Single sample versus multiple sample I have a jar with white and black balls. Total number of balls in the jar is 100000. I want to estimate the proportion of white balls. My constraint is that the sample size for estimation should be low, lets assume 500 balls. I am debating between two approaches.


*

*Draw a single sample of 500 balls, $\hat{p}$ = number of white balls
divided by 500

*Draw 10 samples of 50 balls each. Calculate the proportion of white balls in each sample, i. e., $[r_1, r_2, ..., r_{10}]$. Estimated $\hat{p}$ = average of $[r_1, r_2, ..., r_{10}]$.


Which method should I use so that I am less susceptible to sampling error?
 A: If there is no replacement of balls after drawing, then approaches 1 and 2 are equivalent. With approach 2, you can find the average of the proportions of white balls in each of the 10 samples, or find the total number of white balls in the 10 samples combined and express this as a proportion of 500.  Both calculations will give the same result (given that, as is the case here, the samples are all of the same size).
If however for approach 2 the balls are replaced after each sample of 50, then the sampling error will be slightly higher than with approach 1.  One way to see this is to consider the extreme case in which rather than 100000 there are only 500 balls in the jar. In that case approach 1, which would then be a 100% sample, would be guaranteed to estimate the true proportion correctly.  But approach 2 would still be subject to sampling error because each sample of 50 would be only a 10% sample.  With a much larger number of balls this effect is still present, albeit greatly diminished.
