Select sample size of one group to match error of another group

Question

Say that I have two groups or 'classes' that I want to compare, A and B.

For class A I simulate N events and each of these events contain n measurements (on average), that is I have approximately N x n total measurements (a good approximation for large N).

My question:

If for class B I want to have the same standard error obtained for class A, how many events M must I simulate if each event contains m measurements?

Answer 1

Suppose that the first sample is from a population with standard deviation $\sigma_1$ and the second sample is from a population with standard deviation $\sigma_2.$

Population 1: Let $\bar X_n$ be the sample mean of the $n$ observations in an 'event'. Then $Var(\bar X_n) = \sigma_1^2/n_1.$ Averaging $N$ of these event means to obtain the grand mean $\bar X,$ we have $Var(\bar X)$ $= \frac{\sigma_1^2}{nN}.$ Thus the 'standard error' of $\bar X$ is $\frac{\sigma_1}{\sqrt{nN}}.$

Population 2: Similarly, the standar error of the grand mean $\bar Y$ for Population 2 is $\frac{\sigma_2}{\sqrt{mM}}.$

So, given $\sigma_2$ and $m$, you need to find $M$ such that $\frac{\sigma_1}{\sqrt{nN}} = \frac{\sigma_2}{\sqrt{mM}}.$

If the $\sigma_i$ are unknown, you can take preliminary samples from the two populations and estimate $\sigma_i$ by the preliminary sample standard deviations $S_i.$

Answer 2

$E = z\cdot \frac{\sigma}{\sqrt n}$ so you must also take into account the difference in standard deviations