8(y) = exp V2πσν %3D o2 y>0 Note that u and o are the mean and standard deviation of the underlymg normal distribution; they are not the mean and standard deviation of Y 1iseh. 46. Let Y denote the diameter in millimeters of Styrofoam pellets used in packing. o = .1. Assume that Yhas a log-normal distribution with parameters u = .8 and (a) Find the probability that a randomly selected pellet has a diameter that exceeds 2.7 millimeters. (b) Between what two values will Y fall with probability approximately .95? ad.io L Section 4.5. 47. Verify the normal probability rule. 48. The number of Btu's of petroleum and petroleum products used per person in the United States in 1975 was normally distributed with mean 153 million Btu's and standard deviation 25 million Btu's, Approximately what percent- age of the population used between 128 and 178 million Btu's during that year? Approximately what percentage of the population used in excess of 228 million Btu's? uA,00 49. Reconsider Exercises 40(a), 41(a), and 42(a) in light of the normal probabil- ity rule. 50. For a normal random variable, P[IX – ul < 30] = .997. What value is as- signed to this probability via Chebyshev's inequality? Are the results consis- tent? Which rule gives a stronger statement in the case of a normal variable? 51. Animals have an excellent spatial memory. In an experiment to confirm this statement, an eight-armed maze such as that shown in Fig. 4.21 is used. At the beginning of a test, one pellet of food is placed at the end of each arm. A hun- gry animal is placed at the center of the maze and is allowed to choose freely from among the arms. The optimal strategy is to run to the end of each arm ex- actly once. This requires that the animal remember where it has been. Let X denote the number of correct arms (arms still containing food) selected among its first eight choices. Studies indicate that u = 7.9. (a) Is X normally distributed? (b) State and interpret Chebyshev's inequality in the context of this problem for k = .5, 1, 2, and 3. At what point does the inequality begin to give us µ = %3D %3D some practical information? Section 4.6 52. Let X be binomial withn3D 20 and p = .3. Use the normal approximation to approximate each of the following. Compare your results with the values ob- g tained from Table I of App. A.

8(y) = exp V2πσν %3D o2 y>0 Note that u and o are the mean and standard deviation of the underlymg normal distribution; they are not the mean and standard deviation of Y 1iseh. 46. Let Y denote the diameter in millimeters of Styrofoam pellets used in packing. o = .1. Assume that Yhas a log-normal distribution with parameters u = .8 and (a) Find the probability that a randomly selected pellet has a diameter that exceeds 2.7 millimeters. (b) Between what two values will Y fall with probability approximately .95? ad.io L Section 4.5. 47. Verify the normal probability rule. 48. The number of Btu's of petroleum and petroleum products used per person in the United States in 1975 was normally distributed with mean 153 million Btu's and standard deviation 25 million Btu's, Approximately what percent- age of the population used between 128 and 178 million Btu's during that year? Approximately what percentage of the population used in excess of 228 million Btu's? uA,00 49. Reconsider Exercises 40(a), 41(a), and 42(a) in light of the normal probabil- ity rule. 50. For a normal random variable, P[IX – ul < 30] = .997. What value is as- signed to this probability via Chebyshev's inequality? Are the results consis- tent? Which rule gives a stronger statement in the case of a normal variable? 51. Animals have an excellent spatial memory. In an experiment to confirm this statement, an eight-armed maze such as that shown in Fig. 4.21 is used. At the beginning of a test, one pellet of food is placed at the end of each arm. A hun- gry animal is placed at the center of the maze and is allowed to choose freely from among the arms. The optimal strategy is to run to the end of each arm ex- actly once. This requires that the animal remember where it has been. Let X denote the number of correct arms (arms still containing food) selected among its first eight choices. Studies indicate that u = 7.9. (a) Is X normally distributed? (b) State and interpret Chebyshev's inequality in the context of this problem for k = .5, 1, 2, and 3. At what point does the inequality begin to give us µ = %3D %3D some practical information? Section 4.6 52. Let X be binomial withn3D 20 and p = .3. Use the normal approximation to approximate each of the following. Compare your results with the values ob- g tained from Table I of App. A.

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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