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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

Suppose A and B are mutually exclusive events.Construct a Venn diagram that contains the three events A, B, and C such that P1A ƒ C2 1 and P1B ƒ C2 0?
If , must A B? Draw a Venn diagram to explain your answer.
A maintenance firm has gathered the following information regarding the failure mechanisms for air conditioning systems:The units without evidence of gas leaks or electrical failure showed other
Three containers are selected, at random, without replacement, from the batch.(a) What is the probability that the third one selected is defective given that the first and second one selected were
Continuation of Exercise
A batch of 500 containers for frozen orange juice contains 5 that are defective. Two are selected, at random, without replacement from the batch.(a) What is the probability that the second one
Suppose three castings are selected at random, without replacement, from the lot of 40. In addition to the definitions of events A and B, let C denote the event that the third casting selected is
Continuation of Exercise
A lot contains 15 castings from a local supplier and 25 castings from a supplier in the next state. Two castings are selected randomly, without replacement, from the lot of 40.Let A be the event that
A lot of 100 semiconductor chips contains 20 that are defective. Two are selected randomly, without replacement, from the lot.(a) What is the probability that the first one selected is defective?(b)
Assume that one wafer is selected at random from this set. Let A denote the event that a wafer contains four or more particles, and let B denote the event that a wafer is from the center of the
Consider the data on wafer contamination and location in the sputtering tool shown in Table
The following table summarizes the analysis of samples of galvanized steel for coating weight and surface roughness:(a) If the coating weight of a sample is high, what is the probability that the
The analysis of shafts for a compressor is summarized by conformance to specifications:(a) If we know that a shaft conforms to roundness requirements, what is the probability that it conforms to
Samples of a cast aluminum part are classified on the basis of surface finish (in microinches) and length measurements. The results of 100 parts are summarized as follows:Let A denote the event that
Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows:Let A denote the event that a disk has high shock
The shafts in Exercise 2-53 are further classified in terms of the machine tool that was used for manufacturing the shaft.(a) If a shaft is selected at random, what is the probability that the shaft
A manufacturer of front lights for automobiles tests lamps under a high humidity, high temperature environment using intensity and useful life as the responses of interest. The following table shows
Cooking oil is produced in two main varieties: monoand polyunsaturated. Two common sources of cooking oil are corn and canola. The following table shows the number of bottles of these oils at a
The analysis of shafts for a compressor is summarized by conformance to specifications.(a) If a shaft is selected at random, what is the probability that the shaft conforms to surface finish
Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows:(a) If a disk is selected at random, what is the
Use the axioms of probability to show the following:(a) For any event E, P1E¿2 1 P1E2(b) P1 2 0 (c) If A is contained in B, then P1A2 P1B2
Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows:Let A denote the event that a sample is
Samples of a cast aluminum part are classified on the basis of surface finish (in microinches) and edge finish. The results of 100 parts are summarized as follows:Let A denote the event that a sample
Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows:Let A denote the event that a disk has high shock
A message can follow different paths through servers on a network. The senders message can go to one of five servers for the first step, each of them can send to five servers at the second step, each
Suppose your vehicle is licensed in a state that issues license plates that consist of three digits (between 0 and 9) followed by three letters (between A and Z). If a license number is selected
A sample preparation for a chemical measurement is completed correctly by 25% of the lab technicians, completed with a minor error by 70%, and completed with a major error by 5%.(a) If a technician
If the last digit of a weight measurement is equally likely to be any of the digits 0 through 9,(a) What is the probability that the last digit is 0?(b) What is the probability that the last digit is
Orders for a computer are summarized by the optional features that are requested as follows:(a) What is the probability that an order requests at least one optional feature?(b) What is the
An injection-molded part is equally likely to be obtained from any one of the eight cavities on a mold.(a) What is the sample space?(b) What is the probability a part is from cavity 1 or 2?(c) What
A part selected for testing is equally likely to have been produced on any one of six cutting tools.(a) What is the sample space?(b) What is the probability that the part is from tool 1?(c) What is
The sample space of a random experiment is {a,b, c,d, e} with probabilities 0.1, 0.1, 0.2, 0.4, and 0.2, respectively.Let A denote the event {a,b, c}, and let B denote the event{c,d, e}. Determine
Each of the possible five outcomes of a random experiment is equally likely. The sample space is {a,b, c,d, e}.Let A denote the event {a, b}, and let B denote the event{c,d, e}. Determine the
The rise time of a reactor is measured in minutes(and fractions of minutes). Let the sample space for the rise time of each batch be positive, real numbers. Consider the rise times of two batches.
Counts of the Web pages provided by each of two computer servers in a selected hour of the day are recorded.Let A denote the event that at least 10 pages are provided by server 1 and let B denote the
A sample of two printed circuit boards is selected without replacement from a batch. Describe the (ordered)sample space for each of the following batches:(a) The batch contains 90 boards that are not
A sample of two items is selected without replacement from a batch. Describe the (ordered) sample space for each of the following batches:(a) The batch contains the items {a,b, c, d}.(b) The batch
The rise time of a reactor is measured in minutes (and fractions of minutes). Let the sample space be positive, real numbers. Define the events A and B as follows:Describe each of the following
Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows:Let A denote the event that a sample is
Samples of a cast aluminum part are classified on the basis of surface finish (in microinches) and edge finish. The results of 100 parts are summarized as follows:(a) Let A denote the event that a
Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized below:Let A denote the event that a disk has high shock
A sample of three calculators is selected from a manufacturing line, and each calculator is classified as either defective or acceptable. Let A, B, and C denote the events that the first, second, and
Four bits are transmitted over a digital communications channel. Each bit is either distorted or received without distortion. Let Ai denote the event that the ith bit is distorted, i 1, p ,.(a)
In an injection-molding operation, the length and width, denoted as X and Y, respectively, of each molded part are evaluated. Let A denote the event of 48 X 52 centimeters B denote the event of 9
A digital scale is used that provides weights to the nearest gram.(a) What is the sample space for this experiment?Let A denote the event that a weight exceeds 11 grams, let B denote the event that a
Three events are shown on the Venn diagram in the following figure:Reproduce the figure and shade the region that corresponds to each of the following events. A "D" O. C B
Three events are shown on the Venn diagram in the following figure:Reproduce the figure and shade the region that corresponds to each of the following events. A C B
In a magnetic storage device, three attempts are made to read data before an error recovery procedure that repositions the magnetic head is used. The error recovery procedure attempts three
Calls are repeatedly placed to a busy phone line until a connect is achieved.
An order for an automobile can specify either an automatic or a standard transmission, either with or without air-conditioning, and any one of the four colors red, blue, black or white. Describe the
The time until a tranaction service is requested of a computer to the nearest millisecond.
The following two questions appear on an employee survey questionnaire. Each answer is chosen from the fivepoint scale 1 (never), 2, 3, 4, 5 (always).Is the corporation willing to listen to and
In the manufacturing of digital recording tape, electronic testing is used to record the number of bits in error in a 350-foot reel.
Define ºn(µ) Æ 1 p nPniÆ1 Xi1{Xi · µ} for µ 2 [0,1] where E[X] Æ 0 and E£X2¤Æ 1.(a) Show that ºn(µ) is stochastically equicontinuous.(b) Find the stochastic process º(µ) which has
Find conditions under which sample averages of the following functions are stochastically equicontinuous.(a) g (X,µ) Æ Xµ for µ 2 [0,1].(b) g (X,µ) Æ Xµ2 for µ 2 [0,1].(c) g (X,µ) Æ X/µ
Let g (x,µ) Æ 1{x · µ} for µ 2 [0,1] and assume X » F ÆU[0,1]. Let N1(²,F) be L1 packing numbers.(a) Show that N1(²,F) equal the packing numbers constructed with respect to the Euclidean
Using the asymptotic formula (17.9) to calculate standard errors s(x) for bf (x), find an expression which indicates when bf (x)¡2s(x) Ç 0, which means that the asymptotic 95% confidence interval
You increase your sample from n Æ 1000 to n Æ 2000. For univariate density estimation, how does the AIMSE-optimal bandwidth change? If the sample increases fromn Æ 1000 to n Æ 10,000?
You have a sample of wages for 1000 men and 1000 women. You estimate the density functions bfm(x) and bfw(x) for the two groups using the same bandwidth h. You then take the average bf (x) Æ¡
You estimate a density for expenditures measured in dollars, and then re-estimate measuring in millions of dollars, but use the same bandwidth h. How do you expect the density plot to change? What
Suppose f (x) is the uniform density on [0,1]. What does (17.11) suggest should be the optimal bandwidth h? How do you interpret this?
Show that (17.11) minimizes (17.10).Hint: Differentiate (17.10) with respect to h and set to 0. This is the first-order condition for optimization.Solve for h. Check the second-order condition to
If X¤ is a randomvariable with density bf (x) from (17.2), show that(a) E[X¤] Æ Xn.(b) var[X¤] Æb¾2 Åh2.
Take theU[0,µ] density f (x j µ) Æ 1/µ for 0 · x · µ.(a) Find the conjugate prior ¼(µ). (This may be tricky.)(b) Find the posterior ¼(µ j X) given a random sample.(c) Find the posterior
Take the Pareto density f (x j ®) Æ®x®Å1 for x È 1 and ® È 0.(a) Find the conjugate prior ¼(®). (This may be tricky.)(b) Find the posterior ¼(® j X) given a random sample.(c) Find the
Take the Poisson density f (x j ¸) Æe¡¸¸x x!for ¸ È 0.(a) Find the conjugate prior ¼(¸).(b) Find the posterior ¼(¸ j X) given a random sample.(c) Find the posterior meanb¸Bayes.
Take the exponential density f (x j ¸) Æ 1¸ exp¡¡x¸¢for ¸ È 0.(a) Find the conjugate prior ¼(¸).(b) Find the posterior ¼(¸ j X) given a random sample.(c) Find the posterior meanb¸Bayes.
Let p be the probability of your textbook author Bruce making a free throw shot.(a) Consider the prior ¼¡p¢Æ beta(1,1). (Beta with ® Æ 1 and ¯ Æ 1.) Write and sketch the prior density.Does
Show equation (16.1). This is the same as showing that pº1v2Á¡º1¡x ¡¹1¢¢Á¡º2¡x ¡¹2¢¢Æ c pºÁ¡º¡x ¡¹¢¢where c can depend on the parameters but not x.
Bock (1975, TheoremA) established that for X » N(µ, I K ) then for any scalar function g (x)E£Xh¡X0X¢¤Æ µE[h (QKÅ2)]where QKÅ2 » Â2 KÅ2(¸). Assume bµ » N(µ, I K ) and eµJS Æµ1¡K
Assume bµ » N(0,V ). Calculate the following:(a) ¸.(b) JK in Theorem 15.5.(c) mse£bµ¤.(d) mse£eµJS¤.
For scalar X » N(µ,¾2) use Stein’s Lemma to calculate the following:(a) E£g (X) (X ¡µ)¤for scalar continuous g (x).(b) E£X3 (X ¡µ)¤.(c) E[sin(X) (X ¡µ)].(d) E£exp(tX) (X ¡µ)¤.(e)
Let bµ and eµ be two estimators. Suppose bµ is unbiased. Suppose eµ is biased but has variance which is 0.09 less than bµ.(a) If eµ has a bias of ¡0.1 which estimator is preferred based
Let Xn be the sample mean from a random sample. Consider the estimators bµ Æ Xn, eµ Æ Xn ¡c, and µ Æ cXn for 0 Ç c Ç 1.(a) Calculate the bias and variance of the three estimators.(b) Compare
You work in a government agency supervising job training programs. A research paper examines the effect of a specific job training program on hourly wages. The reported estimate of the effect (in
IfC Æ [L,U] is a 1¡® confidence interval for ¾2 find a confidence interval for the standard deviation ¾.
Let C Æ [L,U] be a 1¡® confidence interval for µ. Consider ¯ Æ h(µ) where h(µ) is monotonically increasing. Set C¯ Æ [h(L),h(U)]. Evaluate the converage probability of C¯ for ¯. Is C¯ a
Take the Bernoullimodelwith probability parameter p. Let b p Æ Xn from a random sample of size n.(a) Find s¡b p¢and a default 95% confidence interval for p.(b) Given n what is the widest possible
A colleague reports a 95% confidence interval [L,U] Æ [0.1, 3.4] for µ and also states “The t-statistic for µ Æ 0 is insignificant at the 5% level”. How do you interpret this? What could
A friend suggests the following confidence interval for µ. They draw a random number U »U[0,1] and set C Æ8
A confidence interval for the mean of a variable X is [L,U]. You decide to rescale your data, so set Y Æ X/1000. Find the confidence interval for themean of Y .
To estimate a variance ¾2 you have an estimate b¾2 Æ 10 with standard error s¡b¾2¢Æ 7.(a) Construct the standard 95% asymptotic confidence interval for ¾2.(b) Is there a problem? Explain the
You have the point estimate bµ Æ 0.45 and standard error s¡bµ¢Æ 0.28. You are interested in¯ Æ exp(µ).(a) Find b¯.(b) Use the Delta method to find a standard error s¡ b¯¢.(c) Use the
You have two independent samples with estimates and standard errors bµ1 Æ 1.4, s¡bµ1¢Æ0.2, bµ2 Æ 0.7, s¡bµ2¢Æ 0.3. You are interested in the difference ¯ Æ µ1 ¡µ2.(a) Find b¯.(b)
You have the point estimate bµ Æ ¡1.73 and standard error s¡bµ¢Æ 0.84.(a) Calculate a 95% asymptotic confidence interval.(b) Calculate a 90% asymptotic confidence interval.
You have the point estimate bµ Æ 2.45 and standard error s¡bµ¢Æ 0.14.(a) Calculate a 95% asymptotic confidence interval.(b) Calculate a 90% asymptotic confidence interval.
You design a statistical test of some hypothesis H0 which has asymptotic size 5% but you are unsure of the approximation in finite samples. You run a simulation experiment on your computer to check
You have two samples (mathematics and literature) of size n of the length of a Ph.D. thesis measured by the number of characters. You believe the Pareto model fits the distribution of
You have two samples (Madison and Ann Arbor) of monthly rents paid by n individuals in each sample. You want to test the hypothesis that the average rent in the two cities is the same. Construct an
The government implements a new banking policy. You want to assess whether the policy has had an impact. You gather information on 10 affected banks (so your sample size is n Æ 10). You conduct a
Take the model X » N(¹,1). Consider testing H0 : ¹ 2 {0,1} against H1 : ¹ Ý {0,1}. Consider the test statistic T Æ min{j pnXnj, j pn(Xn ¡1)j}.Let the critical value be the 1¡® quantile of
You teach a section of undergraduate statistics for economists with 100 students. You give the students an assignment: They are to find a creative variable (for example snowfall in Wisconsin),
In a likelihoodmodel with parameter ¸ a colleague tests H0 : ¸ Æ 1 against H1 : ¸ 6Æ 1. They claim to find a negative likelihood ratio statistic, LR Æ ¡3.4. What do you think? What do you
Test the following hypotheses against two-sided alternatives using the given information.In the following, Xn is the sample mean, s³Xn´is its standard error, s2 is the sample variance, n the sample
Take themodel X » N(¹,¾2). Propose a test for H0 : ¹ Æ 1 against H1 : ¹ 6Æ 1.
Take the Pareto model with parameter ®. We want a test for H0 : ® Æ 4 against H1 : ¸ 6Æ 4.(a) Find the likelihood ratio statistic.
Take the exponential model with parameter ¸. We want a test for H0 : ¸ Æ 1 against H1 :¸ 6Æ 1.(a) Develop a test based on the sample mean Xn.(b) Find the likelihood ratio statistic.
Take the Poisson model with parameter ¸. We want a test for H0 : ¸ Æ 1 against H1 : ¸ 6Æ 1.(a) Develop a test based on the sample mean Xn.(b) Find the likelihood ratio statistic.
Take the Bernoulli model with probability parameter p. We want a test for H0 : p Æ 0.05 against H1 : p 6Æ 0.05.(a) Develop a test based on the sample mean Xn.(b) Find the likelihood ratio statistic.

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