251 3takehome 4 11 07 ECO251 QBA1 THIRD EXAM Apr 18 2005 TAKE 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 like n 20 g make it very clear what value of n you are using Look at the solved problems for Section L the solution to Grass3 and Great Distributions especially the hints on the 3 rd 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 10 g 2 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 1 g units 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 if more than 4 units are defective What is the probability that you will not buy from this supplier again 1 Note If g 9 you 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 1 g units per lot of 100 means that 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 only Your c Lawn fertilizer comes in bags whose weight follows the continuous uniform distribution with a range between 23 8 and 26 0 2 g pounds You need 24 8 pounds to cover your lawn and buy 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 2 0 1g pounds what is the probability that a bag will be between 20 and 24 pounds range will be between 2 and 2 9 pounds 1 d There are only 10 2 g 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 random answer is 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 what is the probability that more than half make over 500 000 Note that the 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 th 3 74 1 iii If a group of 6 g employees 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 2 1h and 0 5 01h where h is the second to last digit of your student number For example if the life of the phillinx is represented by x1 the chance of the phillinx failing in the 3rd month is P 2 x1 3 and the probability of it failing after the third month is P x1 3 For example Ima Badrisk has the number 375292 so her distribution has a mean of 2 1 h 2 9 and a standard deviation of 0 5 01 9 0 59 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 …
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