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Unformatted text preview: 32 Solutions Manual for Statistical Inference 3.5 Let X = number of effective cases. If the new and old drugs are equally effective, then the probability that the new drug is effective on a case is .8. If the cases are independent then X binomial(100, .8), and
100 P (X 85) =
x=85 100 .8x .2100x = .1285. x So, even if the new drug is no better than the old, the chance of 85 or more effective cases is not too small. Hence, we cannot conclude the new drug is better. Note that using a normal approximation to calculate this binomial probability yields P (X 85) P (Z 1.125) = .1303. 3.7 Let X Poisson(). We want P (X 2) .99, that is, P (X 1) = e + e .01. Solving e + e = .01 by trial and error (numerical bisection method) yields = 6.6384. 3.8 a. We want P (X > N ) < .01 where X binomial(1000, 1/2). Since the 1000 customers choose randomly, we take p = 1/2. We thus require
1000 P (X > N ) =
x=N +1 1000 x 1 2 x 1 1 2 1000x < .01 which implies that 1 2
1000 1000 x=N +1 1000 x < .01. This last inequality can be used to solve for N , that is, N is the smallest integer that satisfies 1 2 The solution is N = 537.
1 b. To use the normal approximation we take X n(500, 250), where we used = 1000( 2 ) = 500 1 1 2 and = 1000( 2 )( 2 ) = 250.Then 1000 1000 x=N +1 1000 x < .01. P (X > N ) = P thus, P X  500 N  500 > 250 250 N  500 250 < .01 Z> < .01 where Z n(0, 1). From the normal table we get P (Z > 2.33) .0099 < .01 N  500 = 2.33 250 N 537. Therefore, each theater should have at least 537 seats, and the answer based on the approximation equals the exact answer. ...
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.
 Spring '12
 Dr.Hackney
 Statistics, Binomial, Probability

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