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Unformatted text preview: Department of Management Sciences Faculty of Engineering University of Waterloo MSci 609 Quantitative data Analysis for Management Sciences Winter 2011 Assignment 2 (Due February 15, 2011) Points to consider: • • •
• Include a title page as a separate sheet, and write on it your name, student number, and email address. Work on the assignment independently. Please state all your assumptions and show all your work. Generous partial credit will be given if the reasoning is correct, even if the solution is not. However, none will be given for unsubstantiated claims, even if you provide perfectly correct answer. Clearly define the random variables when applicable. Please do not ask for informal assessment of your work, i.e. do not ask for an assessment of whether you were on the right track on any particular problem. Assignments should be dropped in the drop‐box in CPH (2nd floor) by 1:00 pm on the day it is due. Question 1: (7 marks) According to Industrial Engineering Progress (December 1990), approximately 30% of all pipework failures in industrial plants are caused by operator error. (a) What is the probability that out of the next 20 pipework failures at least 10 are due to operator error? (b) What is the probability that no more than 4 out of 20 such failures are due to operator error? (c) Suppose, for a particular plant, that out of the random sample of 20 such failures, exactly 5 are due to operator error. Do you feel that the 30% figure stated above applies to this plant? Comment. Question 2: (7 marks) A manufacturer company uses an acceptance scheme on items from a production line before they are shipped. the plant is a two‐stage one. Boxes of 25 items are readied for shipment, and a sample of 3 items is tested for defectives. If any defectives are found, the entire box is sent back for 100% screening. If no defectives are found, the box is shipped. (a) What is the probability that a box containing 3 defectives will be shipped? (b) What is the probability that a box containing only 1 defective will be sent back for screening? (c) Suppose that the manufacturer decides to change its acceptance scheme. Under the new scheme, the inspector takes one item at random, inspects it, and then replaces it in the box; a second inspector does likewise. Finally, a third inspector goes through the same procedure. The box is not shipped if any of the three inspectors find a defective. What is the probability that a box containing only 1 defective will be sent back for screening under this new plan? Question 3: (7 marks) The life, in years, for a certain type of electrical switche has an exponential distribution with an average life β=2. If 100 of these switches are installed in different systems, what is the probability that at most 30 fail during the first year? Question 4: (5 marks) A common practice of airline companies is to sell more tickets for a particular flight than there are seats on the plane because customers who buy tickets do not always show up for the flight. Suppose that the percentage of no‐
shows at flight time is 2%. For a particular flight with 197 seats, a total of 200 tickets were sold. What is the probability that the airline overbooked this flight? Question 5: (9 marks) A company rents time on a computer for periods of t hours, for which it receives $600 an hour. The number of times the computer breaks down during t hours is a random variable having the Poisson distribution with λ = (0.8)t, and if the computer breaks down x times during t hours, it costs 50x2 dollars to fix it. How should the company select t in order to maximize the expected profit? 2 ...
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