TPS2e_IM_ch07_124_136

The Practice of Statistics: TI-83/89 Graphing Calculator Enhanced

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124 0.0 12345 Number of rooms in owner-occupied units 6789 1 0 0.1 0.2 Probability 0.3 0.4 7.1 (a) P (less than 3) 5 P (1 or 2) 5 (b)–(c) Answers vary. 7.2 (a) BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG. Each has probability 1/8. (b) Three of the eight arrangements have two (and only two) girls, so P ( X 5 2) 5 3/8 5 0.375. (c) See table. Value of X 0123 Probability 1/8 3/8 3/8 1/8 7.3 (a) 1%. (b) All probabilities are between 0 and 1; the probabilities add to 1. (c) P ( X # 3) 5 0.48 1 0.38 1 0.08 5 1 2 0.01 2 0.05 5 0.94. (d) P ( X , 3) 5 0.48 1 0.38 5 0.86. (e) Write either X $ 4 or X . 3. The probability is 0.05 1 0.01 5 0.06. (f) Read two random digits from Table B. Here is the correspondence: 01 to 48 Class 1, 49 to 86 Class 2, 87 to 94 Class 3, 95 to 99 Class 4, and 00 Class 5. Repeatedly generate 2 digit random numbers. The proportion of num- bers in the range 01 to 94 will be an estimate of the required probability. 7.4 2 6 5 1 3 . 7 6851F_ch07_124_136 20/9/02 11:39 Page 124
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Random Variables 125 0.0 12345 Number of rooms in renter-occupied units 6789 1 0 0.1 0.2 Probability 0.3 0.4 The rooms distribution is skewed to the right for renters and roughly symmetric for owners. This sug- gests that renter-occupied units tend, on the whole, to have fewer rooms than owner-occupied units. 7.5 (a) { X $ 5}. P ( X $ 5) 5 P ( X 5 5) 1 P ( X 5 6) 1 ... 1 P ( X 5 10) 5 0.868. (b) { X . 5} 5 the event that the unit has more than five rooms. P ( X . 5) 5 P ( X 5 6) 1 P ( X 5 7) 1 1 P ( X 5 10) 5 0.658. (c) A discrete random variable has a countable number of values, each of which has a distinct probability ( P ( X 5 x )). P ( X $ 5) and P ( X . 5) are different because the first event contains the value X 5 5 and the second does not. 7.6 (a) P (0 # X # 0.4) 5 0.4. (b) P (0.4 # X # 1) 5 0.6. (c) P (0.3 # X # 0.5) 5 0.2. (d) P (0.3 , X , 0.5) 5 0.2. (e) P (0.226 # X # 0.713) 5 0.713 2 0.226 5 0.487. (f) A continuous distribution assigns probability 0 to every individual outcome. In this case, the probabilities in (c) and (d) are the same because the events differ by 2 individual values, 0.3 and 0.5, each of which has probability 0. 7.7 (a) P ( X # 0.49) 5 0.49. (b) P ( X $ 0.27) 5 0.73. (c) P (0.27 , X , 1.27) 5 P (0.27 , X , 1) 5 0.73. (d) P (0.1 # X # 0.2 or 0.8 # X # 0.9) 5 0.1 1 0.1 5 0.2. (e) P (not [0.3 # X # 0.8]) 5 1 2 0.5 5 0.5. (f) P ( X 5 0.5) 5 0. 7.8 (a) 5 P ( Z $ 2.17) 5 0.0150. (b) 5 P ( Z ,2 2.17) 5 0.0150. (c) P 1 0.35 # ˆ p # 0.45 2 5 P 1 2 2.17 # Z # 2.17 2 5 0.9700. P 1 ˆ p 6 0.35 2 P 1 ˆ p $ 0.45 2 5 P 1 Z $ 0.45 2 0.4 0.023 2 6851F_ch07_124_137 13/09/2002 09*59 PM Page 125
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126 Chapter 7 7.11 (a) { Y . 1}. P ( Y . 1) 5 P ( Y 5 2) 1 P ( Y 5 3) 1 ... 1 P ( Y 5 7) 5 0.75. Or, P ( Y . 1) 5 1 2 P ( Y 5 1) 5 1 2 .25 5 .75. (b) P (2 , Y # 4) 5 P ( Y 5 3) 1 P ( Y 5 4) 5 0.32. (c) P ( Y 2) 5 1 2 P ( Y 5 2) 5 0.68. 0.0 12345 Number of persons in household 67 0.1 0.2 Probability 0.3 0.4 0.5 0.0 Number of persons in family 0.1 0.2 0.3 0.4 0.5 7.9 For a sample simulation of 400 observations from the N(0.4, 0.023) distribution, there were 0 observations less than 0.25, so the relative frequency is 0/400 5 0. The actual probability that is P ( Z ,2 6.52) 3.5 3 10 2 11 , essentially 0. 7.10 (a) Both sets of probabilities sum to 1. (b) Both distributions are skewed to the right; how- ever, the event { X 5 1} has a higher probability in the household distribution. This reflects the fact that a family must consist of two or more persons. Also, the events { X 5 3} and { X 5 4} have slightly higher probabilities in the family distribution, which may reflect the fact that families are more likely than households to have children living in the dwelling unit.
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