Homework #8. Due: Friday, October 24, 2003. IE 230. Textbook: D.C.
Montgomery and G.C. Runger, Applied Statistics and Probability for. Engineers,
John Wiley ...
Homework #8. Due: Friday, October 24, 2003.
IE 230
Textbook: D.C. Montgomery and G.C. Runger, Applied Statistics and Probability for Engineers, John Wiley & Sons, New York, 2003. Sections 4.6–4.12. 1. Exam 1 scores had a minimum of 31, a maximum of 104, a mean of 73.3 and a standard deviation of 15.8. Of 143 students, 28 had scores lying between 71 and 80. Twelve had scores fifty or less. (a) Let X denote the Exam 1 score of a randomly selected student. Assume that X has a normal distribution with the given mean and standard deviation. Sketch the pdf. Label and scale the axes. (b) The Central Limit Theorem says that the normal distribution is a good model for sums of random variables. Explain how a exam score is a sum of random variables (and that therefore exam scores are likely to be approximately normally distributed). (c) In fact, X takes only integer values, so the normal distribution is an approximation. Explain why a reasonable approximation to P(X = 75) is P(74.4 ≤ X ≤ 75.5). (This is called the continuity correction.) (d) Let Y denote the number of scores in [71, 80]. If scores are independent, then Y is binomial with n = 143 and (approximately) p = P(70.5 ≤ X ≤ 80.5). Determine the value of p by (i) shading the area in the sketch of Part (a) and approximating that area. (ii) converting to standard normal and using Table II. (iii) using "normdist" in MSexcel. (e) Determine the value of E(Y). Discuss whether the 28 observed scores in [71, 80] is reasonable if exam scores are normally distributed. 2. (Montgomery and Runger, Problem 4–46) If X is normally distributed with mean 5 and standard deviation 4, determine the value of x that satisfies P(−x < X < x ) = 0.99. 3. (Montgomery and Runger, Problem 4–68) Hits to an Web site are assumed to follow a Poisson distribution with a mean of 10,000 per day. Let X denote the number of hits in a day. (a) Compute a normal approximation for the probability of more than 10,500 hits per day. (b) Compute a normal approximation for x , where P(X > x ) = 0.01. 4. Let X denote your instructor’s commute time. Suppose that X has a gamma distribution with mean µX = 9 minutes and standard deviation σX = 2 minutes. (a) Determine the parameter values λ and r that yield the desired mean and variance. (b) Sketch the resulting pdf. (You can use MSexcel’s function "gammadist" and print, or just trace from the monitor. Notice that help for "gammadist" mistakenly refers to the probability mass function rather than the pdf.)
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Schmeiser
Homework #8. Due: Friday, October 24, 2003.
IE 230
5. (Montgomery and Runger, Problem 4–79) Let T denote the time to failure (in hours) of fans in a personal computer. Assume that T has an exponential distribution with rate λ = 0.00003. (b) What are the units of λ? (b) Sketch the pdf of T . Label and scale the axes. (c) What proportion of the fans will last at least 10,000 hours? (d) Show the answer to Part (c) in your sketch of Part (b). (e) If a particular fan has lasted 20,000 hours, what is the probability that it will last beyond hour 30,000? 6. (Montgomery and Runger, Problems 4–96 and 4–97) Calls to a telephone system arrive according to a Poisson Process at a rate of five calls per minute. (a) What is the name applied to the distribution and the parameter values of the time until the tenth call? (b) What is the mean time until the tenth call? (c) What is the mean time between the ninth and tenth calls? (d) What is the probability that exactly four calls occur within one (randomly selected) minute. 7. (Montgomery and Runger, Problems 4–111) Assume that the life of a roller bearing follows a Weibull distribution with parameters β = 2 and δ = 10,000 hours. (a) Determine the probability that a bearing lasts at least 8000 hours. (If you use the "weibull" function in MSexcel, you need to convert to its parameters α and β.) (b) Determine the mean time to failure of a bearing using (i) the result that Γ(r ) = (r − 1)! when r is a positive integer. (ii) using MSexcel’s function "gammaln", which calculates the natural logarithm of the gamma function. (Notice that the gamma function can yield huge values, since it is a generalization of the factorial function. MSexcel returns the logarithm, which is more tractable. You need to then use "exp(gammaln(r ))".) 8. Let X denote your instructor’s commute time. Suppose that X has a triangular distribution with minimum at 6 minutes, mode at 8.5 minutes, and maximum at 15 minutes. (a) Sketch the resulting pdf. (b) Determine the mean commute time. (c) Determine P(X > 10). (d) Shade in the sketch of Part (a) your answer to Part (c).
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Schmeiser
