TPS2e_IM_ch08_137_153

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

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137 8 8.1 (a) No: There is no fixed n (i.e., there is no definite upper limit on the number of defects). (b) Yes: It is reasonable to believe that all responses are independent (ignoring any “peer pressure”), and all have the same probability of saying “yes” since they are randomly chosen from the popu- lation. Also, a “large city” will have a population over 1000 (10 times as big as the sample). (c) Yes: In a “Pick 3” game, Joe’s chance of winning the lottery is the same every week, so assuming that a year consists of 52 weeks (observations), this would be binomial. 8.2 (a) Yes: It is reasonable to assume that the results for the 50 students are independent, and each has the same chance of passing. (b) No: Since the student receives instruction after incorrect answers, her probability of success is likely to increase. (e) No: Temperature may affect the outcome of the test. 8.3 (a) .2637. (b) The binomial probabilities for x 5 0, . . ., 5 are: .2373, .3955, .2637, .0879, .0146, .0010. (e) The cumulative probabilities for x 5 0, . . ., 5 are: .2373, .6328, .8965, .9844, .9990, 1. Compared with Corinne’s cdf histogram, the bars in this histogram get taller, sooner. Both peak at 1 on the extreme right. 8.4 Let X 5 the number of correct answers. X is binomial with n 5 50, p 5 0.5. (a) P ( X $ 25) 5 1 2 P ( X # 24) 5 1 2 binomcdf (50, .5, 24) 5 1 2 .444 5 .556. (b) P ( X $ 30) 5 1 2 P ( X # 29) 5 1 2 binomcdf (50, .5, 29) 5 1 2 .899 5 .101. (c) P ( X $ 32) 5 1 2 P ( X # 31) 5 1 2 binomcdf (50, .5, 31) 5 1 2 .968 5 .032. 8.5 (a) Let X 5 the number of correct answers. X is binomial with n 5 10, p 5 0.25. The proba- bility of at least one correct answer is P ( X $ 1) 5 1 2 P ( X 5 0) 5 1 2 binompdf (10, .25, 0) 5 1 2 .056 5 .944. (b) Let X 5 the number of correct answers. We can write X 5 X 1 1 X 2 1 X 3 , where X i 5 the number of correct answers on question i . (Note that the only possible values of X i are 0 and 1, with 0 representing an incorrect answer and 1 a correct answer.) The probability of at least one 6851F_ch08_137_153 16/9/02 19:48 Page 137
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138 Chapter 8 correct answer is P ( X $ 1) 5 1 2 P ( X 5 0) 5 1 2 [ P ( X 1 5 0) P ( X 2 5 0) P ( X 3 5 0)] (since the X i are independent) 5 1 25 1 0.6. 8.6 (a) Yes, if the 100 children are randomly selected, it is extremely likely that the result for one child will not be influenced by the result for any other child (e.g., the children are siblings). “Success” in this context means having an incarcerated parent. n 5 100, since 100 children are selected, and p 5 0.02. (b) P ( X 5 0) 5 the probability of none of the 100 selected children having an incarcerated par- ent. P ( X 5 0) 5 binompdf (100, .02, 0) 5 .133. P ( X 5 1) 5 binompdf (100, .02, 1) 5 .271. (c) P ( X $ 2) 5 1 2 P ( X # 1) 5 1 2 binomcdf (100, .02, 1) 5 1 2 .403 5 .597. Alternatively, by the addition rule for mutually exclusive events, P ( X $ 2) 5 1 2 ( P ( X 5 0) 1 P ( X 5 1)) 5 1 2 (.133 1 .271) 5 1 2 .404 5 .596. (The difference between answers is due to roundoff error.) 8.7 Let X 5 the number of players out of 20 who graduate. P ( X 5 11) 5 binompdf (20, .8, 11) 5 .0074. 8.8 (a) n 5 10 and p 5 0.25. (b) (0.25) 2 (0.75) 8 5 0.28157. (c) ...
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