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modern mathematical statistics with applications
Mathematical Statistics For Economics And Business 2nd Edition Ron C. Mittelhammer - Solutions
3. Show that the following probability density functions are members of the exponential class of densities:(a) Binomial family, for a fixed value of n.(b) Poisson family.(c) Negative binomial family for a fixed value of r.(d) Multinomial family, for a fixed value of n.(e) Beta family.
2. The daily price, p, and quantity, demanded, q, of gasoline on a European Wholesale spot market can be viewed (approximately) as the outcome of a bivariate normal random variable, where the bivariate normal density has mean vector and covariance matrix as follows:m ¼ 2:50 100 " #; S ¼ :09 1 1
1. A shipment of 100 DVDs contains k defective disks.You randomly sample 20 DVDs, without replacement, from the shipment of 100 DVDs. Letting p ¼ k/100, the probability that you will obtain less than three defective disks in your sample of 20 disks is then given by Pðx 3Þ ¼ S 3x¼0 20 x! px ð1
49. The bivariate random variable (Y,X) has the following mean vector and covariance matrix:E X Y ¼ 10 5 and Covð Þ¼ X;Y 5 2 2 2 (a) Derive the values of a and b in Y^ ¼ a þ bX that minimize the expected squared distance between Y and Y^, i.e., that produce the best linear predictor
48. The mean vector and covariance matrix of the trivariate random variable X is given by EðXÞ ¼2 42 24 35 and CovðXÞ ¼10 2 1 2 50 1 01 24 35:The random variable Y is defined by Y ¼ c0 X, where c0 ¼ [5 1 3], and the bivariate random vector Z is defined by Z ¼ AX, where A ¼ 111 2 3 4
47. An investor has $10,000 to invest between two investment projects. The rate of return per dollar invested in the two projects can be represented as the outcome of a bivariate random variable ð Þ X1; X2 where E X1 X2 ¼ :20:05 and CovðXÞ ¼ :04 :002:002 :0001 (a) If the investor
46. The weekly number of MACs and PCs sold by a salesperson at the local computer store can be represented as the outcome of a bivariate random variable (X,Y) with the nonzero values of its joint probability density function given by the following table of probabilities:Y 0 1 234 X0 .15 .12 .075
45. A small manufacturing firm produces and sells a product in a market where output prices are uncertain.The owner of the firm wishes to make a short run production decision that will maximize her expected utility, defined by Eð Þ¼ Uð Þ p Eð Þ p a½ varð Þ p where U is utility, p ¼ Pq
44. Derive the moment generating function of each of the random variables below, and use it to define the mean and variance of the random variable.(a) fðxÞ ¼ :2e:2xIð Þ 0;1 ðxÞ(b) fðxÞ ¼ 2xIð Þ 0;1 ðxÞ(c) fðxÞ ¼ :3x:71xIf g 0;1 ðxÞ
43. The bivariate random variable (P,Q) represents the weekly price, in dollars, and the quantity, in number of kegs, of an India Pale Ale beer, HopMeister, sold in the Pacific Northwest market. The moment generating function associated with this bivariate random variable is given by Mð Þ P;Q
42. Given the following three moments about the origin, derive the first three moments about the mean, and determine whether the random variable has a probability density function that is skewed, if you can.(a) EðXÞ ¼ :2; E X2 ¼ :2; E X3 ¼ :2(b) EðXÞ ¼ 1; E X2 ¼ 2; E X3 ¼ 5(c)
41. The daily dollar sales of a large retail “Big Box” store, measured in 1,000 dollar units, is a random variable, D, that has an expectation of 20.(a) Provide an upper bound to the probability that dollar sales exceed 40,000 dollars on a given day.(b) If the variance of D is 4, define an
40. For each probability density function below, determine the mean and variance, if they exist, and define the median and mode.(a) fðxÞ ¼ p 1 þ x2 1 Ið Þ 1;1 ðxÞ(b) fðxÞ ¼ 4x3I½ 0;1 ðxÞ(c) fðxÞ ¼ :3ð Þ :7 x1 If g 1;2;3;::: ðxÞ(d) fðxÞ ¼ x 55 If g 1;2;:::;10 ðxÞ
39. The yield per acre of a certain dwarf watermelon is highly dependent on the amount of rainfall that occurs during the growing season. Following standard cultivation practices, the relationship between tons per acre, Y, and inches of rainfall, R, is given by Y ¼ 25 þ 2R :05R2 for R 2 ½ 0;
38. The regression curve of daily quantity demanded of tablet computers in a Midwestern market, measured in
37. Define the mean, median, mode, and .10 and .90 quantiles of the random variable X defined in the probability models f g RðXÞ;fðxÞ below:(a) fðxÞ ¼ x2 þ 4 50 If g RðXÞ ðxÞ; RðXÞ ¼ f g 0; 1; 2; 3; 4(b) fðxÞ ¼ :5ex=2If g RðXÞ ðxÞ; RðXÞ ¼ ½ Þ 0;1(c) fðxÞ ¼ 3x2If g
36. The daily wholesale price and quantity sold of ethanol in a Midwestern regional market during the summer months is represented by the outcome of a bivariate random variable ð Þ P; Q having the following probability model f g R Pð Þ ; Q ;f pð Þ ; q :f pð Þ¼ ; q :5pepq for ð Þ2 p; q R
35. The Rockbed Insurance Company sells 1-year term life insurance policies for $10,000 of annual coverage, where a 1-year premium is charged to put the policy in force, and then if the insured person does not live through the year, his or her estate is paid $10,000. The mortality tables in the
34.(a) Find the moment-generating function of a random variable X having the density function fðxÞ ¼ 1 83 x Ið Þ 0;1;2;3 ðxÞ:(Hint: Use of the binomial theorem may be helpful in finding a compact representation of this function.)Use the MGF to calculate the first two moments of X about the
33. The length of time in minutes for an individual to be served at a local restaurant is the outcome of a random variable, T, having a mean of 6 and a variance of 1.5. How probable is the event that an individual will be served within 3 to 9 min?
32. An investor wishes to invest $1,000 and is examining two investment prospects. The net dollar return per dollar invested in the two projects can be represented as the outcome of a bivariate random variable (X1, X2)where E X1 X2 ¼ :15:07 and CovðXÞ ¼ :04 :001:001 :0001 :
31. The AJAX Disk Co. manufactures compact disks(CDs) for the music industry. As part of its quality-control program, the diameter of each disk is measured using an electronic measuring device. Letting X1 represent the actual diameter of the disk and X2 represent the measured diameter of the disk,
30. Let X have the moment generating function MX(t).Show that(a) M(X+a) (t) ¼ eat MX(t).(b) MbX(t) ¼ MX(bt).(c) M(X+a)/b(t) ¼ e(a/b)t MX(t/b).
29. The yield/acre of wheat on a given parcel of land can be represented as the outcome of a random variable Y defined by Y ¼ 10x1/3ee for x∈[8,100], where Y ¼ wheat output in bushes/acre x ¼ pounds/acre of fertilizer applied e ¼ is a random variable having the probability density function
28. The season average price per pound, p, and total season quantity sold, q, of sweet cherries in a regional market can be represented as the outcome of a bivariate random variable (P,Q) with the joint probability density function f(p,q) ¼ .5qeq(.5+p)I(0,1)(q) I(0,1) (p)where p is measured in
27. An automobile dealership sells two types of four-door sedans, the “Land Yacht” and the “Mini-Rover.” The number of Land Yachts and Mini-Rovers sold on any given day varies, with the probabilities of the various possible sales outcomes given by the following table:
26. A Seattle newspaper intends to administer two different surveys relating to two different anti-tax initiatives on the ballot in November. The proportion of surveys mailed that will actually be completed and returned to the newspaper can be represented as the outcome of a bivariate random
25. A fruit processing firm is introducing a new fruit drink, “Peach Passion,” into the domestic market. The firm faces uncertain output prices in the initial marketing period and intends to make a short-run decision by choosing the level of production that maximize the expected value of
24. Scott Willard, a famous weatherman on national TV, states that the temperature on a typical late fall day in the upper midwest, measured in terms of both the Celsius and Fahrenheit scales, can be represented as the outcome of the bivariate random variable (C,F) such that EC F" # ¼ 5 41 and
23. A gas station sells regular and premium fuel. The two storage tanks holding the two types of gasoline are refilled every week. The proportions of the available supplies of regular and premium gasoline that are sold during a given week in the summer is an outcome of a bivariate random variable
22. The moment-generating function of the random variable Y is given by MYðtÞ ¼ ð Þ 1 :25t 3 for t < 4.(a) Find the mean and variance of the random variable Y.(b) Is the PDF of Y skewed? Why or why not?(c) It is known that the moment generating function of the PDF fðxÞ ¼ 1 baGð Þa
21. The random variable Y has the PDF f(y) ¼ y2 I[1, 1) (y).(a) Find the mean of Y.(b) Can you find the first 100 moments about the origin(i.e., m0 1; m0 2; ... ; m0 100) for the random variable Y, why or why not?
20. The first three moments about the origin for the random variable Y are given as follows: m0 1 ¼ :5; m0 2 ¼ :5; m0 3¼ :75.(a) Define the first three moments about the mean for Y.(b) Is the density of Y skewed? Why or why not?
19. The total daily dollar sales in the ACME supermarket is represented by the outcome of the random variable S having a mean of 20, where s is measured in thousands of dollars.(a) The store manager tells you the probability that sales will exceed $30,000 on any given day is .75. Do you believe
18. In each case below, calculate the expected value of the random variable Y:(a) Eð Þ¼ Yjx 2x2 þ 3;fXðxÞ ¼ exIð Þ 0;1 ðxÞ:(b) Eð Þ¼ Yjx 3x1x2; Eð Þ¼ X1 5; Eð Þ¼ X2 7; X1 and X2 are independent.
17. The daily price and quantity sold of wheat in a Northwestern market during the first month of the marketing year is the outcome of a bivariate random variable (P,Q)having the probability density function f pð Þ¼ ; q :5pepqI½3;5ðpÞIð0;1ÞðqÞwhere p is measured in $/bushel, and q is
16. The manager of a bakery is considering how many chocolate cakes to bake on any given day. The manager knows that the number of chocolate cakes that will be demanded by customers on any given day is a random variable whose probability density is given by fðxÞ ¼ x þ 1 15 If0;1;2;3g ðxÞ þ 7
15. A game of chance is considered to be “equitable” or“fair” if a player’s expected payoff is equal to zero. Examine the following games:(a) The player rolls a pair of fair dice. Let Z represent the amount of money that the player lets on the game outcome. If the player rolls a 7 or 11,
14. The average price and total quantity sold of an economy brand of ballpoint pen in a large western retail market during a given sales period is represented by the outcome of a bivariate random variable having a probability density function fðp; sÞ ¼ 10 peps I½:10; :20 ðpÞIð0;1Þ
13. The city of Megalopolis operates three sewage treatment plants in three different locations throughout the city. The daily proportion of operating capacity exhibited by the three plants can be represented as the outcome of a trivariate random variable with the following probability density
12. Stanley Statistics, an infamous statistician, wants you to enter a friendly wager with him. For $1,000, he will let you play the following game. He will continue to toss a fair coin until the first head appears. Letting x represent the number of times the coin was tossed to get the first heads,
11. The annual return per dollar for two different investment instruments is the outcome of a bivariate random variable (X1, X2) with joint moment-generating function Mx(t) ¼ exp(u0 t þ :5t0 St), where t ¼ t1 t2 ; u ¼ :07:11 and S ¼ :225 103 :3 103:3 103 :625 103 :(a) Find the
10. The daily quantity of water demanded by the population of a large northeastern city in the summer months is the outcome of a random variable, X, measured in millions of gallons and having a MGF of Mx(t) ¼ (1 .5 t)10 for t < 2.(a) Find the mean and variance of the daily quantity of water
9. The miles per gallon attained by purchasers of a line of pickup trucks manufactured in Detroit are outcomes of a random variable with a mean of 17 miles per gallon and a standard deviation of .25 miles per gallon. How probable is the event that a purchaser attains between 16 and 18 miles per
8. The daily price of a certain penny stock is a random variable with an expected value of $2. Then the probability is .20 that the stock price will be greater than or equation to $10. True or false?
7. For each of the densities below, indicate whether the mean and variance of the associated random variable exist. In addition, find the median and mode, and indicate whether or not each density is symmetric.(a) f(x) ¼ 3x2 I[0, 1](x)(b) f(x) ¼ 2x3 I[1, 1](x)(c) f(x) ¼ [p(1 + x2)]1 I(1, 1)(x)(d)
6. The short-run production function for a particular agricultural crop is critically dependent on the level of rainfall during the growing season, the relationship being Y ¼ 30 þ 3X .075X2, where y is yield per acre in bushels, and x is inches of rainfall during the growing season.(a) If the
5. The daily price/gallon and quantity sold (measured in millions of gallons) of a lubricant sold on the wholesale spot market of a major commodity exchange is the outcome of a bivariate random variable (P,Q) having the joint probability density function fðp; qÞ ¼ 2 pepq I½:5;1 ðpÞIð0;1Þ
4. The yield, in bushels per acre, of a certain type of feed grain in the midwest can be represented as the outcome of the random variable Y defined by Y ¼ 3x:30 l x:45 k eU where xl and xk are the per acre units of labor and capital utilized in production, and U is a random variable with
3. The weekly number of luxury and compact cars sold by “Honest” Abe Smith at the Auto Mart, a local car dealership, can be represented as the outcome of a bivariate random variable (X,Y) with the nonzero values of its joint probability density function given by Y0 1 234 X0 .20 .15 .075 .05 .03
2. A small rural bank has two branches located in neighboring towns in eastern Washington. The numbers of certificates of deposit that are sold at the branch in Tekoa and the branch in Oakesdale in any given week can be viewed as the outcome of the bivariate random variable (X,Y) having joint
1. A small domestic manufacturer of television sets places a three-year warranty on its picture tubes. During the warranty period, the manufacturer will replace the television set with a new one if the picture tube fails.The time in years until picture tube failure can be represented as the outcome
40. The production of a certain volatile commodity is the outcome of a stochastic production function given by Y¼ L:5K:25ev , where v is a random variable having the cumulative distribution function FðvÞ ¼ 1 1þe2ð Þ v1 , L denotes units of labor and K denotes units of capital.a. If labor
39. Let the random variable X represent the product of the number of dots facing up on each die after a pair of fair dice is rolled. Let Y represent the sum of the number of dots facing up on the pair of dice.a. Define a probability model (R(X),f (x)) for the random variable X.b. What is the
38. The daily wholesale price and quantity sold of ethanol in a Midwestern regional market during the summer months is represented by the outcome of a bivariate random variable (P,Q) having the following probability model{R(P,Q),f (p,q)}:f pð Þ¼ ; q :5pepq for ð Þ2 p; q R Pð Þ¼ ; Q ½ 2;
37. For each of the joint PDFs listed below, determined which random variables are independent and which are not.a. f xð Þ¼ ; y eð Þ xþy I½ Þ 0;1 ðxÞI½ Þ 0;1 ðyÞb. f xð Þ¼ ; y xð Þ 1 þ y 300 If g 1;2;3;4;5 ðxÞIf g 1;2;3;4;5 ðyÞc. f xð Þ¼ ; y; z 8xyzI½ 0;1 ðxÞI½
36. The following function is proposed as a cumulative distribution function for the bivariate random variable (X,Y):F xð Þ¼ ; y 1 þ eð Þ x=10þy=20 ex=10 ey=20 Ið Þ 0;1 ðxÞIð Þ 0;1 ðyÞa. Verify that the function has the appropriate properties to serve as a cumulative
35. The BigVision Electronic Store sells a large 73 inch diagonal big screen TV. The TV comes with a standard 1 year warranty on parts and labor so that if anything malfunctions on the TV in the first year of ownership, the company repairs or replaces the TV for free. The store also sells an
34. The daily wholesale price and quantity sold of ethanol in a Midwestern regional market during the summer months is represented by the outcome of a bivariate random variable (P,Q) having the following probability model(R(P,Q),f (p,q)):f pð Þ ; q :5pepq for ð Þ2 p; q R Pð Þ¼ ; Q ½ 2; 4
33. Outcomes of the random variable Z represent the number of customers that are waiting in a queue to be serviced at Fast Lube, a quick stop automobile lubrication business, when the business opens at 9 A.M. on any given Saturday. The probability model ð Þ RðZÞ; fðzÞ for the random variable
32. Let an outcome of the random variable T represent the time, in minutes, that elapses between when an order is placed at a ticket counter by a customer and when the ticket purchase is completed. The following probability model ð Þ RðTÞ; fðtÞ governs the behavior of the random variable T:
31. A small locally-owned hardware store in a western college town accepts both cash and checks for purchasing merchandise from the store. From experience, the store accountant has determined that 2 percent of the checks that are written for payment are “bad” (i.e., they are refused by the
30. The daily quantity demanded of milk in a regional market, measured in 1,000’s of gallons, can be represented during the summer months as the outcome of the following random variable:Q¼ 200 50pþV;where V is a random variable having a probability density defined by fðvÞ ¼ 0:02I½
29. For those functions in (28) that are actually cumulative distribution functions (CDFs), use the duality principle to derive the PDFs (PDFs) that are associated with the CDFs.
28. Given the function definitions below, determine which can be used as cumulative distribution functions(CDFs) and which cannot. Justify your answers.a. FðcÞ ¼ ec 1 þ ec for c 2 1ð Þ ;1b. FðcÞ ¼ 1 x2; for c 2 ð Þ 1;1 0 otherwisec. FðcÞ ¼ 1 ð Þ :5 floorðcÞ for c 1 0
27. Given the function definitions below, determine which can be used as PDFs (PDFs) and which cannot.Justify your answers.a. fðxÞ ¼ 1 4x for x ¼ 0; 1; 2; :::0 otherwiseb. fðxÞ ¼ 1 4 x Ið Þ 0;1 ðxÞc. f xð Þ¼ ; yð Þ 2x þ y =100; for x and y ¼ 0; 1; 2; 3; 4; and y x 0
25. The joint density function of the discrete trivariate random variable (X1, X2, X3) is given by fðx1; x2; x3Þ ¼ :20 If0;1g ðx1ÞIf0;1g ðx2ÞIfj x1 x2 jg ðx3Þþ :05 If0;1g ðx1ÞIf0;1g ðx2ÞIf1j x1 x2 jg ðx3Þ:a. Are (X1, X2), (X1, X3), and (X2, X3) each pairwise independent random
24. If (X1, X2) and (X3, X4) are independent bivariate random variables, are X2 and X3 independent random variables? Why or why not?
23. ACE Rentals, a car-rental company, rents three types of cars: compacts, mid-size sedans, and large luxury cars.Let (x1, x2, x3) represent the number of compacts, mid-size sedans, and luxury cars, respectively, that ACE rents per day. Let the sample space for the possible outcomes of (X1, X2,
22. The Imperial Electric Co. makes high-quality portable compact disc players for sale in international and domestic markets. The company operates two plants in the United States, where one plant is located in the Pacific Northwest and one is located in the South. At either plant, once a disc
21. A committee of three students will be randomly selected from a senior-level political science class to present an assessment of the impacts of an antitax initiative to some visiting state legislators. The class consists of five economists, eight political science majors, four business majors,
20. A package of a half-dozen light bulbs contains two defective bulbs. Two bulbs are randomly selected from the package and are to be used in the same light fixture. Let the random variable X represent the number of light bulbs?
19. Let X be a random variable representing the minimum of the two numbers of dots that are facing up after a pair of fair dice is rolled. Define the appropriate probability density for X. What is the probability space for the experiment of rolling the fair dice and observing the minimum of the two
18. An economics class has a total of 20 students with the following age distribution:# of students age 10 19 4 20 4 21 1 24 1 29 Two students are to be selected randomly, without replacement, from the class to give a team report on the state of the economy. Define a random variable whose outcome
17. For each of the cumulative distribution functions listed below, find the associated PDFs. For each CDF, calculate P(x 6).a. F(b) ¼ (1 eb/6) I (0,1) (b)b. F(b) ¼ (5/3) (.6 .6trunc(b)+1)I{0,1} (b)
16. The daily quantity demanded of unleaded gasoline in a regional market can be represented as Q ¼ 100 10p þ E, where p ∈ [0,8], and E is a random variable having a probability density given by fðeÞ ¼ 0:025I½ 20;20 ðeÞ:Quantity demanded, Q, is measured in thousands of gallons, and
15. The federal mint uses a stamping machine to make coins. Each stamping produces 10 coins. The number of the stamping at which the machine breaks down and begins to produce defective coins can be viewed as the outcome of a random variable, X, having a PDF with general functional form f(x) ¼ a (1
14. The cumulative distribution of the random variable X is given by FðxÞ¼ð1 pxþ1ÞIf0;1;2;:::g ðxÞ;for some choice of p ∈ (0,1).a. Find the density function of the random variable X.b. What is the probability that x 8 if p ¼ .75?c. What is the probability that x 1 given that x 8?
13. The joint cumulative distribution function for (X,Y)is given by Fðx; yÞ ¼ 1 ex=10 ey=2 þ eðxþ5yÞ=10 Ið0;1Þ ðxÞIð0;1Þ ðyÞ:a. Find the joint density function of (X,Y).b. Find the marginal density function of X.c. Find the marginal cumulative distribution function of X.
12. The joint density of the bivariate random variable(X,Y) is given by fðx; yÞ ¼ xy I½0;1 ðxÞI½0;2 ðyÞ:a. Find the joint cumulative distribution function of (X,Y).Use it to find the probability that x .5 and y 1.b. Find the marginal cumulative distribution function of X. What is the
11. A small nursery has seven employees, three of whom are salespersons, and four of whom are gardeners who tend to the growing and caring of the nursery stock.With such a small staff, employee absenteeism can be critical. The number of salespersons and gardeners absent on any given day is the
10. A personal computer manufacturer produces both desktop computers and notebook computers. The monthly proportions of customer orders received for desktop and notebook computers that are shipped within 1 week’s time can be viewed as the outcome of a bivariate random variable (X,Y) with joint
9. The weekly average price (in dollars/foot) and total quantity sold (measured in thousands of feet) of copper wire manufactured by the Colton Cable Co. can be viewed as the outcome of the bivariate random variable(P,Q) having the joint density function:fðp; qÞ ¼ 5 pepq I½:1;:3 ðpÞIð0;1Þ
8. People Power, Inc., is a firm that specializes in providing temporary help to various businesses. Job applicants are administered an aptitude test that evaluates mathematics, writing, and manual dexterity skills. After the firm analyzed thousands of job applicants who took the test, it was found
7. Intelligent Electronics, Inc., manufactures monochrome liquid crystal display (LCD) notebook computer screens. The number of hours an LCD screen functions until failure is represented by the outcome of a random variable X having range R(X) ¼ [0,1) and PDF fðxÞ ¼ :01 exp x 100
6. The ACME Freight Co. has containerized a large quantity of 4-gigabyte memory chips that are to be shipped to a personal computer manufacturer in California. The shipment contains 1,000 boxes of memory chips, with each box containing a dozen chips. The chip manufacturer calls and says that due to
5. Star Enterprises is a small firm that produces a product that is simple to manufacture, involving only one variable input. The relationship between input and output levels is given by q ¼ x5, where q is the quantity of product produced and x is the quantity of variable input used. For any
4. A health maintenance organization (HMO) is currently treating 10 patients with a deadly bacterial infection.The best-known antibiotic treatment is being used in these cases, and this treatment is effective 95 percent of the time.If the treatment is not effective, the patient expires.a. Define a
3. Sparkle Cola, Inc., manufactures a cola drink. The cola is sold in 12 oz. bottles. The probability distribution associated with the random variable whose outcome represents the actual quantity of soda place in a bottle of Sparkle Cola by the soda bottling line is specified to be fðxÞ ¼
2. Graph each of the probability density functions in Problem 1.
42. The BigVision Electronic Store sells a large 73 inch diagonal big screen TV. The TV comes with a standard 1 year warranty on parts and labor so that if anything malfunctions on the TV in the first year of ownership, the company repairs or replaces the TV for free. The store also sells an
41. The Baseball World Series in the U.S. consists of seven games, and the first team to win four games is the winner of the series. Assume that the teams are evenly matched.a. What is the probability that the team that wins the first game of the series will go on to win the World Series?b. What is
40. This problem is the famous “Birthday Problem” in the statistics literature. The problem is the following: In a room of n people, what is the probability that at least two people share the same birthday? You can ignore leap years, so assume there are 365 different birthday possibilities, and
39. A large food processor operates three processing plants on the west coast. The plants, labeled 1, 2, and 3, differ in size, and produce 20, 35, and 45 percent of the food processor’s total output of spinach, respectively.Given past history of USDA inspections for sanitation, the probability
38. A new medical test has been developed by a major pharmaceutical manufacturer for detecting the incidence of a bacterial infection. Of the people who actually have the disease, the test will correctly indicate that the disease is present 95 percent of the time. Among people who do not have the
37. The table below indicates the probabilities of various outcomes with regard to the size of purchases and method of payment for customers that enter to a large New York electronics store:a. Is the event of a customer paying cash independent of the event that the customer spends < $100?b. Given
36. An automobile manufacturer will accept a shipment of tires only if an inspection of 5 percent of the tires, randomly chosen from the shipment, does not contain any defective tires. The manufacturer receives a shipment of 500 tires, and unknown to the manufacturer, five of the tires are
35. A regional airline implements a standard sales practice of “overbooking” their flights, whereby they sell more tickets for a flight then there are seats available for passengers. Their rationale for this practice is that they want to fill all of the seats on their planes for maximum
34. If P (A) ¼ .3, P (B) ¼ .4, P (A|B) ¼ .3, what is the value ofa. P(A\B)b. P (A[B)c. P(A |B)d. P AjBe. P A jBf. P(B|A)g. P A \ B h. P A [ B
33. BuyOnLine is a large internet-based online retailer that maintains four different teams of sales representatives. The ages of unpaid invoices from each of the four sales teams is summarized in the table below.a. If an invoice is selected randomly from the pooled set of invoices, what is the
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