Acceptance Sampling Items are sold as a group (lot) by a producer to a consumer Each lot has a certain fraction p of defective items Consumer will only accept lot if p is small How should consumer decide? A: Inspect a sample from lot Not currently popular QC method

Simple sampling plan Choose a sample size n and critical value c If sample has >c defectives, reject sample P(A) = P( X  c) depends on c,n,p,N Works if n <<N, ( rule of thumb n < 0.5 N) Does not hold if n is large relative to N

Sampling without replacement Lot has N items, Np defective, N(1-p) good P(1 st item is defective) = Np/N = p P(2 nd item is defective) depends on 1 st item Not i.i.d. trials, binomial model does not hold P( X  c) can be calculated using combinatorics

Hypergeometric Distribution An urn contains N balls, M red, rest blue You select n balls as SRSWOR What is the chance k are red? Hypergeometric used in estimating wild-life populations by capture-recapture method

Acceptance probability An acceptance plan is a choice of n and c P(A) cannot be calculated in practice How do we choose a sampling plan? A: look at an OC curve

OC Curve Is this a good OC curve? What does an ideal OC curve look like ?

Optimising the OC curve Decide how close to ideal you want to be Consult a table of sampling plans for n and c More sophisticated sampling plans exist 1 P(A) 1 AQL

Modelling rare events # of misprints in a page of a book # of radioactive emissions from a block of uranium in a minute Binomial expt. n large, p small If np , then This limit is known as the Poisson distribution

Poisson distribution =3  is known as the rate parameter Poisson distr. also used to model events over time X(t) = # events upto time t X(t) is a stochastic(random) process

Poisson process Poisson process (rate = ) P(N(t)=k) = ( t) k e - t / k!

Telephone traffic Traffic at exchange at a rate of 500/hour What is chance of no calls is 2 mins? Avg no. of calls in 10 min