Approximated results

It is interesting to compare the Monte Carlo results of Fig.  to those obtained by the approximated values of expected value and standard deviation given by Eqs. ()-() just putting $\sigma(\pi_1)=\sigma(\pi_2)=0$. The contribution to the uncertainty due to the two binomials of Fig.  is $\sigma_R(n_P)= 30.6$ (rounded to 31 in Fig. ), while those due to $\sigma(p_s)$ are equal to 0, 24.6 and 25.9, for the three population sizes. The combined standard uncertainties are then 30.6, 39.3 and 40.1, in perfect agreement with the results shown in Fig. .