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statistics for business and economics

Questions and Answers of Statistics For Business And Economics

4-18. Can the expression f (x) = -6−|X−7|36 , x = 2, 3, . . . , 12;0 elsewhere serve as a probability mass function? Howis the random variableXdefined?
4-17. Comment on the following statement: IfXis a continuous random variable, then its probability density function need not be continuous; however, its cumulative distribution function will be
4-16. Comment on the following statement: The probability mass function of a discrete random variable X has 1 as an upper bound but the probability density function of a continuous random variable X
4-15. Can the expression F(t) = -0, t < 0;1 − e−t , t ≥ 0 serve as a cumulative distribution function for a continuous random variable X? Find the associated probability density function and
4-14. Let the random variable X have the probability density function f (x) = -x−2, 1< x < +∞;0 elsewhere.For events A = {x|1 < x < 2} and B = {x|4 < x < 5}, find the probability that A or B
4-13. Given the cumulative distribution function F(X) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩0, X < 0;1 2, 0≤ X < 1;3 4, 1≤ X < 4;1, X ≥ 4, determine the associated probability mass function.
4-12. Let a random variable X be defined as the number of heads obtained in two flips of a fair coin. Determine the sample space S and the associated(discrete) probability distribution. Verify that
4-11. Suppose f (x) = -k(3 − x), 0 ≤ x ≤ 3;0 elsewhere.What value of k makes f (x) a legitimate probability density function? Find the cumulative distribution function F(t). Using f (x), find
4-10. For 0 < p < 1, is the expression F(X) = 1 − (1 − p)X,X = 1, 2, . . ., a legitimate (discrete) cumulative distribution function?
4-9. Is the expression F(t) = (1 + e−t)−1 a legitimate (continuous) cumulative distribution function?
4-8. Does F(t) given below satisfy the properties of a cumulative distribution function? Find the associated probability density function and use it to find P(0.30 < X < 0.80).F(t) =⎧⎪⎨⎪⎩0,
4-7. Given the following cumulative distribution function, find:(a) f (0)(b) f (2)(c) f (3)(Hint: f (Xi) = F(Xi) − F(Xi − 1).)F(X) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩0, X < 0;1 2, 0≤ X ≤
4-6. Determine the cumulative distribution function F(X) given the following probability mass function. Use F(X) to determine:(a) P(0 < X ≤ 2)(b) P(X ≤ 1)(c) P(−1 < X ≤ 2)(d) P(−1 < X ≤
4-5. Determine the cumulative distribution function F(X) associated with the following probability mass function.X f(X)0 0.215 1 0.433 2 0.288 3 0.064 1.000
4-4. Let the cumulative distribution function for a random variable X appear as F(t) =⎧⎪⎨⎪⎩0, t < 0;2t − t2 0 ≤ t ≤ 1;1, t > 0.Find:(a) P(X ≤ 1/4)(b) P(X ≥ 1/4)(c) P(1/3 ≤ X ≤
4-3. Determine the constant k such that f (x) = -kx2, −2 ≤ x ≤ 2;0 elsewhere is a probability density function. Then find:(a) The cumulative distribution function(b) P(X > 1/2)(c) P(X ≤ 1)(d)
4-2. For a discrete random variableX, determine the constant k such that f (X) =k/X, X = 1, 2, 3, 4, 5, is a probability mass function. Then determine:(a) The cumulative distribution function(b) P(X
4-1. A random variable X has a probability density function of the form f (x) = -ke−x/3 x > 0;0 elsewhere for a specific constant k. Determine:(a) the value of k(b) the cumulative distribution
3-50. Vessel A1 contains four white (W) marbles, vessel A2 contains two red (R)and two white marbles, and vessel A3 contains four red marbles. Suppose P(A1) = 0.5, P(A2) = 0.4, and P(A3) = 0.1 . If a
3-49. Suppose a gem dealer has a 90% chance of correctly discriminating between natural versus color-enhanced gemstones. If the dealer does appraisals and 75% of the gems that he or she examines are
3-48. A&E Opticians provide free adult screening for glaucoma with the purchase of a pair of glasses. Let G be the event that the customer actually exhibits the early stages of glaucoma and let event
3-47. ACE Metal Products Corp. produces steel pins of a specific diameter. It uses three machines (denoted as A, B, and C) in the initial grinding phase of production. Machines A, B, and C have a
3-46. Passenger cars account for about 25% of all vehicles on the road. If a passenger car is in an accident, there is about a 10% chance of a fatality; the chance of a fatality is only 3% if an
3-45. Sunflower seeds purchased from vendor A1 have an 85% germination rate.Those purchased from vendor A2 have a 90% germination rate, and such seeds purchased from vendor A3 have a 75% germination
3-44. Suppose A, B, and C are events in the sample space S. If events A and B are independent, they might not be independent given C. Comment.
3-43. Suppose events A1, . . . ,An are contained within the sample space S. For r = 3 and 4, verify that 2r − r − 1 equalities must hold for these events to be independent.
3-42. Bayes’ law has often been characterized as involving a reverse process of reasoning; that is, reasoning from effect to cause. Comment.
3-41. Prove that for events A,B ⊆ S, P(A ∩ B) = P(B) − P(A ∩ B).(Hint: Let B = (A ∩ B) ∪ (A ∩ B). )
3-40. Prove that for events A, B, and C within S, P(A ∪ B|C) = P(A|C) + P(B|C) − P(A ∩ B|C).(Hint: Use the distributive law A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). )
3-39. Suppose we select two balls at random without replacement from a vessel containing four red and six blue balls. Find the probability that:(a) Both are red(b) The second ball is red
3-38. Experience dictates that at a particular clinic, about 80% of those attempting to quit smoking by undergoing hypnosis actually have stopped smoking for at least six months. If those patients
3-37. Twenty percent of the students in the freshman class of a certain college have decided on a major and 80% have not. Among those who have already chosen a major, 20% are planning to attend
3-36. Applicants at the ABC employment agency were asked if they had graduated high school. The following table displays their responses to this question.High School Gender Graduate (HSG) Male (M)
3-35. Suppose we toss a single fair six-sided die and we observe the face showing.Given the following events:A: an even number occurs;B: an odd number occurs; and C: a 3 or 4 occurs,determine if:(a)
3-34. Amanufacturer of automatic garage door openers buys circuit boards from three different suppliers: 40% of the total supply comes from Vendor A, 10% comes from Vendor B, and 50% comes from
3-33. SupposeAandBare independent events in S. DoesP(A∩B) = P(A)·P(B)?(Hint: Apply the general addition rule, corollary 1, and DeMorgan’s law.)
3-32. LetA1, . . . ,An be a set of events defined on S. Suppose S = ∪ni=1Ai, P(Ai) >0 for all i and Ai ∩ Aj = φ, i = j. Then for any event B, verify that P(B) = ni=1 P(B|Ai ) · P(Ai). (Hint:
3-31. Four aces are removed from an ordinary deck of 52 playing cards. The remaining 48 cards are put aside and the four aces are shuffled and laid face down on a table. Let us define the events:C:
3-30. Demonstrate that if A,B ⊆ S are independent events, then so are:(a) A and B(b) A and B(c) A and B
3-29. Demonstrate that for a finite sequence of mutually exclusive events A1,A2, . . . ,An within S, P(∪ni=1Ai |B) = ni=1 P(Ai |B).
3-28. A jar contains three white and seven black marbles. Two are drawn at random without replacement. What is the probability that they are both white?What is the probability that they are of a
3-27. If a theorem is specified in terms of n and involves a statement that some relationship holds when n is any positive integer, then a proof of the theorem by mathematical induction proceeds as
3-26. Acoin and a die (each fair) are tossed simultaneously. Determine the simple events within the sample space. Find:(a) P(at least a 4 on the die)(b) P(the coin shows heads)(c) P(not more than 3
3-25. Suppose we toss a fair coin twice in succession (or we toss two fair coins simultaneously). DefineH1 as “heads on the first toss” andH2 as “heads on the second toss.” Verify that H1 and
3-24. Two cards are to be randomly selected from an ordinary deck of 52 playing cards. Determine the probability of drawing two aces if: (a) the first card is not replaced; and (b) the first card is
3-23. Let the proportion of the women in the population with at least one college degree be wc and the proportion of men in the population with at least one college degree be mc. If the proportion of
3-22. A fair single six-sided die is tossed. For the events A1 = {the face shows even}; A2 = {the face shows odd}; A3 = {3 or 4 shows}; A4 = {at least a 3 shows}; A5 = {5 or 6 shows}; and A6 = {at
3-21. Suppose we toss a fair six-sided blue die and a fair six-sided red die simultaneously. What is the probability that:(a) The sum of the faces showing is nine, given that the outcome on the red
3-20. Ared and a blue die (both fair) are tossed simultaneously. Given the events:A1 = {the red die shows odd}A2 = {the blue die shows odd}A3 = {the sum of the faces is odd}determine if A1,A2, and A3
3-19. For events A,B ⊆ S, demonstrate that:(a) if B ⊆ A, then P(B) ≤ P(A)(b) P(A ∩ B) ≥ 1 − P(A) − P(B)
3-18. Ared and a blue die (each fair) are tossed simultaneously. Given the events:A = {the sum of the faces is odd}B = {the blue die shows 1}C = {the sum of the faces is 7}determine if:(a) A and B
3-17. Suppose A, B, and C are events within the sample space S. Demonstrate that P(A∩B∩C) = P(A) ·P(B|A) ·P(CA ∩ B), given that P(A∩B) = 0.
3-16. A fair coin is tossed twice in succession. Find:(a) The probability of two heads given a head on the first toss.(b) The probability of two heads given at least one head.
3-15. Suppose A and B are events within the sample space S and that P(A) =P(B) = 1 2 .Answer the following:(a) If P(B|A) = 1 2 , are A and B independent? Are A and B mutually exclusive?(b) If A and B
3-14. A jar contains six red marbles and four blue ones. Three are drawn at random without replacement. What is the probability that all are blue?What is the probability of a blue on the first draw
3-13. For events A,B ⊆ S, prove that P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
3-12. Forty percent of the subscribers to the Dear Hearts dating service are men(M) and 60% are women (W). Among the men, 70% indicate that they enjoy travel (T ) and 80% of the women state that
3-11. A vessel contains six blue and four red marbles. Three random draws are to be made from the vessel in the following fashion: for each draw a marble is selected and its color is recorded. It is
3-10. Given the following partitioning of the sample space S, find:S A A B 3 4 C 1 5 D 1 1 E 2 3(a) P(A ∩ B)(b) P(A ∪ C)(c) P(B ∪ D)(d) P(C ∩ E)(e) P(A|D)(f) P(D|C)(g) P(A|C)(h) P(A|B)(i)
3-9. In a lot of 200 manufactured items, 35 are defective. Two are drawn at random without replacement. Find the probability that both are defective.
3-8. If A and B are independent events with P(A) = 0.4 and P(B) = 0.3, find:(a) P(A ∪ B)(b) P(A ∩ B)(c) P(A ∪ B)(d) P(A|B)(e) P(B|A)
3-7. For events A,B ⊆ S , the set difference A − B is the set of elements in A that are not in B. Does A−B = A∩B ? Verify that P(A−B) = P(A∩B) =P(A) − P(A ∩ B).
3-6. Two ointments (call themAand B) for a particular type of skin disorder are being tested. Ten percent of patients did not show any improvement after one week. Half of them used ointment A. If
3-5. At a particular university students have either undergraduate (U) or graduate(G) status and either live in a dorm (D) or live off campus (O). One hundred students were chosen at random: 77 were
3-4. A fair pair of (six-sided) dice is tossed and the values of the faces showing are observed. List the points or simple events with coordinates (I, II) in the sample space S, where “I” denotes
3-3. Verify that for S a sample space and A ⊆ S, P(A) = 0:(a) P(S |A) = 1(b) P(φ |A) = 0
3-2. For events A and B within a sample space S, let P(A) = 0.4, P(B) = 0.3, and P(A ∩ B) = 0.1. Are events A and B mutually exclusive? Are they collectively exhaustive? Are they independent
3-1. The nonmanagerial employees of ABC Corp. are classified into the following mutually exclusive groups: production, development, sales, and handling. The results of a recent union poll pertaining
2-15. Measures of dispersion, stated in either absolute or relative terms (e.g., the standard deviation or coefficient of variation, respectively) are standard fare when it comes to analyzing
2-14. Given the sample values X1, . . . ,Xn, let us define:a. Geometric Mean (G.M.)G.M. = n!X1 X2, · · · ,Xn. (2.E.5)(Note that log(G.M.) = 1nni=1 logXi.)b. Harmonic Mean (H.M.)H.M. = n ni=1
2-13. Descriptive Statistical Measures (Grouped Data)Given an absolute frequency distribution involving k classes (as defined in the preceding exercise), we may define the following descriptive
2-12. Absolute/Relative Frequency Distribution (Grouped Data)An absolute frequency distribution shows the absolute frequencies with which the various values of a variable X are distributed among
2-11. Two middle-school teachers were asked to rate N = 10 different essays on the basis of clarity of presentation. They were asked to assign grades of A, B, C, D, or F to each essay, with A being
2-10. Two judges were asked to rate N = 10 different perfumes on a scale from 1 (poor) to 10 (excellent). Given the following rating outcomes, determine if there exists a reasonably strong degree of
2-9. IfY = X/SX, where SX is the standard deviation ofX, prove that Y = X/SX and SY = 1. Also verify that for Z = (X − X)/SX, Z = 0 and SZ = 1.
2-8. Given the following N = 10 data points of the form (Xi,Yi):(1, 1), (2, 3), (3, 1), (4, 3), (6, 3), (6, 5), (6, 6), (8, 6), (8, 8), (9, 10), find ρXY.
2-7. For data set (B) in Exercise 2-2, find v3 and v4. What are the third and fourth standard moments? Interpret their values.
2-6. Transform the following collection of sample observations into a set of Z-scores: 2, 11, 22, 25, 26, 22, 25, 27, 32, 40. Are any of the data values outliers? Find the 20% trimmed mean. What are
2-5. For each of the data sets appearing in Exercise 2-2, findQ1,Q2, P10, P20,D4, and D8. What is the quartile deviation for each of these data sets?
2-4. Use Chebyshev’s theorem to determine the amount of data lying within 2.5 standard deviations of the mean. What is implied interval for data set (A)in Exercise 2-2?
2-3. Which data set in Exercise 2-2 has more variability associated with it?
2-2. For each of the following sample data sets, find the mean, median, mode, and standard deviation:(A) 7, 8, 10, 7, 3, 11, 13, 10, 4, 14(B) 6, 2, 1, 3, 0, 10, 5, 12, 10
2-1. Given the following set of n = 35 observations:10 19 18 21 20 20 18 18 18 16 11 12 16 12 18 19 19 13 15 17 17 16 17 14 15 14 18 19 16 14 15 19 17 17 18 construct an absolute frequency
1-2. For the following variables, determine whether the data are measured on an interval or ratio scale.(a) X: attendance at last night’s concert(b) X: cost of parking your car at the local civic
1-1. For the following variables, determine whether the data are best characterized as categorical (nominal or ordinal) or numerical (continuous or discrete).(a) X: red, white, blue(b) X: poor, fair,
14-14. Using α = 0.05, perform a Lilliefors test for normality using the data presented in Exercise 14-11.
14-13. Using α = 0.05, perform a Lilliefors test for normality using the data presented in Exercise 14-10.
14-12. Given the following absolute frequency distribution, determine if the data set could have been drawn from a normal distribution with μ = 75 andσ = 15. Use α = 0.05.Classes ofX Oi 0–19 3
14-11. The scores obtained on a certain financial analysts certification exam are thought to be normally distributed with μ = 550 and σ = 100. A random sample of n = 10 such scores resulted in the
14-10. The marketing director for ACE Superstores speculates that daily sales in the housewares department are normally distributed with μ = 130 andσ = 20. Individual sales at n = 18 such stores
14-9. A state motor vehicle emission inspection process has been designed so that inspection time is uniformly distributed with limits of 10 and 17 minutes.A sample of n = 10 duration times (taken
14-15. Can we conclude at the α = 0.05 level that the following set of n = 22 data points was extracted from a normal distribution?80 60 90 100 120 160 150 105 160 155 145 120 140 110 175 87 163 98
14-16. Two brands of fertilizer (call them A and B) are being tested on two identical plots of land planted with wheat. Each plot is divided into 10 equal sections. Yields per section have been
14-17. Two different training methods (call them A and B) are being tested at ACE Pharmaceuticals. Two groups of trainees have been selected randomly; one uses Method A and the other uses Method B.
14-18. Use the moments of the following data set to determine if observations can be viewed as having been randomly drawn from a population that follows a normal distribution. Use α = 0.05.10 6 35
15-2. Random samples of high school students from urban, suburban, and rural areas are asked if they are planning to attend an institution of higher learning. Are the categories of planning for
15-3. Adepartment manager at one of theACEDepartment Stores is interested in the relationship between sales force training and customer satisfaction with the service as indicated by customer
15-4. Asociologist is interested in determining if there is a relationship between family income and type of college attended. A random sample of families having at least one child in some sort of
15-5. In a large city four independent random samples of city employees (composed of workers categorized as clerical, sanitation, transportation, and buildings and grounds) were taken in order to
14-8. Atotal of 200 students took the same written portion of a driver’s education exam:Score Oi 70–74 15 75–79 18 80–84 60 85–89 70 90–94 20 95–99 17 If these student drivers can be

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