(a) The time for a consultant to complete a project is uncertain but it is estimated...

 

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(a) The time for a consultant to complete a project is uncertain but it is estimated that the project duration is approximately normally distributed with a mean of 90 days and a standard deviation of 8 days. (i) What is the probability that the consultant takes less than 80 days to complete the project? (ii) The consultant estimated that he would have a 2.5% chance to pay penalty as a result of failing to complete the project on or before the contractual time. What is the contractual time for this project? [4 marks] [4 marks] (b) A manufacturer knows that, on average, 5% of the electronic razors that he produces will require repairs within 1 year after sale. If ten razors are randomly selected, what is the probability that more than one razor will require repairs within 1 year after sale? [5 marks] (c) A manufacturer produces boxes of candies, each containing ten pieces. Two machines, namely I and II, are used for production. Among a large number of boxes of candies that have been produced, 30% of boxes are produced by machine I and the rest are produced by machine II. During production, 10% and 5% of candies produced by machine I and machine II, respectively, have impurity. One box of candies is chosen at random and two pieces of candies are selected from it to test for impurity. It is found that one of the two tested pieces has impurity, calculate the probability that the chosen box of candies was produced by machine I. [7 marks] (a) The time for a consultant to complete a project is uncertain but it is estimated that the project duration is approximately normally distributed with a mean of 90 days and a standard deviation of 8 days. (i) What is the probability that the consultant takes less than 80 days to complete the project? (ii) The consultant estimated that he would have a 2.5% chance to pay penalty as a result of failing to complete the project on or before the contractual time. What is the contractual time for this project? [4 marks] [4 marks] (b) A manufacturer knows that, on average, 5% of the electronic razors that he produces will require repairs within 1 year after sale. If ten razors are randomly selected, what is the probability that more than one razor will require repairs within 1 year after sale? [5 marks] (c) A manufacturer produces boxes of candies, each containing ten pieces. Two machines, namely I and II, are used for production. Among a large number of boxes of candies that have been produced, 30% of boxes are produced by machine I and the rest are produced by machine II. During production, 10% and 5% of candies produced by machine I and machine II, respectively, have impurity. One box of candies is chosen at random and two pieces of candies are selected from it to test for impurity. It is found that one of the two tested pieces has impurity, calculate the probability that the chosen box of candies was produced by machine I. [7 marks]