Suppose that $15$ components are sampled $($n $=15)$ from a very large lot of components that have just been received. Assume that management decides to accept the lot only if no more than two nonconforming items are found in the sample; that is$,$ c $=2.$ Although we do not have the actual percentage of nonconforming components in the lot, let us assume that this value is $5\%.$ What is the probability that the lot will be accepted using this particular sampling plan? What happens if c $=4?$

b$).$ Use the information above and develop the OC curve for the sampling plans $($n$=15,$ c$=2)$ and $($n$=15,$ c$=4)$ in the same chart. If AQL is $3\%$ and LTPD is $15\%,$ what will be the producer and consumer risks for the two sampling plans $($n$=15,$ c$=2)$ and $($n$=15,$ c$=4)$ to be appropriate?

Tom Jones, the mechanic at Golden Muffler Shop, is able to install new mufflers at an average rate of $5$ per hour $($or about $1$ every $12$ minute$),$ according to a negative exponential distribution. Customers seeking this service arrive at the shop on the average of $3$ per hour, following a Poisson distribution. They are served on a first$-$in$,$ first$-$out basis and come from a very large $($almost infinite$)$ population of possible buyers. Calculate the operating characteristics of Golden Muffler$$s queuing system. Make sure to calculate all the operating characteristics.

b$.$ Develop a probability table to calculate f$($x$),$ F$($x$),$ for x $=0,1,2,...,10.$