ages, these erors are often very tortion, erors in measuring distances are often no larger than 000 s Assume that the probability of a serious measureent error is .05, A soch. 150 independent measurements are made, Let X denote the number of serious negligible dis errors made. (a) In finding the probability of making at least one serious error., is the no mal approximation appropriate? If so, approximate the probability usine this method. (b) Approximate the probability that at most three serious errors will he made. 54.) Achemical reaction is run in which the usual yield is 70%. A new process has been devised that should improve the yield. Proponents of the new process claim that it produces better yields than the old process more than 90% of the time. The new process is tested 60 times. Let X denote the number of trials in which the yield exceeds 70%. (a) If the probability of an increased yield is .9, is the normal approximation appropriate? (b) If p = 9, what is E[X]? (c) If p>.9 as claimed, then, on the average, more than 54 of every 60 trials will result in an increased yield. Let us agree to accept the claim if X is at %3D 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? 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. 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- fower than 100 such flights will take place in an hour? do.gobtam

ages, these erors are often very tortion, erors in measuring distances are often no larger than 000 s Assume that the probability of a serious measureent error is .05, A soch. 150 independent measurements are made, Let X denote the number of serious negligible dis errors made. (a) In finding the probability of making at least one serious error., is the no mal approximation appropriate? If so, approximate the probability usine this method. (b) Approximate the probability that at most three serious errors will he made. 54.) Achemical reaction is run in which the usual yield is 70%. A new process has been devised that should improve the yield. Proponents of the new process claim that it produces better yields than the old process more than 90% of the time. The new process is tested 60 times. Let X denote the number of trials in which the yield exceeds 70%. (a) If the probability of an increased yield is .9, is the normal approximation appropriate? (b) If p = 9, what is E[X]? (c) If p>.9 as claimed, then, on the average, more than 54 of every 60 trials will result in an increased yield. Let us agree to accept the claim if X is at %3D 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? 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. 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- fower than 100 such flights will take place in an hour? do.gobtam

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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