Question

For a large population, would a sample collection of n=100 be more accurate than four separate samples of n=25 each? Would the mean of the four n=25 samples equal the mean of the n=100 sample? I would think not necessarily, but how would this look mathematically?

Answer 1

Here n=100 and if we divide the sample into 4 equal sub-groups of sizes without replacement method say n1, n2, n3, n4 such that n1=n2=n3=n4=25

and n1+n2+n3+n4=n i.e each of sample size is 25.

here the mean of 100 samples say and mean for other 4 groups say , , and .

therefore the mean of all subgroups mean is always equal to the mean of 100 samples in without replacement method.

but in with replacement method it is not necessarily. Because in with replacement method there is chance to get select 1 unit more than once in the sample.

In without replacement method, we have to prove

and this is always holds in without replacement method.