I have noticed in textbooks that the standard error of the proportion when using the Finite Population Correction Factor is as following:
$\sqrt {\frac {p(1-p)}n \frac {N-n}{n-1}}$
However, when determining the sample size when using the Finite Population Correction Factor, following formula is used:
${\frac {n_0 N}{n_0 + (N - 1)}}$ where $n_0 = {\frac {z^2 p(1-p)}{e^2}}$
How is the above formula derived? Rearranging the standard error equation gives a different result for n as opposed to the sample size formula.
I also encountered this, but I am unable to find any source that explains the derivation. From the construction of the term:
$$FPC = \frac{n_{0}N}{n_{0}+(N+1)}$$
In this webpage I found some mention, but no derivation: https://select-statistics.co.uk/calculators/sample-size-calculator-population-proportion/#:~:text=X%20%3D%20Z%CE%B1%2F22,N%20is%20the%20population%20size.