The SampSizHPGM.lng Model


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We want to extimate what fraction of items in a 
  certain population are Red.
How big a sample size do we need as a function of:
   M = population size,
   F = actual, but unknown to us, fraction of the population that is Red,
   N = our sample size,
   R = resulting number (a random variable) of items in our sample that are Red,
   P = R/N = our estimate of the fraction items that are Red in the population,
   T = Target standard deviation for P that we specify.
Then, based on the Hypergeometric distribution:
  Var( R) = N*F*M*(M-F*M)*(M-N)/(M*M*(M-1))
                   = N*F*(1-F)*(M-N)/(M-1)
  Var(P) =  Var( R)/( N*N) = F*(1-F)*(M-N)/(N*(M-1))
If we want our estimator to have some specified target standard deviation T or less,
( implies variance of T*T) this means we want to solve:
  T*T = F*(1-F)*(M-N)/(N*(M-1)), or if we multiply through by N:
  N*T*T= F*(1-F)*(M-N)/(M-1), or
  N*(T*T + F*(1-F)/(M-1)) = F*(1-F)*M/(M-1), or
  N  = (F*(1-F)*M/(M-1))/ (T*T + F*(1-F)/(M-1)),
As M goes to infinity, notice that 
   N approaches (from below): (F*(1-F))/(T*T);


Sampling | Hypergeometric distribution | ChartPCurve |