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# Using the law of total probability for the

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Unformatted text preview: (2.12) is on the conTable 2.2: Some choices of a and resulting values of 1 − a 2 5 10 1 . a2 1 − (1/a2 ) 75% 96% 99% servative side; the probability on the left-hand side of (2.12) may be much closer to one than the right-hand side. Often in practice other conﬁdence intervals are used which are based on diﬀerent assumptions (see Example 3.6.10). Example 2.9.2 Suppose the fraction p of telephone numbers that are busy in a large city at a given time is to be estimated by p = X , where n is the number of phone numbers that will be n tested and X will be the number of the tested numbers found to be busy. If p is to be estimated to within 0.1 with 96% conﬁdence, how many telephone numbers should be sampled, based on (2.12)? Solution: For 96% conﬁdence we take a = 5, so the half-width of the conﬁdence interval is a 2. √ = √5 , which should be less than or equal to 0.1. This requires n ≥ ( 2.5 )2 = 625. 0.1 2n n 2.10 The law of total probability, and Bayes formula Events E1 , . . . , Ek are said...
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