I have $100$ coins and I want to test whether they are heterogeneous in their biases.
$H_O$: each coin have the same $p$ for heads
$H_1$: otherwise (heterogeneous $p$'s)
Each coin was tossed $20$ times and so i have the empirical $p$ for each coin ($p_j$) and the empirical $p$ for all $100\times20$ tosses ($p_{all}$).
I estimated for each coin separately, if it is significantly different than $p_{all}$ (using the pdf of a binomial distribution) and got that $\frac{X}{100}$ coins are significantly different from $p_{all}$.
Now I want to estimate the $p$.value of $\frac{X}{100}$, under the null hypothesis that for all coins: $p_j = p_{all}$. How should I do that?
I first thought of using the chi-square $\chi^2$ variance test, but my $p_j$'s are not normally distributed (and got a $p$.value of exactly zero).