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

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
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,
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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 −
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,
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
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,
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
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
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
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
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
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,
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
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%
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
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
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
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
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
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
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%
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
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
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
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
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, . . .
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
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.
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
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
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
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 <
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) =
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.
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
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
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) =
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
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
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
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
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
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
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
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 <
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
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.
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
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
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
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

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