Consider the following problem (which has a real world
problem behind it)
You have 100 coins, each of which has a different
probability of heads (assume that you know that
probability or worse can estimate it).
Each coin is labeled. You ask one person (or machine
if you will) to flip each coin a different number of times,
and you record the number of heads.
Assume that the (known/estimated) probability of heads
is between .01 and .20, and the number of flips for
each coin is between 4 and 40.
The question is how to test that the person/machine
doing the flipping is flipping 'randomly/fairly'. That is,
the person/machine might not flip 'randomly/fairly/...'
and you want to test that hypothesis.
One can easily state the null hypothesis as
p_hat_i = p_know_i for i=1 to 100
where p_hat_i is the observed # heads / # flips for each i.
Since each coin has a different probability of heads,
you can not directly aggregate.
Since the expected number of heads is low, asymptotics for
chi-squares will not apply (each coin has substantial
probability of obtaining 0 heads so empirically you obtain
lots of cells with 0,1,2 etc.).
Given that, I have failed to come up with a statistic to test it.
TIA for any pointers to help.
Bob
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