Stat285.HW.Prob.S1.to.S8.pdf - l 2.7 The following are the...

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Unformatted text preview: l' 2.7 The following are the numbers of customers a restaurant served for lunch on 120 weekdays: so 64 55 5| 60 4| 7| 53 63 64 46 59 66 45. 6| 57 65 62 58 65 55 6| so 55 53 57 58 66 53 56 64 46 59 49 64 60 5.8 64 42 47 59 62 56 63 6| 68 57 5| 6| 5| 6o 59 67 52 52 58 64 43 6o 62 48 62 56 63 55 73 60- 69 53 66 54 52 56 59 65 60 6| 59 63 56 62 56 62 57 57 52 63 48 58 64 59 43 67 52 58 47 63 53 54 67 57. 6| 65 78 6o 66 63_ 58 60 -55 6| s9 74 62 49 63 65 55 6| 54 (a) Group these figures into a table having the classes 40—44, 45—49, 50-54, 55—59, 60-64, 65—69. 79—74, and 75—79. ' (b) Convert the distribution of part (a) into a cumulative “less than" dis- tribution. ' i 2.2] The number of empty seats on flights from Dallas to bilew prleans are grouped into a table with the classes 0—9, 10—19, 20—29, 30—39, and 40 or more. Is it possible to determine from this table the number of flights bn which there were (a) at least 30 empty seats; (c) at least 19 empty seats; (b) more than 30 empty seats; (d) at least 10 empty seats? I 2.27 The following isthe distribution of the actual shelf weight (in ounces) of a sample of 60 "one-pound" sacks of trisodium phosphate, which were filled from bulk stock by a part—time clerk in a hardware store: Weight . Number of sacks 15.60-15.99 8 l 6.00— l 6.39 27 6.40—1 6.79 3 l9 1 630—17.] 9 5 l 720—] 7.59 1 Draw a histogram of this distribution. l 3.7 The “cut-out" syrup density in canned fruits is the percentage (or "degree") ' by weight of sugar in the syrup solution when a can is opened. Examination of samples of eight cans each of three grades of dried prunes yielded the fol- lowing cut-out densities (measured in degrees on the Brix scale): Fancy grade: 33. 35, 32, 32, 35. 30, 33, 34 C [mite grade: 24, 27, 25. 30, 30, 28, 28, 30 Standard grade: 23. 22, l8, I8, 20, 24, 20, 20 g t 3.27 A filling machine in a high-production bakery is set to fill open-face pics with 16 fluid ounces of fill. A sample of four pies from a large production lot shows fills of 16.2, 15.9, 15.8, and l6.l fluid ounces. ' (a) Find the range of these fills. (b) Use the definition formula to calculate the standard deviation of these fills. 3.45 Find the mean and the standard deviation of the following distribution of the ages of the members of a union: Age (years) Frequency 20—24 I I 25—29 24 30-34 30 35-39 " 18 40-44 I 1 45-49 5 50-54 1 3.46 With reference to the distribution of the preceding exercise, find (a) the median; (c) the percentiles P” and P“. (b) the quartiles Q, and Q,; 5.12 A psychologist preparing three-letter nonsense words for use in a memory test chooses the first letter from among the consonants q, w, x, and z; the second letter from among the vowels e, i, and u; and the third letter from among the consonants c, f, p, and v. (a) How many different three-letter nonsense words can she construct? (b) How many of these nonsense words will begin either with a w or an x? (c) How many of these nonsense words will end with a c? - (d) How many of these nonsense words ending in a c begin with an x? 5.25 Calculate the number of ways in which the Internal Revenue Service can choose 4 of 14 income tax returns for a special audit. 5.26 Among the l2 nominees for the Board of Directors of a farm cooperative there are 8 men and 4 women. In how many ways can the members elect as directors ' (a) any two of the nominees; (b) two of the male nominees; (c) one of the male nominees and one of the female nominees? 5.27 The personnel manager of a store wants to fill five openings in its training program With three college graduates and two persons who are not college graduates. In how many ways can these openings be filled if among 2l appli- cants 12 are college graduates? S \ “ fl 5,33 When one card is drawn from a well-shuffled deck of 52 playing cards, what . , ' are the probabilities of getting . 6'13 1“ “EUR 6-4. U Is the event that the unemployment rate will go down and 1 (a) a red king; is the event that the inflation rate will go up. Explain in words what events are (b) a queen. king. or ace of any suit; represented by regions I, 2, 3, and 4. (c) a red card; (d) a 3, 4, 5, or 6 of any suit? . 5.36 A bowl contains l7 red beads, l0 white beads, 20 blue beads, and 3 black beads. if one of these beads is drawn at random, what are the probabilities ' U that it will be (a) red; (c) black; (b) blue or white; (d) neither white nor black? 5.39 If 2 of 20 tires are defective and 4 of them are randomly chosen for inspection . what is the probability that both of the defective tires will be chosen? - 5.4l New York City, LoiiAngeles, and Philadelphia are among the twelve large’s‘ ' cities in the United $‘tates. Assuming that the selection is r'andom (that eacl ' set of three of the twelve cities has the same chance _of being'selected), what art the probabilities that a survey conducted in three of the twelve latgest cities ir the United States will include {I 6:14 With reference to the preceding exercise, what events are represented by (a) New York City; (b) Los Angcle‘i and Philadelphia? (8) regions 3 and 4 together: . . . _ (b) regions 2 and 3 together; 6.3 To construct sample spaces for experiments In which we deal wrth categorlcal (c) regions I, 2, and 3 together? data, we often code the various alternatives by assigning them numbers. For instance, an airline passenger's complaint might be coded 1,2, 3, 4, or 5, depending on whether it is about baggage handling,-ticketing and boarding, seats or leg room, food service, or carry-on facilities. Express each of the fol- ' 6.23 Explain why there must be a mistake in each of the following statements: (a) The probability that a corporation will pay its regular quarterly dividend?“ is 0.83, and the probability that it will not pay its regular quarterly dividend lowing events in words: ‘ is 0 27 = - ‘ = ,2, 3 . ' ' .. . . . :8 f = iii. 3]]. (c) M U l . (b) The probability that a new servrce station wrll lose money during its first . i v _ . . .‘ _ year of operation is 0.33, and the probability that it will break even or 6-4 With reference to the preceding exercise, IN the elements of the sample space make a profit is 0.57. comprising €th 0f the following CVCMS. .and “'50 express the events in words: (c) The probability that on any given 'working day an otfice worker in a certain (a) K ; v (C) K n L: city drives to work is 0.62, and the probability that he either drives to (b) K U L; (d) K ('1 M- work or takes a bus is 0.54. - i (d) The probability that a married student prefers living on campus is 0.29, and the probability that he and his wife both prefer living on campus is 0.43. (e) The probability that an insurance salesman will sell a life insurance policy to a friend is 0.48, the probability that he will sell him automobile insurance but no life insurance is 0.36, and the probability, that he will sell him neither 6.20 Venn diagrams are often used to verify relationships among sets, subsets, or events, without requiring rigorous proofs based on a formal algebra of sets. We simply show that the expressions which are supposed to be equal are repre- sented by the same region of a Venn diagram. Use Venn diagrams 'to show that (c) (A n By = ,4' u 3' gnu (A U By = ,4' n 3'; kind of insurance is 0.12. 6.22 If F is the event that a dishonest land developer is in Tancial difficulties, Tis 6'26 figenjzzrlguzny exclusive events A- and B for WhiCh PM) = 0.37 and the event that he has tax problems, and Q is the event t at he uses questionable (3-) £64,): ' n. ' sales practices, write in symbolic form the probabilities that a dishonest land , ’ (d) PM U 13): developer (b) P(B ); (e) ”11' U B'); (a) has tax problems and uses questionable sales practices; (c) PM D B); (0 PM' 0 B'). (b) is not in financial difficulties but has tax problems; (c) uses questionable sales practices or has tax problems; “ -. (d) has neither financial difficulties nor tax problems. ‘. S - Z 6.27 The probabilities that a review board will rate a given movie X, R, or PC are 0.43, 0.28, and 0.12. What is the probability that the movie will get one or another of these three ratings? 6.31 The probabilities that 0, l, 2, 3, 4, 5, 6, or at least 7 persons will inquire about a piece of industrial property on the first day that it is advertised for sale are 0.002, 0.013, 0.039, 0.08], 0.125, 0.155, 0.160, and 0.425. What are the prob- abilities that - (a) at most 4 persons will inquire about the property on that day; (b) at least 2 persons will inquire about the property all that day; _ . (c) from 3 to 5 persons will inquire about the property on that day? 6.34 The probability that a person stopping at a gas station will ask to have his tires checked is 0.12, the probability that he will ask to have his oil' checked is 0.29, and the probability that he will ask to have them both checked is 0.07. What is the probability that a person stopping at this station will ask to have (a) either his tires or Hils oil checked; (b) neither his tires nor‘his oil checked? 6.35 A businessman has two secretaries. The probability that the one he hired most recently will belabsent on any given day is 0.08, the probability that the other secretary will be absent on any given day is 0.07, and the probability that they will both be absent on any given day is 0.02. What is the probability that (a) either or both secretaries will be absent on any given day; (b) at least one secretary comes to work on any given day; (c) only one secretary comes to work on any given day? 6.49 There are two Porsches in a race, and a driver feels that the odds against their winning are, respectively, 3 to l and 4 to 1. To be consistent, what odds should he assign to the event that neither car will win? . 6.5-1 Given PM) = 0.4, P(B IA) = 0.3, and P(B’|A') = 0.2, find (a) PM); (d) P(A n B): (b) PtBIA’); . (e) HAIB). (:2) PM): - . . ~. 6.56 As part of a promotional scheme in California and Nevada, a company dis- tributing frozen foods will award a grand prize of $100,000 to some person sending in his name on an entry'blank, with the option of including a label from one of the company's products. A breakdown of the 150,000 entries received is shown in the following table: With _ Without label label - California {—80.000 28,000 r .. .———-.- ~ Nevada !; 20.000 221000 11' the winner of the grand prize is chosen by lot, C represents the event that it will be won by an entry from California, and L represents the eventthat it will be won by an entry which included a label, firld each of the following prob- abilities: (b) P(L’): (C) P(C’ n L’); \ (0 mafia): " 6-58 The probability that a bus from BulTalo to Rochester will leave on time is 0.70, and the probability that it will leave on time and also arrive on time is 0.56. What is the probability that if such a bus leaves on time it will also arrive on time? ' 6.63 if the probability that the manager of bank branch A can decipher a garbled teletype message is l and the probability that the manager of bank branch B can decipher it independently of the first manager is §, what is the probability that the message will be deciphered if the tivo managers do, in fact, work on it independently? 6.69 One critical operation in assembling a delicate electronic device requires that a skilled operator fit one part to another precisely. If the operator succeeds in matching the parts on his first attempt, he moves on to the next assembly; otherwise, he repeats his (independent) attempts until he gets a match. What is the probability that an operator with a constant match probability of i will succeed in matching the parts in a given assembly (a) on the fourth attempt and (b) within four attempts? (In this example the set of all possible outcomes is not finite, and when this is the case we must modify the third postulate of probability so that it applies to the union of any number of mutually exclusive events; nevertheless, it is possible to solve thisproblem with the methods discussed in this chapter.) 6.70 The problem of determining the probability that any number of events will occur becomes more complicated when the events are not independent. For three events A, B, and C, for example, the probability that they will all occur is obtained by multiplying the probability of A by the probability of A given B, and then multiplying the result by the probability of C given A n B. For instance, the probability of drawing (without replacement) three aces in a row from an ordinary deck of 52 playing cards is Clearly, thereare only three aces among the 51 cards which remain after the first ace has been drawn, and only two aces among the 50 cards which remain after the first two aces have been drawn. ‘ (c) If S of a company’s 12 delivery trucks do not meet emissidn standards and 4 of the 12 are randomly picked for inspection, what is the probability that none of them meets emission standards? - 8.2 Determine whether the following can be probability distributions, defined in each case for the given values of x, and explain your answers: (a) foo—1% forx=0,l,2,3; (b) f(.\')=Tx§ forx=0,1,2,3,4,5; x— (c)f(x)= 52 forx=0,1,2,3,4,5; 5.3 (d) f(x)=§% rorx=o,t,2,3,}1. $.34 The following table gives the probabilities that a computer will malfunction 0, l, 2, 3, 4, 5,-or 6 times on any given day: Number of 0 l 2' 3 4 5 6 malfunctions _____.___.__......._______—-————-—— Probability 0.15 0.22 0.31 0.18 0.09 0.04 0.01 Calculate the mean and the standard deviation of this probability distribution. 8.35 Using the rules for summations given in Section 3.10, we can derive the follow- ing shortcut formula for the variance of a probability distribution: “ il‘ 01 = XXL/(10“ #2 _ The advantage of !this formula is that we do not have to'work with the devia- tions from the mean. Instead, we subtract It“ from the sum of the products obtained by multiplying the square of each value of the random variable by the corresponding probability. Use this formula to find (a) the variance of the probability distribution of Exercise 8.33; (b) the standard deviation of the probability distribution of Exercise 8.34. 8.37 Find a1 for the distribution of the number of times a fair coin falls heads in ' four flips. using the probabilities on page 212 and (a) the definition formula on page 228; (b) the shortcut formula of Exercise 8.35. . Also compare the results with that obtained on page 229 with the special‘ formula 0' = np(l -— p). 8.3 In a large government agency, illness is given as the reason for 90 percent of all absences from work. Find the probability that three of four absences from work (randomly selected from the agency‘s records) were claimed to be due to illness, by using . (a) the formula for the binomial. distribution; W 8.4 A multiple-choice test consists of eight questions and three answers to each question (onlyone of which is correct). If a student answers each question by rolling a balanced die and checking the first answer if he gets a l or a 2. the second answer if he gets a 3 or a 4, and the third answer if he gets a 5 or a 6, find the probability of getting (at exactly three correct answers; (b) no correct answers; (c) at least six correct answers. H.” The quality-control engineer of an electronics firm claims that 95 percent o the components that are shipped out are in good working condition. Find tl probabilities that 'among 14 components which are shipped out, 0, l, 2, 3, . . 13, or M will be in good working condition, and draw a histogram of this prob bility distribution. to happen on the xth trial it mu . . . , st be preceded b x — t fa probability ls (l —— p)"", and it follows y cess wrll occur on the .rth trial is ilures for which tl that the probability that the first su. (a) When taping a television commercial '“lt‘lll get his lines straight on any one take is 0.40. What is the probabilit _,at this actor Will get hts ltnes straight for the first time on the fourth take 8.25 It is presumed that 1.5 percent of the inhabi immigrants. Use the Poisson a determine the probability that i crty, two are ‘illegal immigrants. . . tants..of a border city are illegal pprommatton to the binomial distribution to n a random sample of 200 inhabitants of the 8.29 The number oflarrivals for service at a tool crib in a plant per quarter hour of firm-shift time IS arlandom variable having the Poisson distribution with A = 2. \ What are the probabilities of 0, I, and 2 arrivals for service in a randomly chosen quarter hour? 8.41 Find the mean and the standard deviation of the distribution of each of the ' following random variables having binomial distributions: (a) The number of heads obtained in 576 flips of a balanced coin. (b) The number of 5's obtained in 405 rolls of a balanced die. , (c) The number of persons (among 400 invited) who will attend the opening of a new branch bank, when the probability is 0.85 that any one of them will attend. . (d) The number of defectives in a sample of 2,400 parts made by a machine, when the probability is 0.04 that any one of the parts is defective. (e) The number of students (among 800 interviewed) who do not like the food served at the university cafeteria, when the probability that any one of them does not like the food is 0.30. ' 8.53 in attempting to simplify purchase controlI a retail hardware store uses a color—code system, putting green tags on stock items bought the first half of a year and yellow tags on items bought the second half of the year. If there are )0 cake pans on a table, 6 tagged green and the rest yellowI and a customer buys 5 of them, selecting them at random, what are the probabilities that she selects A (a) 4 bought the first half of the year; 8 .. Li C, (b) at least 4 bought the first half of the year? . 9.5 A shopper has bought two packs of clove gum, taking them at random either from bin K, which contains four fresh packs and two stale packs of clove gum, or from bin 1., which contains two fresh packs and four stale packs of clove gum, but he is three times as likely to have taken them from bin L as from bin K. ‘ (a) What is the probability that both packs the shopper got are stale? (b) If both packs the shopper got are stale, what is the probability that he took them from bin L? . _ (c) If the shopper got at least one fresh pack, what is the probability that he took them from bin K? . 9.6 1n a cannery, production lines 1,11, and 11! account for 50, 40, and 10 percent of the total output. 11' 0.5 percent of the cans from line I, 0.6 percent from line ll, and 1.5 percent from line 111 have faulty seals, what are the probabilities that a can with a faulty seal, detected at final product inspection, was produced by 11 (a) line 1; H (b) line ll; (c) line 111? 9.7 it is known from experience that in a certain industry 55 percent of all labor- management disputes are over wages, 10 percent are over working conditions, and 35 percent are over fringe issues. Also, 40 percent of the disputes over wages are resolved without strikes, 70 percent of the disputes over working conditions are resolved without strikes, and 45 percent of the disputes over fringe issues are resolved without strikes. if a labor—management dispute in this industry is resolved without a strike, what are the odds that it wasnot over wages ? 10.4 Find the area under thestandard normal curve which lies (a) between 2 = ~0.45 and z = 0.45; (b) to the right of z = —2.20; (c) to the left 012 = —-l.‘35; (d) between 1 = 2.25 and z = 2.65; (e) to the right of z = 1.40; (f) between 2 = —l.90 and z = —0.60. 10.5 Find 2 if the normal-curve area (a) between 0 and z is 0.4484; (d) to the right of z is 0.3300; (b) to the left of z is 0.9868; (c) to the left of 2 IS 0.51085; (c) to the right of z is 0.8413; (f) between —z and 2 15 0.9700. 10.8 A random variable has a normal distribution with the mean [I = 80.0 and the standard deviation 0 = 4.8. What are the probabilities that this random vari~ able will take on a value (a) less than 87.2; (c) between 81.2 and 86.0; (b) greate...
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