7 in 720 app016 p z 00 15 pzi09 1 08621 01379 bpo 14

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?;'~ - -\ / 7 / "': ....... ~~ IN . 7.20 (a)P(p?:0.16) = P( Z?: 0.~~0~~; 15 ) = P(Z?:I.09) = 1-0.8621 =0.1379. (b)p(o 14 < '< 0 16) (0.14-0.15 0.16-0.15) · -P- · =P <Z< =P(-1.09<Z<l09)=08621- 0.0092 0.0092 - - . . 0.1379 = 0.7242. 7.21 Ans;-ve~s w!ll vary. One possibility is to simulate 500 observations from the N(0.15, 0.0092) dJstnbutwn. The required TI-83 commands are as follows: ClrList L 1 randNorm (0.15, 0.0092, 500)---+ L 1 sortA(L 1 ) Scrollin~ through the 500 simulated ob~ervations, we can determine the relative frequency of observatwns that are at least 0.16 by usmg the complement rule. For one simulation, there were 435 observations less than 0.16, thus the desired relative frequency is 1 - 435/500 = 65/500 = 0.13. The actual probability is P(p?: 0.16)= 0.1379. 500 observations yield a reasonably close approximation. 7.22 The table below shows the possible observations of Y that can occur when we roll one standard die and one "weird" die. As in Exercise 7.11, there are 36 possible pairs of faces; however, a number of the pairs are identical to each other. Weird Die Standard Die I 2 3 4 5 6 0123456 0123456 0123456 6 7 8 9 10 11 12 6 7 8 9 10 11 12 6 7 8 9 10 11 12 The possible values ofY are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. Each value ofY has probability 3/36 = 1/12. Random Variables 175 7.23 The expected number of girls is,ux = :~:::X,p, = o(i)+l(i)+2(D+3G) = 1.5 and the variance is a-~ = 2)x,- f'J' p 1 = (0-1.5) 2 G )+(1-1.5) 2 (~)+(2-1.5) 2 (i) + (3-1.5) 2 G)= 0.75 so the standard deviation is ax = 0.866 girls. 7.24 The mean grade is ,u = OxO.Ol + 1 x0.05 + 2x0.30 + 3x0.43 + 4x0.21 = 2.78. 7.25 The mean for owner-occupied units is ,u= (1)(0.003) + (2)(0.002) + (3)(0.023) + (4)(0.104) + (5)(0.210) + (6)(0.224) + (7)(0.197) + (8)(0.149) + (9)(0.053) + (10)(0.035) = 6.284 rooms. The mean for renter-occupied units is J1 = (1)(0.008) + (2)(0.027) + (3)(0.287) + (4)(0.363) + (5)(0.164) + (6)(0.093) + (7)(0.039) + (8)(0.013) + (9)(0.003) + (10)(0.003) = 4.187 rooms. The larger value of ,u for owner-occupied units reflects the fact that the owner distribution was symmetric, rather than skewed to the right, as was the case with the renter distribution. The "center" of the owner distribution is roughly at the central peak class, 6, whereas the "center" of the renter distribution is roughly at the class 4. A comparison of the centers (6.284 > 4.187) matches the observation in Exercise 7.4 that the number of rooms for owner-occupied units tended to be higher than the number of rooms for renter-occupied units. 7.26 If your number is abc, then of the 1000 three-digit numbers, there are six-abc, acb, hac, bca, cab, cba-for which you will win the box. Therefore, you win nothing with probability 994/1000 = 0.994 and \$83.33 with probability 6/1000 = 0.006. The expected payoff on a \$1 bet is J.l = \$Ox0.994 + \$83.33x0.006 = \$0.50. Thus, in the long run, the Tri-State lottery commission will make \$0.50 per play of this lottery game. 7.27 (a) The payoff is either \$0, with a probability of0.75, or \$3, with a probability of0.25. (b) For each \$1 bet, the mean payoff is f1x= (\$0)(0.75) + (\$3)(0.25) = \$0.75. (c) The casino makes 25 cents for every dollar bet (in the long run). 7.28 In Exercise 7.24, we computed the mean grade of ,u = 2.78. Thus, the variance is "~ =co- 2.78)' ( o.ot) + (1- 2.78)' ( o.o5) + (2- 2.78)' (o.3o) + (3- 2.78)' ( 0.43) + c 4- 2.78)' ( o.21) = 0.7516 and the standard deviation is ax= 0.8669. 7.29 The means are: fln =1x0.25 + 2x0.32 + 3x0.17 + 4x0.15 + 5x0.07 + 6x0.03 + 7xO.Ol = 2.6 people for a household and flF =I xO + 2x0.42 + 3x0.23 + 4x0.21 + 5x0.09 + 6x0.03 + 7x0.02 = 3.14 people for a family. The standard deviations are: a~ = (1- 2.6) 2 x0.25 + (2- 2.6) 2 x 0.32 + (3 - 2.6) 2 xO.l7 + ( 4 - 2.6) 2 x0.15 + (5- 2.6) 2 x0.07 + (6- 2.6) 2 x0.03 + (7- 2.6ix0.01 = 2.02, and a" = ~2.02 = 1.421 people for a household and a~ = (1 - 3.14i(O) + (2 - 3.14) 2 (0.42) + (3- 3.14) 2 (0.23) + (4- 3.14i(0.21) + (5- 3.14i(0.09) + (6- 3.14) 2 (0.03) + (7 - 3.14i(0.02) = 1.5604, and a F = ~!.5604 =
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