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statistics for business and economics

Questions and Answers of Statistics For Business And Economics

10-11. Let {X1,X2, . . . ,Xn} constitute a set of sample random variables taken from a general population probability density function with E(Xi) = μ =μ,E(X2 i ) = μ2, and E(X4 i ) = μ4 all
10-10. A set of sample random variables {X1,X2,X3} is taken from a population with mean μ and variance σ2. Find the mean squared error of the following estimators for μ:(a) T1 = (X1 + X2 +
10-9. Let {X1,X2, . . . ,Xn} represent a set of Bernoulli random variables. For X an estimator of unknown p (the probability of a success), find MSE(X, p).
10-8. The statistics X and S2 are unbiased estimators of μ and σ2, respectively,(for all μ and σ2) for {X1,X2, . . . ,Xn} a set of sample random variables.Find the mean square errors of X and
10-7. Let {X1,X2, . . . ,Xn} be a set of independent and identically distributed sample random variables drawn from a binomial population with unknown parameter p. Two possible estimators for p are
10-6. Suppose {X1,X2, . . . ,Xn} depicts a set of sample random variables drawn from a population for which E(Xi) = μ and V(Xi) = σ2. Let T1 =ni=1 Xi n+1 and T2 = ni=1 Xi n−1 be possible
10-5. Suppose that T1 and T2 are independent unbiased estimators of a parameterθ, with V(T1) = σ2 1 and V(T2) = σ2 2. If T3 = αT1 + (1 − α)T2,α constant, is a new unbiased point estimator of
10-4. Suppose {X1,X2, . . . ,Xn} is a set of sample random variables taken from a population that is N(μ, σ). Is T = (X1−X)2 n an unbiased estimator of σ2?
10-3. Suppose {X1,X2, . . . ,Xn} is a set of sample random variables taken from a uniform distribution on the interval (θ, θ +1). Is T1 = X −0.5 an unbiased estimator of θ? Is T2 = 2X an
10-2. Suppose {X1,X2, . . . ,Xn} is a set of sample random variables taken from a population with mean μ and variance σ2 and that T1 = X and T2 = X1+Xn 2 are point estimators of μ. Are T1 and/or
10-1. Let X1, . . . ,Xn constitute a random sample taken from an exponential distribution with probability density function f (x; θ) = -1θ e−x/θ , x > 0,θ > 0;0 elsewhere.Which of the following
9-24. If F is F10,7, find:(a) The F-value with 5% of the area to its right(b) The area to the right of the value 6.62
9-23. If F is F4,5, what is the lower 1% point? Suppose F is F9,7. What is the lower 5% point?
9-22. Suppose the random variable F follows an F-distribution with v1 = 4 and v2 = 8 degrees of freedom, respectively. For α = 0.025, find f1−α,v1,v2 .If α = 0.10, find f1−α,v1,v2 .
9-21. Suppose a random variable F is F-distributed with v1 = 6 and v2 = 8 degrees of freedom, respectively. What is E(F) and V(F)? Determine α3, α4, and the mode of F.
9-20. Suppose a random sample with realizations 1,2,3,4,7, and 10 is drawn from a population that is N(μ, σ) with both μ and σ unknown. Determine the(approximate) probability that X will be
9-19. Find the probability that X does not exceed 290 when a sample of size n = 10 has been taken from a N(305, σ) population and s = 20.
9-18. A sample of size n = 16 is extracted from a N(18, σ) population. If ¯x = 20 and s = 2, find P(X ≥ 19).
9-17. Suppose a random variable X is N(15, σ), where σ is unknown. From a sample of size n = 10 it has been determined that ¯x = 13 with s = 2. Find P(X ≤ 14).
9-16. Suppose a random variable T is t-distributed with v = 10 degrees of freedom. For α = 0.05, find:(a) tα,v(b) tα/2,v.What is V(T)? Find α4. What is P(T ≥ 1.729) when v = 19? For v = 14,
9-15. Suppose a random variable X is N(μ, 1.5). If a random sample of size n = 20 is drawn from the population distribution, determine the probability that S2 > 2.5 (squared units).t Distribution
9-14. From Theorem 9.5 it was determined that Y = (n − 1) S2σ2 is χ2 n−1. Determine the moment-generating function for Y.
9-13. Suppose X1, . . . ,Xn depicts a random sample drawn from a population that is N(μ, σ). Demonstrate that X and S2 are independent random variables.(Hint: First show that if X is independent
9-12. Demonstrate the validity of Theorem 9.2 by using the results offered by Theorems 9.1 and 9.3.
9-11. Demonstrate that Theorem 9.3 is valid by showing that the momentgenerating function for Y = ni=1 Xi corresponds to that of a probability distribution that is χ2 v , where v = ni=1 vi.
9-10. Suppose a random variable Xi is χ2 vi . Determine Xi’s moment-generating function. (Hint: Use the moment-generating function for the gamma probability distribution.)
9-9. Demonstrate that the statement made in Theorem 9.1 is valid.
9-8. Verify that the moment-generating function derived in Exercise 9-7 is a special case of the gamma moment-generating function.
9-7. Suppose a random variable X is χ2 n . Demonstrate that X’s momentgenerating function is of the form mX(t) = (1 − 2t)−n/2, t < 1 2 .
9-6. Demonstrate that if the random variable Z is N(0, 1), then Y = Z2 is χ2 1 .(Hint: We need to show that the moment-generating function of Z2 corresponds to that of a chi-square distribution with
9-5. Suppose X is N(μ, 4), where μ is unknown. For n = 25, find:(a) P(S2 < 18)(b) P(12 < S2 < 20)
9-4. Suppose X is N(30, 3). For a random sample of size n = 40, find P(S2< 10), where S2 is the sample variance.
9-3. For v = 60 degrees of freedom, use the standard normal approximation to determine the following quantiles of the chi-square distribution:(a) χ2 0.90,v(b) χ2 0.95,v
9-2. If X is χ2 v , find:(a) χ2 0.90,10(b) χ2 0.05,15(c) χ2 0.99,5
9-1. Suppose a random variable X is chi-square distributed with ν = 8 degrees of freedom. What is E(X), V(X), α3, and α4 ?
8-39. (Sampling Distribution of the Median) Given a random sample of size n, let us arrange the sample random variables X1, i = 1, . . . , n, in order of increasing numerical magnitude. Let the
8-38. Let X1, . . . ,X25 constitute a set of sample random variables drawn from a population that is N (μ, σ) = N (100, 35). What is the form of the sampling distribution of X? Determine the
8-37. Suppose X1,X2 are independent random variables with X1 ∼ N (4, 6.2), X2 ∼ N (5, 7.1). If we form a new random variable Y = 2X1 − 3X2, find P (Y > 8). What is the form of the
8-36. Determine the moment-generating function of the observed relative frequency of a success ˆP = Xn, whereX = ni=1 Xi is the sum of n independent Bernoulli random variables Xi ( = 0 or 1), i =
8-35. Suppose X1, . . . ,Xn depicts a random sample drawn from the gamma probability density function (7.48). What is the form of the sampling distribution of X?
8-34. Let {X1, . . . ,Xn} be a set of sample random variables drawn from a distribution that is N(σ,μ). Demonstrate that the limiting moment-generating function of Xn is eμt . How do we interpret
8-33. Let a random variable X be binomial b(X; n, p). Demonstrate that the limiting moment-generating function of X is the moment-generating function of a Poisson random variable Y with mean λ.
8-32. Demonstrate that the mean X of a random sample of size n drawn from a population random variable X, which is N(μ, σ), is distributed as N(μ, σ/√n).
8-31. Given the sample values 1, 3, 1, 5, 8, and 2, find M1 and M2. Use them to determine S2. Also find M3 and M4.
8-30. Suppose the sampling distribution of the sample variance has been determined from samples of size n = 4 taken from a population variable X : −2, 0, 1, 2, 3, 4, 5, 7. Find its mean and
8-29. In a mayoral election last year candidate A received 55% of the 8,576 votes cast. If a random sample of n = 200 eligible voters had been taken the day before the election, what would have been
8-28. A particular socioeconomic group has a mean annual income of μ =$35,800 with σ = $3,500. A random sample of n = 250 individuals from this group is taken. What is the probability that X will
8-27. Suppose a random variable X is N(4, 1.5). If a random sample of size n = 25 is drawn, what is the probability that X will exceed 4.5?
8-26. Based upon past experience, it is known that a certain manufacturing process has a defect rate of 3%. Arandom sample of size n = 450 is taken from a large production run and the CEO wants to
8-25. Suppose Xn is N 0, n−1/2. Verify that {Xn} converges to 0 in distribution.
8-24. Let Xn be b(X; n, p). Does 1 − (Xn/n) converge stochastically to 1 − p?
8-23. Let X1, . . . ,Xn be independently distributed N(μ, 1) random variables with −∞ < μ < +∞. Find a real number A such that eXn converges in probability to A.
8-22. Let X1, . . . ,Xn be independently distributed Poisson p(X; λ) random variables with λ > 0 for all i = 1, . . . , n. Demonstrate that Xn converges in probability to E(Xi) = λ.
8-21. Let X1, . . . ,Xn be independently distributed Bernoulli b(Xi; n, p) random variables with 0 < p < 1 for all i = 1, . . . , n, where E(Xi) = p and V(Xi) = p(1 − p). Demonstrate that Xn
8-20. Which of the following statements are true?(a) If X is N (μ, σ), then X is N(μ, √σn ).(b) If X is not N(μ, σ), then X is approximately N(μ, √σn ) for large n.(c) If X is not N
8-19. Suppose that a population variable X has the values 1, 2, 3, 4, 5, and 6.Determine the sampling distribution of the mean for samples of size n = 2.What is its mean and standard deviation?
8-18. N = 8 different items are within a population. If we wish to select a sample of size n = 3, how many ways are there of sampling? What is the probability that any one sample will be chosen? What
8-17. Given the letters A, B, C, and D, determine the number of different samples of size n = 2 that can be selected:(a) With replacement(b) Without replacement If n = 3, recalculate (a) and (b).
8-16. Suppose that X has an unknown mean μ with σ2 = 3. How large of a sample must be taken so that the probability will be at least 0.90 that X will lie within 0.6 of the population mean? How
8-15. Suppose a population variableXhas an unknown mean μ but a known varianceσ2. Arandom sample of size n is selected. Are the following quantities statistics?(a) X(b) X + μ(c) X + σ2(d)
8-14. A nutritionist claims that about 50% of the individuals tasting a new soft drink can detect the presence of a certain artificial sweetener. If the claim is correct, find the probability that
8-13. In a sample of n = 100 observations the number of successes was found to be X = 35. If the population proportion is p = 0.40, find:(a) P(ˆP ≤ 0.38)(b) P(ˆP ≥ 0.50)
8-12. If X1, . . . ,Xn depicts a random sample of size n taken from the probability density function f (x; 2) = 2e−2x, 0 ≤ x < +∞ (see Exercise 8-11), find P (X1 > 10,X2 > 10, . . . ,Xn > 10).
8-11. Let X1,X2, and X3 depict sample random variables drawn from a population with probability density function f (x) = 2e−2x, 0 ≤ x < +∞.What is the joint probability density function of
8-10. Let X be the outcome obtained on the toss of a fair single six-sided die and let Y be the number of heads obtained on three flips of a fair coin.Determine the joint probability mass function.
8-9. A population distribution has an unknown mean with σ = 2.5. How large of a sample must be drawn so that the probability is at least 0.99 that Xn will lie within one unit of the population mean?
8-8. Suppose a population variable X has a mean of 35 and a standard deviation of 10. What is the expression for σ X, the standard error of the mean? If a sample of n = 2000 is taken, what is the
8-7. Suppose a population variable X is N(2800, 2500). If a random sample of size n = 36 is selected, find the probability that the sample mean X will not differ from the population mean μ by more
8-6. If a population variable X is N(200, 12) and the sample size is n = 16, find:(a) P(X > 206)(b) P(X ≤ 209)(c) P(195 < X < 203)
8-5. Let the random variables X1, . . . ,X4 be Poisson distributed with mean λ.Determine the joint probability distribution of the Xi, i = 1, . . . , 4. Are these random variables independent?
8-4. List all possible samples of size n = 2 given the population X : A,B,C, D,E, F. If A = 3, B = 5, C = 6 = D, E = 7, and F = 8, determine the sampling distribution of the mean.
8-3. Suppose that from a population of size N = 20 random samples of size n = 4 are to be selected. How many possible samples can be obtained?What is the probability that any one of them will be
8-2. Suppose as in Exercise 8-1 we know that X is N(μ, 1.5). How large of a sample will be needed if we require that P(|X − μ| ≤ 0.5) = 0.95?7
8-1. Suppose a random variable X is N(μ, 1.5). Given n = 10, determine the probability that X will not differ from μ by more than 0.5 units. (Hint:Z = (X − μ)/σX is N(0, 1).)
7-61. Suppose a random variable X is beta distributed with probability density function given by (7.56). Use this density function to obtain μr , the rth moment about zero. Use the resulting
7-60. Verify that if a random variableXis beta distributed with probability density function (7.56), then (7.56) is a legitimate probability density.
7-59. Let a random variable X be beta distributed with α = 2 and β = 3. Find:(a) P(X < 0.1)(b) The probability that X will assume a value within one standard deviation of the mean
7-58. The proportion of defective parts produced by a metal stamping machine follows a beta distribution with α = 1 and β = 10. Find:(a) The probability that the proportion of defective units
7-57. Bill and Alice’s Country Store has propane tanks that are filled every Wednesday. Experience indicates that the proportion of gas used between fills is beta distributed with α = 3 and β =
7-56. Suppose that for a beta distribution n = 12, α = 6 and p = 0.30. Using(7.68), find the probability that X is less than or equal to p. What is the probability that X is less than or equal to
7-55. Suppose a random variableXis beta distributed with parameters α = β = 3.Find X’s probability density function. Then determine P(X ≥ 0.75).
7-54. Suppose a random variable X follows a gamma distribution with cumulative distribution function given by (7.49). Demonstrate that (7.49.2) can be obtained from (7.49) by successive integration
7-53. Suppose a random variable Xis gamma distributed with probability density function (7.48). Demonstrate that the rth moment about zero, μr , can be written as μr= θr(α + r)/(α). Then use
7-52. Suppose a random variable X follows a gamma distribution with probability density function given by (7.48). Determine X’s moment-generating function.
7-51. Suppose a random variable Xis gamma distributed with probability density function given by (7.48). Demonstrate that E(X) = αθ, V(X) = αθ2.
7-50. Let X be gamma distributed with α = 2 and θ = 50. Find:(a) The probability that X assumes a value within two standard deviations of the mean(b) The probability that X assumes a value below
7-49. Cars arrive at a toll booth at an average rate of 10 cars every 20 minutes via a Poisson process. Determine the probability that the toll booth operator will have to wait longer than 30 minutes
7-48. Given the gamma function (7.42), use integration by parts to demonstrate that:(a) (α) = (α − 1)(α − 1)(b) If α is a positive integer, (α) = (α − 1)(α − 2) . . . 2 · 1(1)(c)
7-47. Let a random variable X be gamma distributed with probability density function given by (7.48). What is the role of θ? Demonstrate that if X is gamma distributed with parameters α and θ,
7-46. Given that a random variable X is gamma distributed with probability density function (7.48), verify that (7.48) is a legitimate density function.
7-45. If X is gamma distributed with θ = 4 and α = 2, find P(X < 8).
7-44. Suppose that 20 customers per hour arrive at a gas station, where arrivals are assumed to follow a Poisson process. If a minute is one unit, find λ.What is the probability that the station
7-43. Suppose a random variableXis gamma distributed with parameters α = 2,θ = 10. Find:(a) X’s probability density function(b) E(X) and V(X)(c) X’s cumulative distribution function(d) P(15 < X
7-42. Let a random variable X have a probability density function given by(7.39). Determine X’s moment-generating function and use it to determine E(X),V(X).
7-41. For a random variable X with exponential probability density function(7.39), verify that (7.41) holds.
7-40. Suppose that X is an exponential random variable with probability density function (7.39). Demonstrate that E(X) = 1λ ; V(X) = 1λ2 .
7-39. Verify that if a random variable X follows an exponential probability distribution, then (7.39) is a legitimate probability density function. (Note:Readers not familiar with the gamma function
7-38. The amount of time required for a customer at Big Bank to cash a check is exponentially distributed with a mean time of 1.35 minutes. Find:(a) λ(b) The probability that a customer will cash a
7-37. Let X follow an exponential distribution with parameter λ = 3. Find:(a) P(X > 1)(b) P(2 < X ≤ 5)(c) E(X) and V(X)
7-36. The distribution of length of operating life before the first malfunction of a certain electrical component is exponential with E(X) = 350 hours. What is the form of X’s probability density

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