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applied statistics and multivariate
Applied Statistics And Probability For Engineers 5th Edition Douglas C. Montgomery, George C. Runger - Solutions
Consistent Estimator. Another way to measure the closeness of an estimator to the parameter 0 is in terms of consistency. If O, is an estimator of 9 based on a random sample of observations, O, is consistent for if lim PO,
A collection of a randomly selected parts is measured twice by an operator using a gauge. Let X and Y, denote the measured values for the ith part. Assume that these two random variables are independent and normally distributed and that both have true mean and variance (a) Show that the maximum
When the sample standard deviation is based on a random sample of size n from a normal population, it can be shown that S is a biased estimator for dr. Specifically, E(S)=2/(-1) (a/2)/[(n - 1)/2] (a) Use this result to obtain an unbiased estimator for o of the formc, when the constante, depends on
A lot consists of N transistors, and of these. M (MN) are defective. We randomly select two transis- tors without replacement from this lot and determine whether they are defective or nondefective. The random variable 1.if the ith transistor is nondefective 0, if the ath transistor is defective
An electric utility has placed special meters on 10 houses in a subdivision that measures the energy consumed (demand) at each hour of the day. They are interested in the en- ergy demand at one specific hour, the hour at which the system experiences the peak consumption. The data from these 10 me-
You plan to use a rod to lay out a square, each side of which is the length of the rod. The length of the rod is , which is unknown. You are interested in estimating the area of the square, which is u. Because is unknown, you measure it a times, obtaining observations X X X Suppose that each
A random variable x has probability density function f(x)= 28
Let X be a random variable with mean and variancea. Given two independent random samples of sizes w, and w with sample means, and, show that +(-a) < < is an unbiased estimator for. If and are independent. find the value of a that minimizes the standard error of X.
A manufacturer of semiconductor devices takes a random sample of 100 chips and tests them, classifying each chip as defective or nondefective. Let Xi 0 if the chip is nondefective and Xi 1 if the chip is defective. The sample fraction defective is What is the sampling distribution of the random
A random sample of 36 observations has been drawn from a normal distribution with mean 50 and standard deviation 12.Find the probability that the sample mean is in the interval. Is the assumption of normality important? Why?
A procurement specialist has purchased 25 resistors from vendor 1 and 30 resistors from vendor 2.Let X1,1, X1,2, , X1,25 represent the vendor 1 observed resistances, which are assumed to be normally and independently distributed with mean 100 ohms and standard deviation 1.5 ohms.Similarly, let
The time between failures of a machine has an exponential distribution with parameter . Suppose that the prior distribution for is exponential with mean 100 hours. Two machines are observed, and the average time between failures is hours.(a) Find the Bayes estimate for .(b) What proportion of
The weight of boxes of candy is a normal random variable with mean and variance pound. The prior distribution for is normal, with mean 5.03 pound and variance pound. A random sample of 10 boxes gives a sample mean of pounds.(a) Find the Bayes estimate of .(b) Compare the Bayes estimate with the
Suppose that X is a normal random variable with unknown mean and known variance 2 9.The prior distribution for is normal with 0 4 and 1.A random sample of n 25 observations is taken, and the sample mean is(a) Find the Bayes estimate of .(b) Compare the Bayes estimate with the maximum
Suppose that X is a Poisson random variable with parameter . Let the prior distribution for be a gamma distribution with parameters m 1 and .(a) Find the posterior distribution for .(b) Find the Bayes estimator for .
Suppose that X is a normal random variable with unknown mean and known variance 2. The prior distribution for is a normal distribution with mean 0 and variance .Show that the Bayes estimator for becomes the maximum likelihood estimator when the sample size n is large.
Reconsider the oxide thickness data in Exercise 7-29 and suppose that it is reasonable to assume that oxide thickness is normally distributed.(a) Compute the maximum likelihood estimates of and 2.(b) Graph the likelihood function in the vicinity of and , the maximum likelihood estimates, and
Consider the Weibull distribution(a) Find the likelihood function based on a random sample of size n. Find the log likelihood.(b) Show that the log likelihood is maximized by solving the equations(c) What complications are involved in solving the two equations in part (b)?
Let X1, X2, , Xn be uniformly distributed on the interval 0 toa. Recall that the maximum likelihood estimator of a is .(a) Argue intuitively why cannot be an unbiased estimator for a.(b) Suppose that . Is it reasonable that consistently underestimates a? Show that the bias in the estimator
Consider the probability density function(a) Find the value of the constant c.(b) What is the moment estimator for ?(c) Show that is an unbiased estimator for .(d) Find the maximum likelihood estimator for .
Let X1, X2, , Xn be uniformly distributed on the interval 0 toa. Show that the moment estimator of a is Is this an unbiased estimator? Discuss the reasonableness of this estimator.
Consider the probability density function Find the maximum likelihood estimator for .
Consider the shifted exponential distribution When 0, this density reduces to the usual exponential distribution. When , there is only positive probability to the right of .(a) Find the maximum likelihood estimator of and , based on a random sample of size n.(b) Describe a practical situation in
Let X be a random variable with the following probability distribution:Find the maximum likelihood estimator of , based on a random sample of size n.
Of, randomly selected engineering students at ASU, X, owned an HP calculator, and ofa, randomly selected engineering students at Virginia Tech, X, owned an HP calcu- lator. Let p and p; be the probability that randomly selected ASU and Virginia Tech engineering students, respectively, own HP
Two different plasma etchers in a semiconductor fac- tory have the same mean etch rate. However, machine 1 is newer than machine 2 and consequently has smaller variabil- ity in etch rate. We know that the variance of etch rate for machine I is of and for machine 2 it is dad. Suppose that we have n,
T, and S are the sample mean and sample variance from a population with mean, and variance o. Similarly, X, and Sare the sample mean and sample variance from a sec- ond independent population with mean p, and variance o The sample sizes are, and respectively (a) Show that X-X, is an unbiased
Suppose that X is the number of observed "successes" in a sample of a observations, where p is the probability of success on each observation. (a) Show that P=X/n is an unbiased estimator of p. (b) Show that the standard error of is p(1 - p)/n. How would you estimate the standard error?
Data on oxide thickness of semiconductors are as follows: 425,431,416, 419, 421, 436, 418, 410,431,433,423, 426, 410, 435, 436, 428, 411, 426, 409, 437, 422,428,413,416. (a) Calculate a point estimate of the mean oxide thickness for all wafers in the population. (b) Calculate a point estimate of
Let three random samples of sizes n = 20, n = 10, and w;= 8 be taken from a population with mean and variance . Let S., S. and S be the sample variances. Show that (205+105 +85/38 is an unbiased estimator of a
Suppose that, and, are estimators of 9.We know that E() = E() = 8, E(0) = 0,1'(0) = 12.V()=10. and E(-9)=6. Compare these three estimators. Which do you prefer? Why?
Suppose that, and, are estimators of the parame ter 8.We know that E) = 0, E() = 0/2, (e) = 10, V(+) = 4.Which estimator is best? In what sense is it best?
Suppose that, and, are unbiased estimators of the parameter. We know that (e) 10 and (e) 4.Which estimator is best and in what sense is it best? Calculate the relative efficiency of the two estimators.
Let XXX denote a random sample from a population having mean and variance o. Consider the following estimators of: 2 (a) is either estimator unbiased? (b) Which estimator is best? In what sense is it best? Calculate the relative efficiency of the two estimators.
Suppose we have a random sample of size 2n from a population denoted by X, and E(X) = and (X) = r. Let Xx and x 2n be two estimators of . Which is the better estimator of pa? Explain your choice.
Suppose that we have a random sample X. XX from a population that is M. ). We plan to use ---/c to estimate o. Compute the bias in as an estimator of or as a function of the constant c.
A consumer electronics company is comparing the brightness of two different types of picture tubes for use in its television sets. Tube type A has mean brightness of 100 and standard deviation of 16, while tube type B has unknown mean brightness, but the standard deviation is assumed to be
A random sample of size n1 16 is selected from a normal population with a mean of 75 and a standard deviation of 8.A second random sample of size n2 9 is taken from another normal population with mean 70 and standard deviation 12.Let and be the two sample means. Find:(a) The probability that
Suppose that X has a discrete uniform distribution 1(x)= 1 x=1,2,3 0, otherwise A random sample of - 36 is selected from this population. Find the probability that the sample mean is greater than 2.1 but less than 2.5, assuming that the sample mean would be measured to the nearest tenth.
Suppose that the random variable X has the continu ous uniform distribution f(x)= 11.0x1 10, otherwise Suppose that a random sample of 12 observations is selected from this distribution. What is the approximate probability distribution of X-6? Find the mean and vari- ance of this quantity.
Consider the concrete specimens in the previous exercise. What is the standard error of the sample mean?
The compressive strength of concrete is normally distributed with j = 2500 psi and or = 50 psi. Find the prob- ability that a random sample of n = 5 specimens will have a sample mean diameter that falls in the interval from 2499 psi to 2510 psi.
Consider the synthetic fiber in the previous exercise. How is the standard deviation of the sample mean changed when the sample size is increased from a = 6 to = 49?
A synthetic fiber used in manufacturing carpet has ten- sile strength that is normally distributed with mean 75.5 psi and standard deviation 3.5 psi. Find the probability that a ran- dom sample of = 6 fiber specimens will have sample mean tensile strength that exceeds 75.75 psi.
Suppose that samples of size n = 25 are selected at random from a normal population with mean 100 and standard deviation 10.What is the probability that the sample mean falls in the interval from pr-1.80 to +1.0?
Consider the hospital emergency room data from Exercise 6-104.Estimate the proportion of patients who arrive at this emergency department experiencing chest pain.
Know how to construct a point estimator using the Bayesian approach
Know how to compute and explain the precision with which a parameter is estimated
Know how to construct point estimators using the method of moments and the method of maximum likelihood
Explain important properties of point estimators, including bias, variance, and mean square error
Understand the central limit theorem
Explain the important role of the normal distribution as a sampling distribution
Explain the general concepts of estimating the parameters of a population or a probability distribution
Trimmed Mean. Suppose that the sample size a is such that the quantity a7/100 is not an integer. Develop a procedure for obtaining a trimmed mean in this case.
Trimmed Mean. Suppose that the data are arranged in increasing order, 1% of the observations are removed from each end, and the sample mean of the remaining numbers is calculated. The resulting quantity is called a trimmed mean. The trimmed mean generally lies between the sample mean I and the
Suppose that we have a sample x, y and we have calculated, and for the sample. Now an (+1)st observation becomes available. Let X-1 and be the sample mean and sample variance for the sample using all n + 1 observations. (a) Show how (b) Show that can be computed using and... (n 1) + (x) #+1 (c) Use
An experiment to investigate the survival time in hours of an electronic component consists of placing the parts in a test cell and running them for 100 hours under elevated temperature conditions. (This is called an "accelerated" life test.) Fight components were tested with the following
Consider the sample XX... sample mean I and sample standard deviations. Let =(x, x=1,2..., . What are the values of the sample mean and sample standard deviation of the ? with
A sample of temperature measurements in a furnace yielded a sample average (F) of 835.00 and a sample standard deviation of 10.5.Using the results from Exercise 6-108, what are the sample average and sample standard deviations expressed in "C?
Coding the Data. Let y, a + boc, 1 = 1.2...... where a and bare nonzero constants. Find the relationship between and , and between 1, and 5,-
Using the results of Exercise 6-106, which of the two quantities (-) (will be smaller, provided that and ja?
Consider the quantity (x-a). For what value of a is this quantity minimized?
Consider the airfoil data in Exercise 6-18.Subtract 30 from each value and then multiply the re- sulting quantities by 10.Now computes for the new data. How is this quantity related to s for the original data? Explain why.
Patients arriving at a hospital emergency department present a variety of symptoms and complaints. The following data were collected during one weekend night shift (11:00 PM. to 7:00A.M.): Chest pain Difficulty breathing 8 7 Numbness in extremities 3 Broken bones 11 Cuts Abrasions Stab wounds 16 21
In their book Introduction to Time Series Analysis and Forecasting (Wiley, 2008), Montgomery, Jennings, and Kulabci presented the data on the drowning rate for children between one and four years old per 100,000 of population in Arizona from 1970 to 2004. The data are: 19.9, 16.1, 19.5, 19.8, 21.3,
In 1879, A. A. Michelson made 100 determinations of the velocity of light in air using a modification of a method proposed by the French physicist Foucault. He made the measurements in five trials of 20 measurements each. The ob- servations (in kilometers per second) follow. Each value has 299,000
Transformations. In some data sets, a transformation by some mathematical function applied to the original data, such as Vy or log y, can result in data that are simpler to work with statistically than the original data. To illustrate the effect of a transformation, consider the following data,
Reconsider the golf ball overall distance data in Exercise 6-33.Construct a box plot of the yardage distance and write an interpretation of the plot. How does the box plot com- pare in interpretive value to the original stem-and-leaf diagram?
Construct normal probability plots of the cold start ignition time data presented in Exercises 6-53 and 6-64.Construct a separate plot for each gasoline formulation, but arrange the plots on the same axes. What tentative conclusions can you draw?
Construct a normal probability plot of the effluent dis- charge temperature data from Exercise 6-92.Based on the plot, what tentative conclusions can you draw?
Reconsider the data in Exercise 6-88.Construct nor- mal probability plots for two groups of the data: the first 40 and the last 40 observations. Construct both plots on the same axes. What tentative conclusions can you draw?
Reconsider the golf course yardage data in Exercise 6.9.Construct a box plot of the yardages and write an inter- pretation of the plot.
A communication channel is being monitored by recording the number of errors in a string of 1000 bits. Data for 20 of these strings follow: Read data across. 3 10 3 2 41 3 1 1 2 3 3 2 02 0.I (a) Construct a stem-and-leaf plot of the data. (b) Find the sample average and standard deviation. (c)
A manufacturer of coil springs is interested in imple- menting a quality control system to monitor his production process. As part of this quality system, it is decided to record the number of nonconforming coil springs in each production batch of size 50.During 40 days of production, 40 batches of
The following data are the temperatures of effluent at discharge from a sewage treatment facility on consecutive days: 43 45 49 45 529 47 51 48 52 50 46 52 46 51 44 49 46 51 44 50 48 50 49 50 (a) Calculate the sample mean, sample median, sample vari- ance, and sample standard deviation. (b)
Reconsider the data from Exercise 6-88.Prepare comparative box plots for two groups of observations: the first 40 and the last 40.Comment on the information in the box plots.
The total net electricity consumption of the US. by year from 1980 to 2007 (in billion kilowatt-hours) follows. Net consumption excludes the energy consumed by the gener ating units. Read left to right. 1980 20944 1981 2147.1 1982 2086.4 1983 2151.0 1984 2285.8 1985 2324.0 1986 2368.8 1987 2457.3
An article in Quality Engineering (Vol. 4, 1992, pp. 487-495) presents viscosity data from a batch chemical process. A sample of these data follows on p. 219:(a) Reading down and left to right, draw a time series plot of all the data and comment on any features of the data that are revealed by this
Consider the following two samples: Sample 1: 10,9,8,7,8,6,10,6 Sample 2: 10, 6, 10, 6, 8, 10, 8, 6 (a) Calculate the sample range for both samples. Would you con chade that both samples exhibit the same variability? Explain. (b) Calculate the sample standard deviations for both samples. Do these
A sample of six resistors yielded the following resis tances (ohms): x = 45, x = 38, x=47,4 = 41.x5 = 35, and x = 43.(a) Compute the sample variance and sample standard deviation. (b) Subtract 35 from each of the original resistance measure ments and computes and s. Compare your results with those
The table below shows unemployment data for the US. that are seasonally adjusted. Construct a time series plot of these data and comment on any features (source: U.S. Bureau of Labor Web site, http://data.bls.gov).
The concentration of a solution is measured six times by one operator using the same instrument. She obtains the follow- ing data: 63.2, 67.1, 65.8, 64.0, 65.1, and 65.3 (grans per liter). (a) Calculate the sample mean. Suppose that the desirable value for this solution has been specified to be
It is possible to obtain a "quick and dirty" estimate of the mean of a normal distribution from the fiftieth percentile value on a normal probability plot. Provide an argument why this is so. It is also possible to obtain an estimate of the stan- dard deviation of a normal distribution by
Construct two normal probability plots for the height data in Exercises 6-30 and 6-37.Plot the data for female and male students on the same axes. Does height seem to be normally distributed for either group of students? If both populations have the same variance, the two normal probability plots
Construct a normal probability plot of the sus- pended solids concentration data in Exercise 6-32.Does it seem reasonable to assume that the concentration of suspended solids in water from this particular lake is normally distributed?
Construct a normal probability plot of the cycles to failure data in Exercise 6-23.Does it seem reasonable to as- sume that cycles to failure is normally distributed?
Construct a normal probability plot of the octane rat- ing data in Exercise 6-22.Does it seem reasonable to assume that octane rating is normally distributed?
Construct a normal probability plot of the O-ring joint temperature data in Exercise 6-19.Does it seem reasonable to assume that O-ring joint temperature is normally distributed? Discuss any interesting features that you see on the plot.
Construct a normal probability plot of the solar inten sity data in Exercise 6-12.Does it secremonable to assume that solar intensity is normally distributed?
Construct a normal probability plot of the visual accommodation data in Exercise 6-11.Does it seem reasonable to assume that visual accommodation is normally distributed?
Construct a normal probability plot of the insulating fluid breakdown time data in Exercise 6-8.Does it seem reasonable to assume that breakdown time is normally distributed?
Construct a normal probability plot of the piston ring diameter data in Exercise 6-7.Does it seem reasonable to assume that piston ring diameter is normally distributed?
An article in Nature Genetics ("Treatment-specific Changes in Gene Expression Discriminate in Vivo Drug Response In Human Leukemia Cells" (2003, Vol. 34(1), pp. 85-90)] studied gene expression as a function of treatments for leukemia. One group received a high dose of the drug while the control
In Exercise 6-53, data were presented on the cold start ignition time of a particular gasoline used in a test vehicle. A second formulation of the gasoline was tested in the same ve hicle, with the following times (in seconds): 183, 1.99, 3.13.3.29. 2.65,2.87,3.40, 2.46, 1.89, and 3.35.Use these
Use the data on heights of female and male engineer- ing students from Exercises 6-30 and 6-37 to construct comparative box plots. Write an interpretation of the informa- tion that you see in these plots.
Reconsider the semiconductor speed data in Exercise 6-34.Construct a box plot of the data and write an interpretation of the plot. How does the box plot compare in interpretive value to the original stem-and-leaf diagram?
Reconsider the weld strength data in Exercise 6-31.Construct a box plot of the data and write an interpretation of the plot. How does the box plot compare in interpretive value to the original stem-and-leaf diagram?
Reconsider the water quality data in Exercise 6-32.Construct a box plot of the concentrations and write an interpre- tation of the plot. How does the box plot compare in interpretive value to the original stem-and-leaf diagram?
Reconsider the energy consumption data in Exercise 6-29.Construct a box plot of the data and write an interpreta- tion of the plot. How does the box plot compare in interpective value to the original stem-and-leaf diagram?
Reconsider the motor fuel octane rating data in Exercise 6-20.Construct a box plot of the data and write an interpretation of the plot. How does the box plot compare in in- terpretive value to the original stem-and-leaf diagram?
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