variable whose distribution function is given by F (t ) = 1 − exp{−αtβ}, t ≥ 0 is said to have a Weibull distribution with parameters α, β. Compute its failure rate function.
Test the hypothesis, at the 5 percent level of significance, that the lifetimes of items produced at two given plants have the same exponential life distribution if a sample of size 10 from the first plant has a total lifetime of 420 hours whereas a sample of 15 from the second plant has a total
Suppose that 20 items having an exponential life distribution with an unknown rate λ are put on life test at various times. When the test is ended, there have been 10 observed failures — their lifetimes being (in hours) 5, 7, 6.2, 8.1, 7.9, 15, 18, 3.9, 4.6, 5.8. The 10 items that did not fail
A company claims that the mean lifetimes of the semiconductors it produces is at least 25 hours. To substantiate this claim, an independent testing service has decided to sequentially test, one at a time, the company’s semiconductors for 600 hours.If 30 semiconductors failed during this period,
A producer of batteries claims that the lifetimes of the items it manufactures are exponentially distributed with a mean life of at least 150 hours.To test this claim, 100 batteries are simultaneously put on a test that is slated to end when the 20th failure occurs. If, at the end of the
A sample of 50 transistors is simultaneously put on a test that is to be ended when the 15th failure occurs. If the total time on test of all transistors is equal to 525 hours, determine a 95 percent confidence interval for the mean lifetime of a transistor.Assume that the underlying distribution
One often hears that the death rate of a person who smokes is, at each age, twice that of a nonsmoker. What does this mean? Does it mean that a nonsmoker has twice the probability of surviving a given number of years as does a smoker of the same age?
Repeat Problem 18, this time using a cumulative sum control chart with d = 1 and B = 2.49.
Repeat Problem 17, this time using a cumulative sum control chart with(a) d = .25, B = 8;(b) d = .5, B = 4.77.
Explain why a moving-average control chart with span size k must use different control limits for the first k−1moving averages, whereas an exponentiallyweighted moving-average control chart can use the same control limits throughout. [Hint:Argue that Var(Mt ) decreases in t , whereas Var(Wt )
Analyze the data of Problem 18 with an exponential weighted moving-average control chart having α = 29.
Redo Problem 17 by employing an exponential weighted moving average control chart with α = 13.
The data shown below give subgroup averages and moving averages of the values from Problem 17. The span of the moving averages is k = 8. When in control the subgroup averages are normally distributed with mean 50 and variance 5. What can you conclude Xt Mt 50.79806 50.79806 46.21413 48.50609
The following data represent 25 successive subgroup averages and moving averages of span size 5 of these subgroup averages. The data are generated by a process that, when in control, produces normally distributed items having mean 30 and variance 40. The subgroups are of size 4. Would you judge
Surface defects have been counted on 25 rectangular steel plates, and the data are shown below. Set up a control chart. Does the process producing the plates appear to be in statistical control? Plate Number Number of Defects Plate Number 14 Number of Defects 4 10 11 12 2343-25025-7 123 56989 15 16
The following data represent the number of defective chips produced on the last 15 days: 121, 133, 98, 85, 101, 78, 66, 82, 90, 78, 85, 81, 100, 75, 89. Would you conclude that the process has been in control throughout these 15 days? What control limits would you advise using for future production?
Suppose that when a process is in control each item will be defective with probability.04. Suppose that your control chart calls for taking daily samples of size 500.What is the probability that, if the probability of a defective item should suddenly shift to .08, your control chart would detect
The following data represent the results of inspecting all personal computers produced at a given plant during the past 12 daysDoes the process appear to have been in control? Determine control limits for future production. Day Number of Units Number Defective 1 80 5 234567 110 7 90 4 4 80 9 100 12
The following data present the number of defective bearing and seal assemblies in samples of size 100.Does it appear that the process was in control throughout? If not, determine revised control limits if possible. Number Sample Number of Defectives Sample Number Number of Defectives 1234567890
Samples of n = 6 items are taken from a manufacturing process at regular intervals.A normally distributed quality characteristic is measured, and X and S values are calculated for each sample. After 50 subgroups have been analyzed, we have(a) Compute the control limit for the X - and S-control
The following data refer to the amounts by which the diameters of 14 inch ball bearings differ from 14 inch in units of .001 inches. The subgroup size is n = 5.(a) Set up trial control limits for X - and S-control charts.(b) Does the process appear to have been in control throughout the
Control charts for X and S are maintained on resistors (in ohms).The subgroup size is 4. The values of X and S are computed for each subgroup. After 20 subgroups, X i = 8,620 andSi = 450.(a) Compute the values of the limits for the X and S charts.(b) Estimate the value of σ on the assumption
Control charts for X and S are maintained on the shear strength of spot welds.After 30 subgroups of size 4,X i = 12,660 andSi = 500. Assume that the process is in control.(a) What are the X -control limits?(b) What are the S-control limits?(c) Estimate the standard deviation for the process.(d)
The following are X and S values for 20 subgroups of size 5.(a) Determine trial control limits for an X -control chart.(b) Determine trial control limits for an S-control chart.(c) Does it appear that the process was in control throughout?(d) If your answer in part (c) is no, suggest values for
In Problem 4, determine the control limits for an S-control chart.
Determine the revised X - and S-control limits for the data in Example 13.3a.
Samples of size 5 are taken at regular intervals from a production process, and the values of the sample averages and sample standard deviations are calculated.Suppose that the sum of the X and S values for the first 25 samples are given by(a) Assuming control, determine the control limits for an X
If Y has a chi-square distribution with n − 1 degrees of freedom, show thatNow make the transformation x = y/2.) (Hint: Write E[Y] = 2 (n/2) T[(n - 1)/2] E[] = (1) dy 0 = 6.00 e-y/2y(n-1)/2-1 dy (n-1) 2(-1)/2 2 00 = e-y/2yn/2-1 dy (n-1) 2(n-1)/2 2
Suppose that a process is in control with μ = 14 and σ = 2. An X -control chart based on subgroups of size 5 is employed. If a shift in the mean of 2.2 units occurs, what is the probability that the next subgroup average will fall outside the control limits? On average, how many subgroups will
A repair shop will send a worker to a caller’s home to repair electronic equipment. Upon receiving a request, it dispatches a worker who is instructed to call in when the job is completed. Historical data indicate that the time from when the server is dispatched until he or she calls is a normal
The following data represent the number of defects discovered at a factory on successive units of 10 cars each.Does it appear that the production process was in control throughout? Cars Defects Cars Defects Cars Defects Cars Defects 12345 141 162 150 819 74 11 63 16 68 85 12 74 17 95 95 13 103 18
Let us reconsider Example 13.2a under the new supposition that the process is just beginning and so μ and σ are unknown. Also suppose that the sample standard deviations were as follows:Since X = 3.067, S = .122, c(4) = .9213, the control limits areSince all the X i fall within these limits, we
A manufacturer produces steel shafts having diameters that should be normally distributed with mean 3 mm and standard deviation .1 mm. Successive samples of four shafts have yielded the following sample averages in millimeters.What conclusion should be drawn? Sample X Sample X 1 3.01 2 2.97 3. 3.12
The following table (taken from Quinn, W. H., Neal, T. V., and Antu˜nez de Mayolo, S. E., 1987, “ElNi˜no occurrences over the past four-and-a-half centuries,”Journal of Geophysical Research, 92 (C13), pp. 14,449–14,461) gives the years and magnitude (either moderate or strong) of major El
Can we use the runs test if we consider whether each data value is less than or greater than some predetermined value rather than the value s-med?
The following data represent the successive quality levels of 25 articles: 100, 110, 122, 132, 99, 96, 88, 75, 45, 211, 154, 143, 161, 142, 99, 111, 105, 133, 142, 150, 153, 121, 126, 117, 155. Does it appear that these data are a random sample from some population?
A production run of 50 items resulted in 11 defectives, with the defectives occurring on the following items (where the items are numbered by their order of production): 8, 12, 13, 14, 31, 32, 37, 38, 40, 41, 42. Can we conclude that the successive items did not constitute a random sample?
The m sample problem: Consider m independent random samples of respective sizes n1, . . . , nm from the respective population distributions F1, . . . , Fm, and consider the problem of testing H0 : F1 = F2 = · · · = Fm. To devise a test, let Ri denote the sum of the ranks of the ni elements of
In a 10-year study of the dispersal patterns of beavers (Sun, L. andMuller-Schwarze, D., “Statistical resampling methods in biology: A case study of beaver dispersal patterns,” American Journal of Mathematical and Management Sciences, 16, pp. 463–502, 1996) a total of 332 beavers were trapped
In a 1943 experiment (Whitlock, H. V., and Bliss, D. H., “A bioassay technique for antihelminthics,” Journal of Parasitology, 29, pp. 48–58, 10), albino rats were used to study the effectiveness of carbon tetrachloride as a treatment for worms.Each rat received an injection of worm larvae.
The following are the burning times in seconds of floating smoke pots of two different types:We are interested in testing the hypothesis that the burning time distributions are the same.(a) Determine the exact p-value.(b) Determine the p-value yielded by the normal approximation.(c) Run a
Determine the p-value in Problem 13 by(a) using the normal approximation;(b) using a simulation study.
Fifteen cities, of roughly equal size, are chosen for a traffic safety study. Eight of them are randomly chosen, and in these cities a series of newspaper articles dealing with traffic safety is run over a 1-month period. The number of traffic accidents reported in the month following this campaign
In a study of bilingual coding, 12 bilingual (French and English) college students are divided into two groups. Each group reads an article written in French, and each answers a series of 25 multiple-choice questions covering the content of the article. For one group the questions are written in
Let X1, . . . , Xn be a sample from the continuous distribution F having median m;and suppose we are interested in testing the hypothesis H0 : m = m0 against the one-sided alternative H1 : m > m0. Present the one-sided analog of the signed rank test. Explain how the p-value would be computed.
Ten pairs of duplicate spectrochemical determinations for nickel are presented below.The readings in column 2were taken with one type of measuring instrument and those in column 3 were taken with another type.Test the hypothesis, at the 5 percent level of significance, that the two measuring
An engineer claims that painting the exterior of a particular aircraft affects its cruising speed. To check this, the next 10 aircraft off the assembly line were flown to determine cruising speed prior to painting, and were then painted and reflown.The following data resulted.Do the data uphold the
Twelve patients having high albumin content in their blood were treated with a medicine. Their blood content of albumin was measured before and after treatment.The measured values are shown in the table.Is the effect of the medicine significant at the 5 percent level?(a) Use the sign test.(b) Use
Determine the p-value when using the signed rank statistic in Problems 1 and 2.
An experiment was initiated to study the effect of a newly developed gasoline detergent on automobile mileage. The following data, representing mileage per gallon before and after the detergent was added for each of eight cars, resulted.Find the p-value of the test of the hypothesis that mileage is
In 2004, the national median salary of all U.S. financial accountants was $124,400.A recent random sample of 14 financial accountants showed 2007 incomes of (in units of $1,000)125.5, 130.3, 133.0, 102.6, 198.0, 232.5, 106.8, 114.5, 122.0, 100.0, 118.8, 108.6, 312.7, 125.5 Use these data to test
To test the hypothesis that the median weight of 16-year-old females from Los Angeles is at least 110 pounds, a random sample of 200 such females was chosen. If 120 females weighed less than 110 pounds, does this discredit the hypothesis? Use the 5 percent level of significance. What is the p-value?
The published figure for the median systolic blood pressure of middle-aged men is 128. To determine if there has been any change in this value, a random sample of 100 men has been selected. Test the hypothesis that the median is equal to 128 if(a) 60 men have readings above 128;(b) 70 men have
An engineering firm is involved in selecting a computer system, and the choice has been narrowed to two manufacturers. The firm submits eight problems to the two computer manufacturers and has each manufacturer measure the number of seconds required to solve the design problem with the

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