Industrial filling process

The characteristics of an industrial filling process in which an expensive liquid is injected into a container was investigated. The quanitity injected per container is approximately normally distributed with mean 10 units and standard deviation .2 units. Each unit of fill costs $20 per unit. If a container contains less than 10 units it must be reprocessed at a cost of $10. A properly filled container sells for $230.

a. find the probability that a container is under-filled. Not under-filled.
b. a container is initially under-filled and must be reprocessed. Upon refilling, it contains 10.60 units. How much profit will the company make on this container?
c. the operations manager adjusts the mean of the filling process upward to 10.10 units in order to make the probability of under-filling approximately zero. Under these conditions, what is the expected profit per container?

As part of a project targeted at improving the services of a local bakery a management consultant monitored customer arrivals for several saturdays and sundays. using the arrival data, she estimated the average number of customer arrivals per 10 minute period on saturday to be 6.2. she assumed that arrivals per 10 min interval followed the Poisson distribution shown in the table.
x 0 1 2 3 4 5 6 7 8 9 10 11 12 13
p(x) .002 .013 .- .081 .125 .155 - .142 .110 .076 - .026 .014 .007

a. compute the missing probabilities
b. plot the distribution
c. find mean (mu) and standard deviation (sigma) and plot the intervals (mu) +/- (sigma), (mu) +/- 2(sigma), and (mu) +/- 3(sigma) on your plot of part b.
d. the owner of the bakery claims that more than 75 customers per hour enter the store on saturdays. based on the consultant's data, is this likely? explain.