**Question**

I am developing an internal audit process within our supply chain to determine packaging and Finalizing SOP’s are being followed. I need to determine what will be the sample size needed to accurately represent the population. We are currently shipping out 650k cartons a day. How do I determine how many audits I need a day for statistical significance?

**Answer**

Statistical sampling theory shows that for large populations, the sample size is not a function of the population size, assuming all units in the population have an equal probability of being selected for the sample. To ensure a representative sample, stratified random sampling is employed to represent in the audit sample. This method requires that each category (or stratum) is specified, and that none of them overlap (i.e., items to be audited must fall in only one category). For example, you can break the packaging records in groups of 25,000 (26 stratum for 650,000 records), sampling 1/26^{th} of the sample from each stratum.

To determine the sample size, we employ the binomial distribution where a records is either confirming or nonconforming.

The basic formula for the binomial confidence interval is

For a given sample size (n) with a given number of defects (x), the probability of the sample coming from a population with probability (p) is given by the value alpha (a). The above equation can be solved for probability (p) at a given a level or can be solved for a at a given population probability (p).

In other words, you specify the percent defective in the population you can accept. The only when to ensure 0% defective is 100% sampling. You solve the equation for n by setting 1-alpha (1-a) equal to a high probability (i.e. 95%). If you desire to accept zero (0) defects in the sample then set x equal to zero. In this case, the equation reduces to ln(1- a)/ln(1-p).

Hope this helps with the question.

Thanks

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