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We have 2 coins that we don't know whether they are biased or not. M observers, each flip each coin N times and get fraction of heads to total flips of $p_m^1$ and $p_m^2$. Therefore, for each coin i we have:

\begin{eqnarray*} P_i = \frac{\sum_{m = 1}^{M}p_m^i}{M} \end{eqnarray*}

However, each observer has biased fingers (in other words, even if the coins are fair they would get head more than tails, for example).

How can we test whether $P_1$ and $P_2$ are statistically different from each other? And what is the SEM for each coin?

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