Stat285.prob.done.in.lecture.pdf

Stat285.prob.done.in.lecture.pdf - 2.9 During the...

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Unformatted text preview: \ 2.9 During the instruction period referred to in lowing are the grades made by the members of the group: 73 65 82 70 45 50 70 54 32 75 75 67 65 60 75 87 83 40 72 64 58 75 89 70 73 55 61 78 89 93 43 51 59 38 65 71 75 85 65 85 49 97 55 60 76 75 69 35 45 63 (a) Group these grades into a di 50—59, . . . , and 90-99. (b) Convert the distribution of part (a) into a percentage distribution. stribution having the classes 30—39, 40—49, . as u 1‘ a.” .. ‘0 u ' ' i ’ bin 'obs in one city turned t‘ 18 éwl ompleted home plum ‘ g 1 . _ 3.10 angnipgcgog % l, 0‘: 4, (ii, 5.1, 1, 2,. 0l 1, 0, and 2 violations of the city build- in code. Find (a)g the mean; (c) the mode. (b) the median; 5.1 A person with $2 in his pocket bets $1, even money, on the flip of a coin, and he continues to bet $1 so long as he has any money. Draw a tree diagram to show the various things that can happen during the first threeflips ot' the coin. In how many of the cases will he be (it) exactly $1 ahead; (b) exactly 81 behind? ' , 5.6 A woman can buy of five colors. How many choices does she have? 5.13 A true—false test consists of 15 questions. In how many ways can a stude'nt mark her answers to these questions? 5.8 It“ the NCAA has applicatiOns from {our universities for hosting its inter- collegiate swimming championships in 1984 and 1985, in how many ways can it select the sites for these championship meets (a) it' they are not to: be held at the same university; (b), if they may be held at the same university? 5.24 Calculate the number of ways in which a discount chain can choose 2 of 12 locations for the construction of new stores. . 5.38 If two different cards are drawn from a well-shuffled deck of 52 playing cards, what are the probabilities of getting - 811)) two hearts; (c) a‘king and a queen? two aces; the preceding exercise, the study group was given a test on number systems and Boolean algebra, and the f0]- .7 Erma 'Paoeusns covey-(J '\'v'\ V {Jukni-Ixes P.‘ 09 Z 6.19 With reference to the preceding exercise and the Venn diagram of Figure 6.6, express in words what events are represented by the following regions: (a) region 1; ((1) regions 3 and 5 together; (b) region 5; (e) regions 1 and 4 together; (c) region 8; (f) regions 6 and 8 together. 6.25 1! Fund Tare the events that a one car owner in a certai drive a Ford or a Toyota, and abilities that such a person will (a) not drive a Ford; (b) drive a Ford or a Toyota; (c) drive neither a Ford nor a Toyota. n income bracket will HF) = 0.34 and P0) = 0.08, find the prob 6.65, Among the 80 pieces of luggage loaded on a plane in San Francisco, 55 are destined for Seattle and 25 for Vancouver. if two of the pieces of luggage are ..sent to i’ortland by mistake and the “selection" is random, what are the probabilities that ' ' (a) _ both should have gone to Seattle; (b) both should have gone to Vancouver; (c) one should have gone to Seattle and one to Vancouver? 9|. 8.1 In each case determine whether the given values can be looked upon as the values of a probability distribution of a random variable which can take on only the values 1, 2, 3. and 4, and explain your answers: (a) f(1) = 0.24, [(2) = 0.24, f(3) = 0.24, f(4) = 0.24; (b) f0) = l. f(2) = %.f(3) = 42'. f(4) '= %: (c) f(1) = 0.15, f(2) = 0.38, f(3) = 0.24, [(4) = 0.23; (d) f(1) = 0.23, f(2) = 0.42, f(3) = —0.03, [(4) = 0.38; (e) m) = l. f(2) = 3. 1(3) = 1.. f(4) = 11.. - I 8.7 In the game of f‘chuck-a-luck" three dice are thrown and a player bets on the . occurrence of a number which he can choose. If he wins 8 i if his number appears on only one die, 52 if his number appears on two dice, $3 if his number appears on all three dice, and he loses 51 if his number appears on none of the dice, find the probabilities of his inning $1, $2, $3, and the probability of his losing 31.. Also determine the player's mathematical expectation. i 8.8 A bank knows that 60 percent of its checking-account customers will pay extra for personalized cheeks. Find the probability that at most two of eight ran- domly selected checking-account customers will pay extra for personalized checks. 8.26 It is known from experience that 2.8 percent of the calls received at a company switchboard are wrong numbers. Use the Poisson approximation to the binomial distribution to determine the probability that among 250 calls received at the switchboard, 6 are wrong numbers. 8.28 If 'the number of complaints which a laundry receives per day is a random variable hating the Poisson distribution with 1. = 3.5, find the probabilities that on any given day the laundry will receive (a) no complaints; (c) two complaints; (b) at least one complaint; (d) four complaints. 9.6 In a cannery, production lines I, II, and III account for 50, 40, and 10 percent of the total output. If 0.5 percent of the cans from line I, 0.6 percent from line II, and 1.5 percent from line III have faulty seals, what are the probabilities that a can with a faulty seal, detected at final product inspection, was produced by (a) line I; (b) line II; (c) line III? 12.4 The management of a resta'urant found on the basis of a random sample of size n =1 50 that it took its head chef on the average 52 = 14.7 minutes to pre- pare a particular cooked-to-order entree; the standard deviation of the sample is s = 4.8 minutes. What can we assert with 98 “A confidence about the possible size of the error in the estimate 55 = 14.7 minutes of the true average time it takes the chef to prepare this item? 9.26:2— 12.11 It is desired to estimate the mean lifetime of a certain kind of electronic com- ponent. Given that a' = 65 days, how large a sample is needed to be able to assert with probability 0.95 that the estimate will be off by at most 20 days? 12.25 Suppose that in the filling example the criterion is changed so that the hypothe- sis p = 300 grams is accepted if the sample mean falls between 298 and 302 grams; otherwise, the hypothesis is rejected. ' ' (a) Show that this will increase the probability of a Type I error from 0.0026 to 0.0456. (b) Show that this will decrease the probability of a Type II error when ,1: =3 ' 296 from 0.1587 to 0.0228. 12.34 The security department of a factory wants to know whether the true average time required by the night watchman to walk his round is 30 minutes. If, in a random sample of 45 rounds, the night watchman averaged 29.2 minutes with a' standard deviation of 1.6 minutes, determine at the level of significance 0.05 whether this is sufiicient evidence to reject the null hypothesis [1 = 30 minutes. 12.35 In trying to rent a location for a restaurant in a shopping mail, a rental agent assures a restaurant owner that on an average business day at least 3,600 persons pass through the mall. Being a cautious businessman, the restaurant owner hires a firm to conduct a survey, and this survey shows that on the average 3,453 persons, with .a standard deviation of 428 persons, passed through the mall on 32 randomly selected days. Use the 0.05 level of signifi- cance to test the null hypothesis fl = 3,600 against the alternative hypothesis [l < 3,600. 12.39 The police chief of a large city claims that the mean age of bicycle thieves is 9.5 years. A research worker who feels that this figure is too low takes a random sample of 80 cases of bicycle theft from police files, and finds the mean age of the thieves to be l0.8 years with a standard deviation of 4.3 years. What can the research worker conclude from these figures at the level of significance at = 0.0l ? . \ - LL ...
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