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applied statistics and multivariate

Questions and Answers of Applied Statistics And Multivariate

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 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
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
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
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
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
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)
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
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
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
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
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
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
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)
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
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
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
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
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,
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
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
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
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
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 =
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
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
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
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.
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
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
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
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
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
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
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
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
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
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)
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
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)
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
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
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
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
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,
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)
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
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
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
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
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
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
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
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
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
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
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
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
Exercise 6-19 presented the joint temperatures of the O-rings (F) for each test firing or actual launch of the space shuttle rocket motor. In that exercise you were asked to find the sample mean and
Exercise 6-18 presents drag coefficients for the NASA 0012 airfoil. You were asked to calculate the sample mean, sample variance, and sample standard deviation of those coefficients. (a) Find the
The nine measurements that follow are furnace tem- peratures recorded on successive batches in a semiconductor manufacturing process (units are F): 953, 950, 948,955,951, 949,957,954,955. (a)
An article in Transactions of the Institution of Chemical Engineers (Vol. 34, 1956, pp. 280-293) reported data from an experiment investigating the effect of several process variables on the vapor
The "cold start ignition time" of an automobile engine is being investigated by a gasoline manufacturer. The follow- ing times (in seconds) were obtained for a test vehicle: 1.75, 1.92, 2.62, 2.35,
The Pareto Chart. An important variation of a his- togram for categorical data is the Pareto chart. This chart is widely used in quality improvement efforts, and the categories usually represent
Construct a histogram for the pinot noir wine rating data in Exercise 6-35.Comment on the shape of the histogram. Does it convey the same information as the stem-and-leaf display?
Construct a histogram for the semiconductor speed data in Exercise 6-34.Comment on the shape of the histogram. Does it convey the same information as the stem-and-leaf display?
Construct a histogram for the overall golf distance data in Exercise 6-33.Comment on the shape of the histogram. Does it convey the same information as the stem-and-leaf display?
Construct a histogram for the water quality data in Exercise 6-32.Comment on the shape of the histogram. Does it convey the same information as the stem-and-leaf display?
Construct a histogram for the spot weld shear strength data in Exercise 6-31.Comment on the shape of the his- togram. Does it convey the same information as the stem-and- leaf display?

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