# Determination of sample size for a proportion

SCENARIO

The parent company has other business units (think of these as franchises owned by the parent company). In order to quality assure procedures the parent company samples employees’ work at other business units. The employee’s work is then checked and if a breach of procedure is discovered the franchise receives a penalty. The parent company doesn’t just want to sample employees in one department of a business unit as there might be no breaches in that department and in another department there might be a very high level of breaches. So the idea is to sample departments from each business unit and within each sampled department sample the employees. Some units have a very large number of departments, some not so much departments; other units have small numbers of employees and departments – size of business units vary. So the question is. what is the minimum sample size that can be obtained in order to be confident that if the sample shows no breach of procedure then a breach existing in the whole company is very unlikely?

THE SOLUTION?

Consider the small sample of five employees work from a single company.

The type of data that would be collected would look as follows where a zero means the employee’s work has been inspected and there is no breach in procedure (and a 1 in the column headed breach would mean the employee breached the procedures).

Employee ID Breach 101 0 102 0 103 0 104 0 105 0

If the employee breached the procedure then the company receives a penalty and we are successful in detecting the breach (not bothered about this scenario). However, as in the above table if no breach was detected we still are not sure that a breach does not exist in the company (because we only sampled five people). From the structure of the data you can see that the statistic of interest is the proportion of employees with a breach. So what I would like to be able to say is that in those instances where the sample size shows no breach I am also 95% confident that there is no breach in that company.

I have looked at the Wikipedia article that talks about determining the sample size for a proportion https://en.wikipedia.org/wiki/Sample_size_determination#Proportions

This formula reduces to 1 divided by the square of the margin of error. So if I wanted to estimate my sample size of employees with a margin of error of 0.03 at a 95% confidence interval I would need around 1000 employees sampled. My problem? Some companies do not have 1000 employees (some might have 250 employees for example so the formula is in effect telling me to sample the whole company).

Also I need to work out the minimum number of departments to sample and again using this formula it tells me I need to sample roughly 1000 departments. Obviously some companies will not have 1000 departments. The formula is independent of population size so how can I modify my calculations or otherwise determine the sample size so I don’t end up with silly results like ‘sample the whole company’.

Essentially it works out to multiplying the estimated standard error by $$\sqrt{\frac{N-n}{N-1}}$$ where $$N$$ is the size of the population and an $$n$$ is the size of the sample.