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statistics for business and economics

Questions and Answers of Statistics For Business And Economics

7-35. The length of time required to wash and dry a car at ACE Hand Wash is exponentially distributed with a mean of 13 minutes. What percentage of people will have their car ready within 10 minutes?
7-34. Imperfections in a certain grade of sail canvas occur randomly with a mean of one flaw per 70 square feet of canvas. What is the probability that a piece of sail canvas with dimensions 30 feet
7-33. System failures at an industrial facility are approximately Poisson distributed with λ = 0.25 per hour (the system experiences a failure every four hours). If system startup begins at 7 A.M.
7-32. Suppose that four calls per minute arrive at a switchboard and that the arrival of calls follows a Poisson process. What is the probability that the next call will arrive within seven minutes?
7-31. Suppose a random variable X depicts the life of a circuit board component with probability density function f (x; 0.0005) = -0.0005e−0.0005x, x > 0;0 elsewhere.What is the probability that
7-30. Let X be Poisson random variable with mean λ = 20. Approximate the following probabilities using the normal distribution:(a) P(16 ≤ X ≤ 20)(b) P(16 < X ≤ 20)(c) P(16 < X < 20)
7-29. Traffic safety records indicate that the number of accidents occurring along a particular stretch of road is Poisson distributed with a mean of two accidents per week. Determine the approximate
7-28. Suppose X is a Poisson random variable with parameter λ = 45. Use (7.35)to find P(40 ≤ X ≤ 50). Recalculate this probability by employing the continuity correction P(a ≤ X ≤b) →
7-27. Suppose that 60% of the customers of a local department store that receive a circular in the mail announcing a special sale actually respond to the announcement. If 1000 circulars are mailed,
7-26. Let X be distributed as b(X; 20, 0.3). Use the normal distribution to find the approximate probabilities:(a) P(X ≤ 5)(b) P(X ≥ 10)(c) P(3 ≤ X ≤ 7)
7-25. About 10% of all purchasers of the Zapper microwave oven fill out a marketing survey card that accompanies the warranty statement. What is the approximate probability that at least 20 cards
7-24. Suppose that X is binomially distributed b(X; 15, 0.5). Use the normal approximation to the calculation of binomial probabilities to determine P(X = 8) and P(5 ≤ X ≤ 10).
7-23. Given that a random variableXisN(μ, σ) with probability density function(7.7), use (7.7) to determine X’s central moment-generating function.
7-22. Given that a random variable X is N(μ, σ) with probability density function(7.7), use (7.7) to determine X’s moment-generating function. Once this function is obtained, use it to determine
7-21. Suppose a random variable Z is N(0, 1). Rewrite (7.A.2) in terms of even powers of t.(1) Verify that mZ(t) = ∞n=0 antn, where an = -0 if n is odd;1 2n/2(n2)! if n is even.(2) Using (4.36),
7-20. Rewrite (7.A.5) in terms of even powers of σt. Then demonstrate that m(2)X−μ(0) = σ2.
7-19. Suppose a random variable X is N(μ, σ). Which (if any) of the following statements are true?(a) Since the normal probability density function f (x;μ, σ) is symmetrical about x = μ and the
7-18. Suppose that a random variable Z is N(0, 1). (1) Demonstrate that E(Z) = 0,V(Z) = 1 using (7.13). (2) Suppose Z is not N(0, 1) but, instead, is a continuous random variable with σ > 0. For Z =
7-17. Suppose a random variable X is N(μ, σ). Find E(X),V(X) using (7.7).
7-16. A real-valued function y = f (x) is said to be an odd function if f (−x) =−f (x). For any such function 4 a−a f (x)dx = 0. Using this concept, demonstrate that 4+∞−∞ xe−1 2 x2dx =
7-15. Verify that if a random variable X is N(μ, σ), then (7.7) is a legitimate probability density function.
7-14. Suppose that the amount of cereal X (measured in ounces) placed in a box is N(16, 0.06). Let a denote the amount of cereal such that 95% of all boxes contains at least a ounces. Find a.
7-13. The semester stipend given to college interns at Big Bucks Investments Inc.is normally distributed with a mean of $4,000 and a standard deviation of$600. If the bottom 5% of interns is to
7-12. Given that X is N(19, σ) and P(X < 22) = 0.65, find σ.
7-11. If X is N(μ, 10) and P(X ≤ 90) = 0.95, find μ.
7-10. Find the 20th percentile of the distribution that is N(60, 20).
7-9. If X is N(50, 10), find a number b such that P(−b ≤ X ≤b) = 0.90.
7-8. Chebyshev’s Theorem informs us that the probability that X will deviate from μ by not more than 2σ is at least 0.75. Is this statement valid if X is N(40, 2)? Verify your answer.
7-7. X is N(10, 8). Find:(a) P(|X| ≤ 5)(b) P(|X| > 8)
7-6. X is N(50, 10). Find:(a) P(X ≤ 60)(b) P(X ≥ 55)(c) P(40 ≤ X ≤ 49)(d) γ0.70
7-5. Given that a random variable X has a uniform probability distribution with probability density function (7.1), find X’s moment-generating function.Use it to determine E(X),V(X).
7-4. Given that a random variable X is uniformly distributed with probability density function (7.1), find E(X),V(X).
7-3. If X is uniformly distributed on (α, β), find a value of k such that f (x) = -k, α < x < β;0 elsewhere is a legitimate probability density function.
7-2. The life X of a new miniature battery is uniformly distributed between 48 and 55 hours of continuous use. Law enforcement use requires that it last at least 51 hours. What is the probability
7-1. If a continuous random variableXis uniformly distributed over the interval(3,9), find its probability density function. Whatis its cumulative distribution function? What is P(3.5 ≤ X < 7)?
6-59. For each of the following moment-generating functions, find the associated probability mass function:(a) mX(t) = 0.0156(1 − 0.75et)−3, t < 0.2877(b) mX(t) = 0.45et(1 − 0.55et)−1, −∞
6-58. Verify that (6.60) is a legitimate probability mass function.
6-57. Suppose that in a group of 20 consumers, five prefer brand A espresso, six prefer brand B espresso, and six prefer brand C espresso. A sample of 10 consumers is selected at random without
6-56. Seated at a table are five individuals, four of which are registered Democrats.If two individuals are selected at random without replacement, what is the probability that the non-Democrat will
6-55. A vessel contains 10 chips of which six are red and four are blue. Three are drawn at random without replacement. If X represents the number of red chips drawn, find E(X), V(X) .
6-54. Suppose that out of 120 applicants for a particular job only 80 of them have prior experience at operating a certain piece of milling machinery. If 10 applicants are randomly chosen for an
6-53. Suppose that out of 50 property owners from a given city it is known that 30 support a bond issue for the addition of a new wing to the public library but 20 do not. If five property owners are
6-52. For a Poisson random variable X with probability mass function given by equation (6.47), find E(X), V(X) using equation (6.47).
6-51. Given that a random variable X follows a Poisson probability distribution, determine X’s moment-generating function.
6-50. Given that a random variable X is Poisson distributed, find X’s probabilitygenerating function. What is E(X), V(X)?
6-49. Verify that equation (6.47) is a legitimate probability mass function.
6-48. A night watchman (based on his past experience) estimates that there is only about a 1% chance of completing his rounds early and having to wait to punch in his code on the hour at a security
6-47. Suppose that in a wooded area the airborne particles of a certain type of pollen occur at an average rate of six per cubic foot of air and that the number of particles X found in a cubic foot
6-46. Suppose that the average number of incoming telephone requests for emergency service received by staff members of the regional auto club office is 30 per hour. What is the probability that at
6-45. Suppose the number of suicides in a certain locale is Poisson distributed with parameter λ = 2 per day, where X is the number of suicides per week.What is the probability of exactly 10
6-44. Suppose that the 911 emergency facility of a particular metropolitan area experiences 300 calls per hour. Let X be the number of calls that arrive in a one-minute period. Then X is Poisson
6-43. Suppose that the intensity of the process generating incoming calls at a switching station is two per minute. What is the probability that the station receives exactly 12 calls in a 5-minute
6-42. Suppose that the number of moving violations captured by a traffic camera at a given traffic light is Poisson distributed with three moving violations recorded per day. Let the random variable
6-41. Suppose a random variableXfollows a negative binomial distribution. Find E(X), V(X), using the moment-generating function.
6-40. Given that a random variable X follows a negative binomial probability distribution, determine X’s moment-generating function.
6-39. A girl scout is selling American flags door to door and has a quota of 10 flags to be sold. Let the probability of a sale be 0.25. What is the probability that her quota will be made at the
6-38. A fair coin is tossed a larger number of times. What is the probability that the seventh head is obtained on the tenth toss? What is the probability that it is obtained on the fifteenth toss?
6-37. Suppose that a basketball player has a 75% of chance of making a freethrow shot. The player shoots free throws until a total of 10 are made. What is the probability that 12 shots will be
6-36. Suppose that for a large number of red and blue marbles within a jar the probability of obtaining a red marble is 1 4 and the probability of getting a blue one is 3 4 on any given draw. Marbles
6-35. Given that a random variableXfollows a geometric probability distribution, find X’s moment-generating function.
6-34. Given that a random variableXfollows a geometric probability distribution, find X’s probability-generating function. Use it to obtain E(X), V(X).
6-33. Given that a random variableXfollows a geometric probability distribution, verify that (6.27) is a legitimate probability mass function.
6-32. The caps on soda bottles are examined with a scanning device in order to determine if they are properly set. Experience dictates the probability of detecting an improperly set cap is 0.01. What
6-31. Amajor candy company markets 35 different products. Past experience dictates that there is a 2% chance that in any given year a dissatisfied customer will file a complaint with his or her
6-30. Suppose that 1 in 1000 bottle cups of Zesty Cola contains the letterZprinted on its underside. This special bottle cap is redeemable at any store selling this cola for a special prize. If X is
6-29. Suppose an individual stands at the free throw line on a basketball court and shoots until she makes a basket. Let the free throws be independent. If experience dictates that this person has an
6-28. Suppose an individual is going to roll a bowling ball until he or she get a strike. If, under independent rolls, this person has a probability of 0.20 of making a strike and if X is the number
6-27. Customers at the Big Deal Outlet Store pay for their purchases by check(10%), cash (5%), major credit card (70%), or store credit card (15%).What is the probability that among the next eight
6-26. Twelve six-sided fair dice are tossed simultaneously. What is the probability that each odd number appears exactly two times and that each even number appears exactly two times?
6-25. Suppose that in a particular game a play renders 1, 2, 3 or 4 points with associated probabilities p1 = 0.60, p2 = 0.30, p3 = 0.08, and p4 = 0.02.Under the assumption that these probabilities
6-24. Suppose that in a certain locale 30% of the households purchase Brand A soap powder, 50% purchase Brand B soap powder, and 20% of the households use Brand C soap powder. For a random sample of
6-23. Suppose that a random variable X follows a binomial distribution with E(X) = np, V(X) = np(1−p) . HereXis the number of successes obtained in n independent trials of a simple alternative
6-22. For X a binomial random variable with probability mass function given by equation (6.15), find E(X), V(X) using equation (6.15).
6-21. Let X be a binomial random variable with probability mass function provided by equation (6.15). Determine the moment-generating function forX.
6-20. Verify that the probability generating function for a binomial random variableXisφX(t) = [(1 − p) + pt]n. Use it to determine the mean and variance of X.
6-19. Verify that (6.15) is a legitimate probability mass function.
6-18. Verify that the probability-generating function for the Bernoulli random variable X is φX(t) = (1 − p) + pt. Use it to find E(X), V(X).
6-17. Let the random variable X be binomially distributed with probability mass function b(X; 3, p) = 3 XpX(1 − p)3−X, X = 0, 1, 2, 3. Determine the cumulative distribution function P(X ≤ t)
6-16. Experience dictates that a certain basketball player makes 80% of his free throws and that those free throws are independent. If he gets eight free throws in a particular game, what is the
6-15. Suppose that30%of all persons taking a particular driver’s education course fail the written exam for a driver’s license. If the written exam is given to five randomly chosen individuals,
6-14. Suppose a fair coin is tossed 10 times in succession. What is the probability of obtaining at least seven heads?
6-13. A fair pair of dice is rolled n = 4 times. Define a success as the sum of the faces is nine. Determine the implied probability distribution. What is the probability of at least two nines? What
6-12. A true or false exam has 20 questions. If a student guesses at the answer to each question, what is the probability that he or she guesses correctly on more than half of the questions?
6-11. Let the proportion of successes in a sample of size n be Y = X/n. Since Y = y and X = nY = ny are equivalent events (there exists a one-to-one correspondence between X and Y), it follows that
6-10. Afair coin is tossed five times. Using the binomial formula, what is the probability of X successes in the five trials? Determine the resulting probability distribution. Find E(X) and V(X).
6-9. Astudent takes a multiple choice exam that contains 15 questions, each with four possible answers. If a student guesses at the answer to each question, what is the probability that he or she
6-8. We toss a fair coin n = 10 times. What is the probability of observing four heads followed by six tails?
6-7. Suppose a dart player has a probability of 8/9 of hitting the bullseye and that his throws are independent. If the player is given three darts, what is the probability that:(a) he hits the
6-6. Suppose that for a Bernoulli process p = 0.15. Also, suppose that n = 10 items are drawn from this process and we find that the first, third, and eighth items have the same particular
6-5. Suppose a vessel contains 10 identical marbles of which four are red and six are blue. For each of four random draws from the vessel a marble’s color is recorded and the marble is returned to
6-4. Verify that if a random variable X follows a discrete uniform distribution with probability mass function given by (6.9), then E(X) = k+1 2 , V(X) = k2−1 12 .
6-3. Let a random variable X follow a discrete uniform distribution with probability mass function given by (6.9). Find X’s moment-generating function.
6-2. Let a random variable X follow a discrete uniform distribution with probability mass function given by (6.9). Determine X’s probability-generating function.
6-1. For a particular variety of electronic bulletin board the number of components having unacceptable reliability coefficients was evenly distributed between 7 and 30. Determine the probability of
5-57. (Bayes’ Rule for continuous random variables) We may posit the multiplication theorem for probability density functions as f (x, y) = g(x)h(y|x) =h(y)g(x|y). Then using this expression, along
5-56. (Bayes’ Rule for discrete random variables) Using (5.4) and (5.13), express(5.9) as g(X|Y) = g(X)h(Y|X)i g(X)h(Y|X).Similarly, show that (5.11) can be written, via (5.3) and (5.13), as
5-55. Suppose that (X,Y) is a bivariate random variable and that X and Y are independent continuous random variables with moment-generating functions mX(t) and mY(t), respectively. Verify that for
5-54. Suppose that X1, . . . ,Xn are independent continuous random variables, where the probability density function of each Xi appears as f i(x) = -1λ e−xi /λ, xi > 0, λ > 0, i = 1, . . . , n;0
5-53. Suppose that X and Y are independent continuous random variables with bivariate probability density function f (x, y). Verify that for U = X + Y, mU(t) = mX(t) · mY(t). (Hint: Use equation
5-52. Given the following bivariate probability density functions, find the joint moment-generating functions for the random variables X and Y:(a) f (x, y) = -4xy, 0< x < 1, 0 < y < 1 0 elsewhere(b)

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