How to test if the proportion of subjects being significant - is significantly different than expected by chance?

Question

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).

Answer 1

If all $p$'s are the same, then you have one large sample (size $100 \times 20$) from the same binomial distribution, under the alternative hypothesis, not. So you could estimate the variance and see if that is consistent with a binomial variance of $n p (1-p)$ or is different. You could use a goodness of fit test for the binomial. Or, you could just use logistic regression and see if the model with a 100-levels factor fits better than the null model ...