Q1. A local hospital estimates that the number of patients admitted daily to the emergency room has a Poisson probability distribution with a mean of 4.0.

What is the probability that on a given day

a) only 2 patients will be admitted?

b) at most 6 patients will be admitted?

c) no one will be admitted?

d) What is the standard deviation of the number of patients admitted?

e) For each patient admitted, the expected daily operational expenses to the hospital are $800. If the hospital wants to be 94.1% sure of meeting daily expenses, how much money should it retain for operational expenses daily?

Q2. The specifications for the thickness of nonferrous washers is 1.0 (+ or -) 0.04 mm. From the process data, the distribution of the washer thickness is estimated to be normal with a mean of 0.98 mm and a standard deviation of 0.02 mm. The unit cost of rework is $0.10, and the unit cost of scrap is $0.15. For a daily production of 10,000 items:

a) What proportion of the washers is conforming? What is the total daily cost of rework and scrap?

b) In its study of constant improvement, the manufacturer changes the mean setting of the machine to 1.0 mm. If the standard deviation is the same as before, what is the total daily cost of rework and scrap?

c) The manufacturer is trying to further improve on the process and reduces its standard deviation to 0.015 mm. If the process mean is maintained at 1.0 mm, what is the percent decrease in the total daily cost of rework and scrap compared to that of part (a)?

Q3. A component is known to have an exponential time-to-failure distribution with a mean life of 10,000 h.

a) What is the probability of the component lasting at least 8000 h?

b) If the component is in operation at 9000 h, what is the probability that it will last another 6000 h?

c) Two such components are put in parallel, so that the system will be in operation if at least one of the components is operational. What is the probability of the system being operational for 12,000 h? Assume that the components operate independently.