Question

4-22) A process is known to produce 5% nonconforming items. A sample of 40 items is selected from the process. (a) What is the distribution of the nonconforming items in the sample? (b) Find the probability of obtaining no more than 3 nonconforming items in the (o (d) Compare the answers to parts (b) and (c). What are your observations? sample. Using the Poisson distribution as an approximation to the binomial, calculate the probability of the event in part (b).

please help ...

Answer 1

a) Let X = number of nonconforming items produced in a particular process

p = 0.05 = probability of nonconforming items produced in a process

n = sample size = 40

So X follows binomial distribution with parameters n = 40 and p = 0.05

b) Here we want to find P( X <= 3)

Let's use excel:

P( X <= 3) = "=BINOMDIST(3,40,0.05,1)" = 0.8619

c) Let $\lambda$ = n*p = 40*0.05 = 2

Therefore X follows approximately Poisson distribution with parameter   $\lambda$ = 2

Let's use excel to find P( X <= 3)

P( X <= 3) = "=POISSON(3,2,1)" = 0.8571

d) From part a) and b) the difference is = 0.8619 - 0.8571 = 0.0047 which is very small.

Because for large n and very small p ; we can use Poisson approximation to the Binomial.

And here n = 40 , p = 0.05

that is both the conditions are satisfied. So we get good approximated probability when we use Poisson distribution.