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I am looking to conduct an epidemiological study on a population of 1000 individuals and need to find the appropriate sample size. So far, I have the parameters to calculate my sample size but I am a bit unsure about whether I am using the correct formula(s). The situation is as follows:

population: 1000
test sensitivity: ~100%
test specifity: 99,81%
Z = 1,96 (two-tailed 95%)
disease prevalence: between 0,4% and 8,4%

To calculate my sample size, I used (from [1]): $$n=\frac{N\cdot Z^{2}\cdot P(1-P)}{d^{2}(N-1)+Z^{2}\cdot P(1-P)}$$

With N = population = 1000
n = sample size
Z = 1.96
P = expected prevalence
and d = precision = P/5, which I got from [2]

If I plug in the numbers for 8.4% prevalence, I get a sample size of 506, but we suspect that the prevalence will actually be at around 0.4%, which gives me a sample size of 960 individuals. Now, my question would be: am I doing this right? Am I using the right formulas? Is my thinking for the sample size calculation correct? Thank you all for your help!

[1] NAING, L., WINN, T. & RUSLI, B. 2006. Practical issues in calculating the sample size for prevalence studies. Archives of orofacial Sciences, 1, 9-14.

[2] POURHOSEINGHOLI, M. A., VAHEDI, M. & RAHIMZADEH, M. 2013. Sample size calculation in medical studies. Gastroenterology and hepatology from bed to bench, 6, 14-17.

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IMHO, the reason you are getting such a large sample size (compared to the population), is that you want a very tight precision. You set d (which I translate as a form of statistical error ) as 1/5 of the actual value you are expecting. When P is 0.4% that means 0.08% statistical error which is quite low.

If you indeed need such a high precision, then you are probably better of sampling the entire population. However, since 1000 is unusually round number for population, I am highly sceptical regarding your application.

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