251 3takehome 4/11/07 ECO251 QBA1 THIRD EXAMApr 18, 2005TAKE HOME SECTION- Name: _________________________ Student Number: _________________________Throughout this exam show your work! Please indicate clearly what sections of the problem you are answering and what formulas you are using.Part III. Do all the Following (19+ Points) Show your work! Neatness counts!1. (Webster) Identify the distribution that you are using in each problem. If you have a number likegn 20, make it very clear what value of nyou are using. Look at the solved problems for Section L, the solution to Grass3 and ‘Great Distributions’ (especially the hints on the 3rd page) before you start.a) The average number of calls that come into a switchboard is 1 each minute. Let g be the last digit of your student number. Assume that the operator is away from her desk for 210 g minutes (so that the time that you use will be between 5 and 9.5 minutes). What is the average number of calls that will come into the switchboard in that time? (0)(i) What is the probability that at least one call is missed while the operator is away? (1)(ii) What is the probability that between 10 and 20 (inclusive) calls are missed while the operator is away? (1)b) You purchase parts from a supplier who has an average defect rate of g1units per lot of 100. (Your average for a lot of 100 will be between 1 and 10.)(i) You order 150 units (1.5 lots) , but say that you will not buy from this supplier again ifmore than 4 units are defective. What is the probability that you will not buy from this supplier again? (1) (Note: If ,9gyou order 200 units.)(ii) The next time you order (if there is a next time), you resolve to take a sample of ten items from the thousand in this order. You will return the whole mess if more than two items are defective. If an average of g1 units per lot of 100 meansthat the probability of an item being defective is %1 g, what is the probability that you will return the order? (2) Note: To keep us on a level playing field, you must work out this answer by hand. Though it is strongly recommended that you check your answer against Table 14 and/or Table 15, no answer using tables will be accepted![5]c) Lawn fertilizer comes in bags whose weight follows the continuous uniform distribution with a range between 23.8 and g2.026 pounds. You need 24.8 pounds to cover your lawn and buy only one bag. (Your upper number will be between 26 and 28.9 pounds.)(i) What is the probability that you will not have enough fertilizer in the bag? (1)(ii) If you buy two bags, what is the mean and standard deviation of their total weight? (1)[7](iii) If the bags are distributed with a mean of 25.5 pounds and a range of g1.02 pounds, what is the probability that a bag will be between 20 and 24 pounds? (Your range will be between 2 and 2.9 pounds) (1)d) There are only g210 officers in your firm that are eligible to be on the Compensation Committee. Of these half make over $500,000.00 a year.251 3takehome 4/11/07(i) If you pick 4 officers at random, what is the probability that exactly three make over $500,000? (2)(ii) The shareholders will be very unhappy if the see that more than half of the members of the Compensation Committee make over $500,000. If you pick 4 officers at random, what is the probability that more than half make over $500,000? (Note that the answer is not the same as the answer to (i).) (1) [11]e) Only 65% of the many workers in a high security scientific installation seem to be capable of remembering to carry their badges. If they are among the 35% that forget they will be sent home to get them!(i) If the installation opens at 6am, what is the probability that the second employee to arrive is the first one sent home? (1)(ii) On the average, if we number the employees by their arrival order, what is the average number of the first employee to be sent home? (The third? The 4th? The 3.74th?)(1)(iii) If a group of g6employees arrive at 9am, what is the probability that at least one will not be sent home? (1)(iv) In the same group, what is the probability that at least one will be sent home? (1) (v) In the same group, what is the probability that more than half will not be sent home? (1)[16](vi) (Extra Credit) By late morning, the employees are trickling is at only 12 per hour. The guard decides that if the probability of someone arriving in the next 5 minutes after you come in is less than 50%, he will hide behind the warning sign and take a few nips from his bottle. He asks you whether the probability is less than 50%. Do not use the Poisson distribution for this problem. (2)2. As everyone knows, a jorcillator has two components, a phillinx and a flubberall. It seems that the jorcillator only works as long as both components work (so that it fails in the first month if either component fails). Note change. The probability of the phillinx failing is given by a Normal distribution with h1.2 andh01.5.0 , where h is the second-to-last digit of your student number. For example, if the life of the phillinx is represented by 1x, the chance of the phillinx failing in the 3rd month is 321xP and the probability of it failing after the third month is 31xP. For example: ImaBadrisk has the number 375292, so her distribution has a mean of 9.21.2 h and a standard deviation of 59.0901.5.0 . Your mean will be between 2.0 and 2.9 and your standard deviation will be between 0.50 and 0.59.The probability of the flubberall failing is: in the first month 40%; in the second month 30%; in the third month 20% and after the third month 10%. , We will divide time into four periods with the last period being ‘beyond the third month.’ The jorcillator is guaranteed to fail in one of the four periods. Failure of components is assumed to be independent, so if the probability of the phillinx failing in the first month is .1, and the probability of the flubberall failing in the first month is .4, the probability of both components failing in the first month is (.1) (.4) =.04 (This is not the necessarily the probability that the jorcillator will fail in the first month!) In order to maintain my sanity, use the following events. Failure of the phillinx in period 1, 2, 3, 4 are events ,1A,2A,3A and 4A.Failure of the flubberall in period 1, 2, 3, 4 are events ,1B,2B,3B and 4B.251
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