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We modify our sample size based on lot size only when the LOT size is less than ~about 50. This choice is rather arbitrary as it will depend on how accurate we wnat our estimate to be.
When lot sizes get 'small', attribute (categorical, pass/fail) sampling results will follow a hypergeometric distribution and not the general approximation of the Binomial which applies to 'large' lot sizes (theoretically infinite, in practice greater than ~50). The other rule of thumb for adjusting yoru sampel size for lot size is when the sample size is ~ 5% of the lot size or greater...I rarely use this approach as I've found in practice that it really make no practical difference but I do use the lot size <50 rule...
Technically speaking the use of the hypergeometric applies when we sample without replacement from a finite lot. This is because the probability of finding something in a lot changes as the lot size decreases when we pull a sample and don't replace it. (this is the probability that we all despised in high scholl or college: you are getting dressed in the dark and you 20 pairs of socks randomly scattered in your sock drawer. 10 pairs are black, 5 pairs are blue, 4 pairs are brown and 1 pair is pink. what are the odds that you pick at least 1 pink sock? my answer was always: none becuase I am going to turn the lights on)
the Binomial applies perfectly when we are sampling from an infinite population: if you flip a coin...this also works when you sample with replacement.
Neither of these situations fit the practical industrial world. Fortunately in practice once the lot gets over~50 the differences in teh resutlsbetweent he two distribtuions is so small as to have no practical meaning; the differences will have no effect on the ultimate decision we make.
The hypergeometric is complicated to calculate so the 'rule of thumb' was established years ago and is mostly accepted without the mathematical curves since they're a bit of work to generate. Not that I'm against the propagation of knowledge I just would rather re-derive other more impactful rules than this one...perhaps one of our other statisticians has this derivation handy for posting?
The formula used to modify the sample size (calculated the 'standard way' without regard to lot size) is generally taken to be: N/(N+n) where N = lot size and n = sample size.
When lot sizes get 'small', attribute (categorical, pass/fail) sampling results will follow a hypergeometric distribution and not the general approximation of the Binomial which applies to 'large' lot sizes (theoretically infinite, in practice greater than ~50). The other rule of thumb for adjusting yoru sampel size for lot size is when the sample size is ~ 5% of the lot size or greater...I rarely use this approach as I've found in practice that it really make no practical difference but I do use the lot size <50 rule...
Technically speaking the use of the hypergeometric applies when we sample without replacement from a finite lot. This is because the probability of finding something in a lot changes as the lot size decreases when we pull a sample and don't replace it. (this is the probability that we all despised in high scholl or college: you are getting dressed in the dark and you 20 pairs of socks randomly scattered in your sock drawer. 10 pairs are black, 5 pairs are blue, 4 pairs are brown and 1 pair is pink. what are the odds that you pick at least 1 pink sock? my answer was always: none becuase I am going to turn the lights on)
the Binomial applies perfectly when we are sampling from an infinite population: if you flip a coin...this also works when you sample with replacement.
Neither of these situations fit the practical industrial world. Fortunately in practice once the lot gets over~50 the differences in teh resutlsbetweent he two distribtuions is so small as to have no practical meaning; the differences will have no effect on the ultimate decision we make.
The hypergeometric is complicated to calculate so the 'rule of thumb' was established years ago and is mostly accepted without the mathematical curves since they're a bit of work to generate. Not that I'm against the propagation of knowledge I just would rather re-derive other more impactful rules than this one...perhaps one of our other statisticians has this derivation handy for posting?
The formula used to modify the sample size (calculated the 'standard way' without regard to lot size) is generally taken to be: N/(N+n) where N = lot size and n = sample size.