# Sample size with finite population and max error rate

I have a problem I need to solve.

I have on average about 5,000 new documents that are created every month, and I need to determine what would be my optimal sample size so that I don't need to check all of them.

However, there is a requirement that no more than 1.25% of all document can have mistakes in them.

Meaning that in this case if I were to inspect all 5,000 documents there should not be more than 63 (1.25%*5,000=63) that had mistakes in them.

Hopefully, someone can walk me through the process of determining the optimal sample size for my problem. And how to transform the data back to the full population measures.

P.S. I am familiar with the formula for finding sample size with a finite population, but my upper limit of error rate is what I am not able to figure out.

Thanks

Edit:

Additional information. As part of the agreement, if the error rate is greater than 2.25% then we need to check 100% of the documents. So I guess we never want to get to the 2.25% threshold.

Sorry for not being clear in my earlier explanation.

• What specifically do you want to know? What the true error rate is? With what degree of precision?
– mkt
Jul 18 '17 at 16:59
• I am interested in finding out what would be sufficient Sample size of random document I need to check in order to determine if my error rate is below 1.25%. Jul 18 '17 at 17:02
• The answer is that you need to check $100-1.25\%$ of the documents every month. However, if you are willing to take a sample of the documents, estimate the total number with mistakes, and to have some chance that the estimate will be incorrect, then much cheaper solutions are available. The missing pieces of information concern how much risk you can afford (in not detecting a situation where too many documents are in error) and how you wish to quantify that risk, given this is an ongoing procedure: it's not a single study, and therefore standard sample size calculations do not apply.
– whuber
Jul 18 '17 at 17:05
• @whuber Can you elaborate a bit further 1) how you came to 100 document to check 2) it's not a single study,... standard sample size calculation do not apply Jul 18 '17 at 17:10
• Not 100; I wrote $100-1.25\% = 98.75\%$ of all of them. That would be $4,938$. It's not a single study due to what you say: you make this decision "every month." It's an ongoing procedure.
– whuber
Jul 18 '17 at 18:01

• As I argue in comments to the question, this is not a "typical sampling plan problem," because it's not a one-off question: it concerns an ongoing sample. The usual solutions will be inadequate because they do not account for the rate of errors over time. Even a single sample of size $200$ would usually not be considered adequate, because a random sample (without replacement) of that size will fail to detect any one of $63=1.25\%$ of $5000$ defective items with a chance of more than $7.5\%$. Unfortunately, the OP has not yet indicated what error risk they can tolerate.
• Well, as I reported in the first comment, it will not meet the spec. It risks overlooking more than $1.25\%$ defective items and this risk recurs each month. A more appropriate framework would be that of quality control rather than a one-off sample design. It's impossible to be much more specific because the OP has not disclosed their risks: what would be the cost of failing to detect more than $1.25\%$ defectives? What are the costs of testing the samples? Because those have to be considered in any adequate solution, ISO 2859 cannot be blindly applied.