Sample size has to be less or equal to 5 % of population, np(1-p) is bigger or equal to 10
The bigger the sample size the smaller standard deviation. You cut the standard deviation of p^ in half by increasing the sample size by a factor of 4.
Normal, 0.4, 0.022
Normal, 0.7, 0.026
Normal, 0.35, 0.015
0.0039
0.0228
Normal, 0.42, 0.0129
0.0102
0.0606
Normal, 0.39, 0.022
0.3228
0.3198
0.0838 it depends how low a probability you find unusual