# Samples size with sample proportion close to 0 or 1

For a future monitoring program on small water bodies we want to calculate the sample size. The bodies of water are so small that their number easily exceeds 100.000 in the monitoring area and therefore we assume the sample population to be infinite. With the future monitoring data we want to be able to assess whether 99% of the water bodies are unaffected by pollutants. From former monitoring programs we know that 99% are unaffected.

My approach calculating the sample size:

P = 0.01 # sample proportion
e = 0.01 # margin of error
level = 0.95 # confidence level
N = Inf # infinite population
q = qnorm((1 + level)/2)

n = P * (1 - P) / (e^2/q^2 + P * (1 - P)/N)
n
 380.3044


Using this formula I get 380 samples. However I read several times that for a sample proportion close to 0 or 1 using the approach of the Normal Approximation is not valid.

1) What could I use instead?

2) changing the margin of error only a little bit has considerable influence on the number of samples. E.g. for e = 0.02 I already get an n of 95.07. Is this the result of using a close to 0 sample proportion?

• You can use a binomial model! – kjetil b halvorsen May 3 '17 at 15:57
• Search this site for "binomial sample size" there are lots of posts some should be helpful – kjetil b halvorsen May 3 '17 at 16:03
• This actually is a reasonably accurate estimate: the Rule of Three, which is based on a Binomial model, says you need a sample size of 3/(100%-99%) = 300. This will be more accurate than the Normal approximation and it requires no preliminary estimate of a margin of error. – whuber May 3 '17 at 16:22
• @whuber thanks for your answer! Concerning question 2: It puzzles me that changing the margin of error by 1% the sample size already changes so dramatically. Is this the effect of P being close to 0? – andschar May 3 '17 at 16:28
• The sample size will be inversely proportional to the square of the MoE. This phenomenon shows that you are missing some crucial elements in the formulation of your question. The most important are these: if the proportion of unaffected waterbodies is 99%, (1) it's unlikely the proportion in a sample will be exactly 99% and (2) you will probably want to construct a lower confidence limit of the unaffected proportion (called a "tolerance limit," or LTL) based on the sample. How small could you allow the LTL to be and what level of confidence do you need in it? – whuber May 3 '17 at 16:55