Consider the following protocol. We want to test metallic bars for brittleness. The bars were molded under similar conditions and tested at 5 places. Under the assumption that bars have uniform composition, the $\#$ of breaks/bar should follow $\text{Bin}(5,p)$, with probability of failure $p$ unknown. If all bars have uniform strength, $p$ should be the across bars (under the previous assumption it is also the same within a bar). If they are of different strength the failure probability should vary.
I have tested the null hypothesis (using given data which are not important for my question -- the data are basically the number of bars out of 280 that exhibited $i$ breaks/bar, where $0 \le i \le 5$) that $p$ is the same for all bars and found that the binomial model is unsuccessful in modeling this protocol. I am trying to find a model that better represents the physical mechanism that generates the data.
Since the Binomial model does not really fit the data well, I am assuming that modeling the process as a Bernoulli process where we have $5\times 280$ trials with equal probability of failure would yield the same results. Poisson is usually used for modeling rare events so my intuition tells me to avoid it here. A normal distribution seems to be unlikely, too, since the number of breaks is not a continuous quantity. But this pretty much exhausts the discrete distributions I am aware of, for modeling the process. Am I missing something fundamental?
