problemsheet_3

# problemsheet_3 - Q1 The following probability distribution...

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Q1 The following probability distribution of the monthly incomes (x in \$1000) of account executives has been estimated for a big accounting firm. x 12 14 20 22 30 p( x ) .15 .15 .4 .2 .1 (a) Find the expected value and the standard deviation of the monthly incomes. (b) A random sample of 64 executives is taken. What is the probability that the sample mean value exceeds \$20,500? Q2 The life of a nickel-cadmium battery produced by a manufacturer is normally distributed with a mean of 20 hours and a standard deviation of 4 hours. (a) What proportion of batteries have a life of less than 15 hours? (b) The manufacturer is considering a warranty for the battery. How many hours of use should the warranty cover so that 5% of the batteries have to be replaced? (c) Suppose that a battery with life of less than 15 hours is considered to be defective. If 8 batteries are selected at random from a large batch, what is the probability that at most 2 batteries are defective? Q3 Suppose the score of the ISMT 111 mid-term exam is a normal random variable with mean 65 and standard deviation 10. The instructors decide to offer 4 grades to the students. The best group get As, the second best group get Bs, the third best group get Cs and the worst group get Ds. Only 5% of the students get As and 10% of the students get Ds. (a) Calculate the cut-off score for the grade ”A”. (b) If we sample 5 students independently from the class, what is the probability that there is at least one student getting a ”D”? (c) If we sample 100 students independently from the class, what is the probability that there are more than 8 students getting an ”A”? (Hint: use Normal Approximation.) (d) Suppose the instructors decide to fail those students who score less than 40 in the mid-term exam. If we sample 80 students independently from the class, what is the expected number of students who fail the mid-term? Q4 For a certain aptitude test, it is known from past experience that scores are distributed normally with an average of μ = 650 and a standard deviation of _ = 12. If a randomly

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selected individual takes the test, what is the probability that his score will lie in the interval 630 to 680? Assume the current population distribution is identical to that of previous populations. Q5 The project of planning and scheduling the budget requires a sequence of tasks including forecasting demand, preparing the production schedule, estimating costs, and preparing the budget. Project schedulers have found that the time required to complete a budget is normally distributed with a mean of 8 days and a standard deviation of 2 days. To arrange financing at the current rate of interest, the treasurer must know the budget within 5 days of beginning the process. Otherwise, a penalty will be imposed by the bank. (a) What is the probability that a budget is finished within 5 days?
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