Advanced Statistics From An Elementary Point Of View 1st Edition Michael J Panik - Solutions

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11-45. Confidence Interval for a Population Total:Let {X1, . . . ,Xn} constitute a set of sample random variables taken from a population of sizeN.Weare interested in determining an interval estimate for the population total τ = Nj=1 Xj . Let NX serve as an estimator forτ . Then for N¯x the
11-44. Confidence Interval for the Coefficient of Variation:We previously expressed the population coefficient of variation as CV =σμ. Given a collection of sample random variables X1, . . . ,Xn, let us denote the realization of CV as cv = s¯x . Then for n ≥ 25 and cv < 0.40, a 100(1 − α)%
11-43. Confidence Intervals for Quantiles:Let X1, . . . ,Xn represent a set of sample random variables taken from a population of size N. For the quantile P, let γp be its sample realization, where p depicts the proportion of the population below γp (e.g., for quartiles, p = 0.25j, j = 1, 2, 3;
11-42. Suppose X1, . . . ,Xn is a random sample drawn from a Poisson probability mass function f (x; nλ) = e−nλ (nλ)X X! ,X = 0, 1, 2, . . . ,λ > 0. If Y = ni=1 Xi ∼ f (x; nλ) and y denotes the sample realization of Y, then it can be shown that a 100(1 − α)% confidence interval for λ
11-41. Suppose ˆPX, ˆPY are sample proportions determined by drawing nX, nY items, respectively, from two separate binomial populations with parameters pX, pY. In addition, let nX, nY be sufficiently large so that ˆPX, ˆPY are approximately normally distributed. Verify that ˆPX − ˆPY is
11-40. Suppose X, Y are the sample means determined from random samples of sizes nX, nY, respectively, taken from two separate population distributions represented by the random variables X,Y. Verify that if X, Y are normally and independently distributed, then X − Y is N(μX−Y,
11-39. Suppose X1, . . . ,Xn depicts random sample drawn from an exponential probability density f (x; λ) = λe−λx, x > 0,λ > 0. It can be demonstrated that a 100(1−α)% confidence interval for λ has the form (1, 2) =χ2α/2 2n¯x ,χ2 1−(α/2)2n¯x. Given this result, let us assume
11-38. A random sample of size n = 18 is taken from a normal population with the result that ¯x = 15 and s2 = 5.3. Determine a set of Bonferroni joint confidence limits for μ and σ2 for a family confidence coefficient of 1 − α = 0.95.
11-37. Random samples of sizes nX = nY = 5 were taken from normal distributions having unknown variances σ2X and σ2Y, respectively. Find:(a) A 95% confidence interval for σ2X(b) A 99% confidence interval for σ2Y(c) A 95% confidence interval for σ2X/σ2Y Sample 1: 3.7, 8.6, 4.2, 5.5, 6.2 Sample
11-36. A machine designed to dispense a specific amount of soap powder had a sample variance of s2 1= 0.714 for n1 = 17 fills. A similar machine was tested for n2 = 25 fills and it was determined that s2 2= 0.438. (Assume the fill amounts for these machines are approximately normally
11-35. Suppose that two independent random samples of sizes nX = 10 and nY =15, respectively, are drawn from normal populations and that s2X= 0.238 and s2Y= 0.149. Find a 95% confidence interval for σ2X/σ2Y. What is the 95% confidence interval for σX/σY?
11-34. A group of 10 enrollees are weighed at the beginning and at the end of a special combined diet and exercise program. Let the random variable(expressed in pounds) depict the weight difference (postprogram less preprogram).Suppose D is approximately normally distributed with values:−5.7,
11-33. An experiment was conducted to compare the reaction times of a group of 10 volunteers (from a psychology class) to a bright light versus the sound of a loud air horn. When signaled with either the light or horn, the individual was asked to press a button to turn off the light or to turn off
11-32. In a set of nX = 150 trials of a random experiment, x = 75 successes were observed; in a second set of nY = 250 trials (performed independently of the first set of trials), y = 105 successes were observed. Determine a 95%confidence interval for pX − pY.
11-31. A recent World Health Services poll revealed that x = 976 out of nX = 1900 African women had their first child before the age of 15 years, and y = 732 out of nY = 1850 Asian women had their first child before they were 15 years old. If pX and pY are the respective proportions of African and
11-30. Two spray-bottle rug cleaners were tested for their ability to remove a specific type of stain. The first was deemed successful in x = 37 out of nX = 70 independent trials and the second was judged successful in y = 62 out of nY = 85 independent trials. If pX (respectively, pY) is the true
11-29. A time and motion expert is interested in comparing two new methods of completing a certain task. The participants are divided into two groups of equal size and, for each group, their times (in minutes) to completion of the task will be monitored. It is anticipated that the completion times
11-28. A horticulturalist has been using brand A fertilizer on a certain variety of plant. A new brand B fertilizer has just been marketed, which purports to be superior to brand A. Two independent groups of n1 = n2 = 8 new plantings are to be treated (one with the brand A fertilizer and the other
11-27. Two brands of comparable electric drills (call them A and B) are each guaranteed for 1 year under normal use. For n1 = 45 brand A drills, 10 failed to operate in under 1 year and for n2 = 48 brand B drills, 16 failed before the warranty period ended. Determine a 99% confidence interval for
11-26. Independent random samples of sizes nX = 8 and nY = 10 were extracted from normal populations having unknown means. If it is known that σ2X=2.5 and σ2Y= 4.8, find a 95% confidence interval for μX − μY.Sample 1: 3, 8, 6, 5, 10, 7, 7, 6 Sample 2: 1, 5, 8, 8, 6, 7, 6, 9, 4, 3 What is the
11-25. Two soda vending machines in two separate wings of an office building are tested monthly for average fill. Each machine is set to dispense 8 oz. of soda. On a particular day a service technician obtains the following results (in ounces):Machine 1 8.5 8.1 8.3 8.7 8.4 8.5 Machine 2 8.2 7.9 8.1
11-24. The ABC corporation claims that its new gasoline additive for passenger cars will enhance the mileage per tankfull of gasoline if a can of their product is added to a full tank of gasoline. Five vehicles were tested, yielding the following results (miles per tankfull):Vehicle 1 2 3 4 5
11-23. Resolve the preceding exercise if it is known that σ2X= σ2Y= σ2.
11-22. Assume that miles per gallon of gasoline for imported passenger cars(of a given engine displacement) are normally distributed with a known standard deviation of 3.1 miles per gallon, and a comparable figure for domestic vehicles having the same displacement is 4.7 miles per gallon.Suppose
11-21. Suppose that nX = 10, nY = 8, ¯x = 70.5, ¯y = 77.3, σX = 22, and σY = 19.If the samples are independent and drawn from two normal populations, find a 95% confidence interval for μX − μY.If σ2X and σ2Y are unknown and estimated by s2X= 5 and s2Y= 3.7, respectively, with nX = 20 and
11-20. A researcher wants to check the variability of a device used to measure the light intensity (in lumens) emitted from a certain welding machine.Nine independent measurements are recorded: 546, 609, 550, 585, 570, 577, 580, 540, and 561. If the measurements recorded by this instrument are
11-19. A random sample of 17 high school students was selected from the senior class. It was found that the standard deviation of their summer hourly wage rates was $4.35. Find a 90% confidence interval for the true variance of the hourly wage rate for all seniors who had summer jobs. What
11-18. From a sample of size 20 taken from an approximately normal population it was determined that s2 = 5.36. Find a 90% confidence interval for σ.
11-17. Suppose the diameters of steel rods X (expressed in hundreds of millimeters)are approximately normally distributed. Given the following set of sample observations: 533, 532, 535, 540, 544, 536, 541, 538, 533, 537, determine a 95% confidence interval for σ2. What is the 95% confidence
11-16. Suppose a random sample of size n = 15 is taken from a normally distributed population. It is determined that 15 i=1(xi − ¯x) = 131.43. Find a 95% confidence interval for σ2. What is the 95% confidence interval for σ?
11-15. If the population proportion p is to be estimated by a 100(1 − α)% confidence interval and we are sampling from a binomial population of size N without replacement, then a measure of how precisely we have estimated p is determined by ± error bound, where the error bound is w2= zα/2p(1
11-14. If the population mean μ is to be estimated by a 100(1 − α)% confidence interval and we are sampling without replacement from a finite population and n/N > 0.05, then a measure of how precisely we have estimatedμ is provided by ± error bound, where the error bound is w2= zα/2√σn?N
11-13. Suppose the sponsor of a popular television show wants to determine the proportion of the viewing public watching the show to within 3% with 95% reliability. How large of a sample should be taken? Assume that past experience dictates that p = 0.45. If this known value of p is not available,
11-12. A psychology experiment is structured so that either response A or response B is forthcoming from a certain stimulus. If p is the true proportion of subjects providing response B, determine the sample size needed to estimate p to within ±0.87 units with 90% reliability. Suppose a pilot
11-11. Under random polling of the eligible voters of a certain city it was determined that out of n = 350 voters, 200 of them favor candidate A. Find an approximate 95% confidence interval for p, the true fraction of eligible voters favoring this candidate.
11-10. About 9 10 of n = 400 persons of voting age interviewed were in favor of funding a new public safety program. Determine a 95% confidence interval for the fraction of the population who are in favor of the program.
11-9. In a sample of n = 100 high pressure castings, 15 were found to exhibit one or more defects. Find a 99% confidence interval for the population percent of all defective castings.
11-8. Suppose we wish to estimate the average daily yield (in terms of telephone inquiries) of a new magazine ad for a certain product. We want a degree of precision of ±3 telephone calls with 95% reliability. If it is known that σ = 10 calls, find the required sample size. If σ is unknown but
11-7. A random sample is to be drawn from a population that is N(μ, σ), withσ2 known. One objective is to use X to estimate μ to within k units with 100(1−α)% reliability. Solve for the sample size n required to meet these conditions from the following probability statement: P(|X − μ|
11-6. A certain type of electronic measurement has a standard deviation of 9 units. How large of a sample should be taken so that the 95% confidence interval for the mean will not exceed 4 units in width?
11-5. A random of sample size n = 16 from a distribution of the form N(μ, 25)yielded a mean of ¯x = 75. Find a 99% confidence interval for μ. What is the confidence interval if σ is unknown and s = 25?
11-4. Let X be the amount of distillate per day (in pounds) produced by a chemical process at ACE Laboratories and suppose that X is approximately normally distributed. A sample of n = 10 consecutive days yielded the following amounts (in pounds): 2.3, 4.1, 5.6, 3.1, 4.0, 5.1, 2.7, 3.6, 4.2, and
11-3. A manufacturer of adhesives has developed a new super latex cement that was tested on n = 12 different industrial rubber compounds. The drying times to adhesion (in seconds) were as follows:81 103 100 67 63 71 90 91 85 88 71 73 Find a 95% confidence interval for the true mean drying time for
11-2. Let X ( the length of life in hours of a certain brand of 100-watt frosted light bulb) be N(μ, 500). A random sample of n = 20 bulbs was tested(until they all burned out), yielding a mean life of ¯x = 1357 hours. Find a 95% confidence interval for μ. Explain your result. Suppose X is not
11-1. The times spent at the pump were recorded for n = 65 randomly selected customers at a small independent gas station. It was found that ¯x = 4.6 minutes with s = 2.06 minutes. Find a 95% confidence interval for the true average time spent at the pump. How precisely have we estimated μ?
10-66. Suppose X1, . . . ,Xn is a random sample drawn from a probability density function that is N(μ, σ). Find a constant c such that cS is an unbiased estimator for σ. (Hint: c =n−1 2 5 n−1 2 6(n2).)
10-65. LetX1, . . . ,Xn be a random sample taken from the probability density function f (x; θ) = (θ + 1)xθ, 0 < x < 1. Find a method-of-moments estimator for θ.
10-64. Let {X1,X2, . . . ,Xn} represent a set of sample random variables drawn from the probability mass function f (x; θ) = θx(1 − θ)1−x, x = 0, 1 and 0 ≤ θ ≤ 1 2 . Find the method-of-moments estimator of θ.
10-63. If {X1,X2, . . . ,Xn} is a set of sample random variables taken from the probability density function f (x; θ) = θe−θx, 0 < x < +∞, find the methodof-moments estimator of θ.
10-62. Let {X1,X2, . . . ,Xn} depict a set of sample random variables from the Poisson probability mass function f (x; λ) = λxe−λx! , x = 0, 1, 2, . . . , 0 < λ < +∞.Determine the method-of-moments estimator of λ.
10-61. Let {X1,X2, . . . ,Xn} correspond to a set of sample random variables taken from a distribution that is N(μ, σ). Find estimates of μ and σ using the method of moments. (Hint: First determine method-of-moments estimates of μ and σ2.)
10-60. Demonstrate that the method-of-moments estimator for θ determined in Exercise 10-59 is a consistent estimator of θ.
#!# 10-59. Suppose X1, . . . ,Xn represents a random sample drawn from a uniform distribution with probability density function u(x; 0, θ) = 1θ , 0 < x
10-58. Method of Moments Technique for Finding Point Estimators:Let {X1,X2, . . . ,Xn} be a set of samplerandomvariables taken from a distribution with a probability density function f (x; θ1, . . . , θk). Let μr= E(Xr)denote the rth moment of the distribution withMr= ni=1 Xr in representing
10-57. Suppose X1, . . . ,Xn is a random sample taken from a binomial probability mass function b(X; n, p). Is the statistic X = ni=1 Xi complete for p?
10-56. Suppose X1, . . . ,Xn is a random sample drawn from the probability mass function f (x; θ) = P(X = x) = θ(1 − θ)x−1, x = 1, 2, . . . , 0 < θ < 1. Is this family of distributions complete?
10-55. Let {X1,X2, . . . ,Xn} be a set of sample random variables taken from a geometric distribution with probability mass function f (X; p) = (1 −p)X−1p,X = 1, 2, 3, . . . , 0 < p < 1. Determine the maximum likelihood estimator of p.
10-54. Let {X1,X2, . . . ,Xn} be a set of sample random variables drawn from the Cauchy probability density function f (x; θ) = 9π 31 + (x − θ)22:−1,−∞ < x, θ < +∞. Verify that the Cramér-Rao lower bound is 2n.
10-53. LetX1, . . . ,Xn depict a random sample drawn from the probability density function f (x; θ) = θxθ−1, 0 < x < 1,θ > 0. Find the minimum variance bound for an unbiased estimator of θ.
10-52. Let {X1,X2, . . . ,Xn} depict a set of sample random variables drawn from a N(θ, 1) population. Find a best unbiased estimator of θ2. Does the variance of this estimator attain the Cramér-Rao lower bound?
10-51. Suppose X1, . . . ,Xn depicts a random sample drawn from the probability density function f (x, θ) = 1θ e−x/θ , x ≥ 0,θ > 0. Find a minimum variance bound estimator for θ.
10-50. Suppose {X1,X2, . . . ,Xn} constitutes a set of sample random variables drawn from the gamma probability density function (7.4.8). Determine maximum likelihood estimates of α, θ. (Note: Do not expect to find a closed form solution.)
10-49. Let {X1,X2, . . . ,Xn} represent a set of sample random variables drawn from the probability density function f (x; θ1, θ2) = - 1θ2 e−(x−θ1)2/θ2 , −∞ < θ1 ≤ x < +∞, 0 < θ2 < +∞, 0 elsewhere.Find maximum likelihood estimates for θ1 and θ2. Are these estimates jointly
10-48. Suppose {X1,X2, . . . ,Xn} represents a set of sample random variables drawn from each of the following probability density functions. Find a maximum likelihood estimator of θ:(a) f (x; θ) = 1θ x 1θ−1, 0 < x < 1,θ > 0(b) f (x; θ) = θxθ−1, 0 ≤ x ≤ 1,θ > 0(c) f (x; θ) = 1θ2
10-47. Suppose X1, . . . ,Xn constitutes a random sample drawn from a population that is N(μ, 1). Find the maximum likelihood estimator of μ.
10-46. Suppose we draw a random sample of size n from a multinomial distribution with probability mass function (6.24). Additionally, suppose there occurs xj outcomes of type Aj , j = 1, . . . , k. Determine the maximum likelihood estimators for the parameters pj , j = 1, . . . , k. (Hint: Maximize
10-45. Consider a family of probability density functions of the form f (x; θ) =1θ , 0 < x < θ. Is this family complete?
10-44. Let {X1,X2, . . . ,Xn} be a set of sample random variables drawn from the probability density function f (x; θ) = -(1 + θ)xθ, 0≤ x ≤ 1,θ > 0;0 elsewhere.Find the maximum likelihood estimator of θ.
10-43. Let {X1,X2, . . . ,Xn} constitute a set of sample random variables from a Poisson probability density function with parameter (mean) λ. Find the maximum likelihood estimator of θ.
10-42. Let {X1,X2, . . . ,Xn} be a collection of sample random variables drawn from the probability mass function f (x; θ) = θxe−θx! , x = 0, 1, 2, . . . , 0 < θ < +∞.Verify that the family of distributions of X = ni=1 Xi is complete.
10-41. Let S2 1= 1nni=1(Xi − X)2 denote the variance of a random sample of size n(>1) drawn from a N(μ, θ) probability density function. The estimator T = n n−1S2 1 is unbiased since E(T) = θ. Is the statistic T the most efficient estimator for θ?
10-40. LetX1, . . . ,Xn depict a random sample drawn from the Poisson probability density function f (X; θ) = θXe−θ /X!, x = 0, 1, 2, . . . . It is known that T = ni=1 Xi is a sufficient statistic for the mean θ > 0. Demonstrate that Tn= X is an efficient statistic for θ.
10-39. Let X1, . . . ,Xn depict a random sample drawn from the exponential probability density function f (x; θ) = 1θ e−x/θ , x ≥ 0,θ > 0. Find an efficient or minimum variance unbiased estimator for V(Xi).
10-38. For X1, . . . ,Xn a random sample drawn from the probability density function f (x; θ) = 1θ e−x/θ , x ≥ 0,θ > 0, find a minimal sufficient statistic for θ.
10-37. LetX1, . . . ,Xn represent a random sample drawn for the exponential probability density function f (x; λ) = λe−λx, x ≥ 0, λ > 0. Find an asymptotically efficient estimator for λ.
10-36. Suppose X1, . . . ,Xn depicts a random sample drawn from a binomial population.In addition, let X = ni=1 Xi success be observed. Is ˆP = Xn aconsistent estimator for the binomial parameter p?
10-35. Suppose X1, . . . ,Xn constitutes a random sample drawn from a N(μ, σ)population. Does S2 = 1 n−1 ni=1(Xi − X)2 converge to σ2 in probability?
10-34. It is known that forXaN(μ, σ) random variable, the maximum likelihood estimator of σ2 is ˆT = S2 1= 1nni=1(Xi − X)2 and that T is a biased estimator of σ2. Demonstrate that the bias goes to zero as n→∞.
10-33. Let X1, . . . ,Xn represent a random sample taken from a population with meanμand variance σ2. Let Xn = 1nni=1 Xi and S2n= 1(n−1) ni=1(Xi−X)2 depict sequences of estimators for μ and σ2, respectively. Verify that {Xn}and {S2n } are mean–squared–error–consistent sequences
10-32. Let {X1,X2, . . . ,Xn} be a set of sample random variables drawn from the probability density function f (x; θ) = θ(1 − θ)x−1, x = 1, 2, . . . ; 0 < θ < 1.Find a minimal sufficient statistic for θ.
10-31. Suppose {X1,X2, . . . ,Xn} is a set of sample random variables taken from a population distribution that is N(μ, σ). Demonstrate thatni=1(Xi − X)2 is a minimal sufficient statistic for σ2. Use this statistic to find an efficient estimator for σ2.
10-30. Given the probability density function presented in Exercise 10–12, find a minimal sufficient statistic for θ.
10-29. Let X1, . . . ,Xn be a random sample taken from a Bernoulli probability distribution P(Xi = 1) = p, P(Xi = 0) = 1 − p, with p unknown). Find a minimal sufficient statistic for p.
10-28. Suppose X1, . . . ,Xn is a random sample taken from a population with known mean μ = μ0 and unknown variance σ2. Is S2 0= 1nni=1(Xi − μ0)2 an unbiased estimator of σ2?
10-27. Given that X1, . . . ,Xn depicts a random sample drawn from a variable X that is N(0, σ), find a sufficient statistic for σ2.
10-26. Suppose {X1, . . . ,Xn} is a set of sample random variables drawn from the probability density function:f (x; θ) = θ/(1 + x)θ+1, 0 < x < θ < +∞.Find a sufficient statistic for θ.
10-25. Let {X1,X2, . . . ,Xn} depict a set of sample random variables drawn from the probability density function f (x; θ) = 2θ−2x, 0 < x < θ. Find a best unbiased estimator for θ2.
10-24. Suppose a random variable X has a probability density function of the form f (x; λ) = -λe−λx, x ≥ 0,λ > 0;0 elsewhere.For X1, . . . ,Xn a random sample drawn from this exponential distribution, let T = ni=1 Xi n be an estimator for λ. Find MSE(T, λ).
10-23. Suppose X1, . . . ,Xn is a random sample taken from a variable X that is N(μ, σ). Express P|Xn − μ| < ε as an integral that converges to 1 as n→∞.
10-22. Let {X1,X2, . . . ,Xn} be a set of sample random variables for each of the following probability density functions. Find a sufficient statistic forθ > 0:(a) f (x; θ) = 2θ−2x, 0 < x < θ(b) f (x; θ) = θxθ−1, 0 < x < 1, θ > 0
10-21. Suppose {X1,X2, . . . ,Xn} is a set of independent sample random variables drawn from the probability mass function g(x; θ) = θ(1 − θ)x−1, x = 1, 2, . . . , 0 < θ < 1. Find a sufficient statistic for θ.
10-20. Let X1, . . . ,Xn constitute a random sample drawn from a uniform distribution on 3θ − 1 4 , θ + 1 4 2. Is Xi an unbiased estimator for θ?
10-19. Let X1, . . . ,Xn be a random sample drawn from a N(μ, σ) probability density function. Maximum likelihood estimators for μ and σ2 are X and S2 1= 1nni=1 (Xi − X)2, respectively. Demonstrate that X is an unbiased estimator for μ and S2 1 is a biased estimator for σ2. Find the
10-18. Suppose {X1,X2, . . . ,Xn} is a set of sample random variables drawn from the exponential probability density function f (x; θ) =1θ e−x/θ, 0 < x
10-17. Determine a lower bound for the probability that ˆP= Xn lies within anε-neighborhood of p(= E(ˆP)). For what value of n will this probability exceed the value 1 − δ?
10-16. Suppose {X1,X2, . . . ,Xn} is a set of sample random variables drawn from a N(μ, σ) population. Consider two estimators for μ:T1 =ni=1 Xi n + 1, T2 = 1 2Xi + 1 2n ni=2 Xi.Which of these two estimators has the largest bias? Are these estimators efficient? Are T1 and T2 asymptotically
10-15. Verify that if X1, . . . ,Xn depicts a random sample from a population for which both E(Xk) and V(Xk) exist, then 1nni=1 Xk i is a consistent estimator of E(Xk). Use this result to demonstrate that S2 1= 1nni=1 (Xi − X)2 is a consistent estimator of σ2.
10-14. Let {X1,X2, . . . ,Xn} depict a set of sample random variables drawn from a N(μ, 1) population. For each random variable Yn presented, determine if there exists a real number c such that Yn p −→ c as n→∞:(a) Yn = eXn (b) Yn = eX2 n−2Xn
10-13. Let {X1,X2, . . . ,Xn} be a collection of sample random variables taken from the probability density function f (x; θ) = 1 2 (1 + θx),−1 < x,θ < 1. Find a consistent estimator for θ.
10-12. Let {X1,X2, . . . ,Xn} depict a set of sample random variables drawn from the probability density function f (x; θ) = -θxθ−1, 0< x < 1,θ > 0;0 elsewhere.Find a consistent estimator for θ.

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Advanced Statistics From An Elementary Point Of View 1st Edition Michael J Panik - Solutions

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