CONTINUOUS DISTRIBUTIONS 149 least 59. What is the probability that we will accept the claim if p is really only .9? (d) What is the probability that we shall not accept the claim (X < 58) if it is true, and p is really .95? 55. Opponents of a nuclear power project claim that the majority of those living near a proposed site are opposed to the project. To justify this statement, a ran- dom sample of 75 residents is selected and their opinions are sought. Let X de- note the number opposed to the project. (a) If the probability that an individual is opposed to the project is .5, is the normal approximation appropriate? (b) If p = .5, what is E[X]? (c) If p> .5 as claimed, then, on the average, more than 37.5 of every 75 in- dividuals are opposed to the project. Let us agree to accept the claim if X is at least 46. What is the probability that we shall accept the claim if p is really only .5? (d) What is the probability that we shall not accept the claim (X < 45) even though it is true and p is really .7? %3D 56. (Normal approximation to the Poisson distribution.) Let X be Poisson with parameter As. Then for large values of As, X is approximately normal with mean As and variance As. (The proof of this theorem is also based on the Cen- tral Limit Theorem and will be considered in Chap. 7.) Let X be a Poisson random variable with parameter As = 15. Find P[X < 12] from Table II of App. A. Approximate this probability using a normal curve. Be sure to employ the half-unit correction factor. %3D 57. The average number of jets either arriving at or departing from O’Hare Air- port is one every 40 seconds. What is the approximate probability that at least 75 such flights will occur during a randomly selected hour? What is the prob- ability that fewer than 100 such flights will take place in an hour? Gecoqolicent2 esem inohangout aber 58. The length of time in hours that a rechargeable calculator battery will hold its charge is a random variable. Assume that this variable has a Weibull distribu- tion with a = .01 and B = 2. (a) What is the density for X? (b) What are the mean and variance for X? Hint: It can be shown that IT(@) = (a-1)T(@ - (c) What is the reliability function for this random variable? Section 4.7 %3D 1) for any a > 1. Furthermore, T(1/2) = V. %3D Lin the reliability of such a battery at t = 3 hours? At t = 12 hours?