Binomial - probability from average result?

I'm looking at recovery rates for stolen cars. Across the whole State there were 15,000 cars stolen, of which 62% were recovered.

I now want to see if any particular sub-regions are significantly different to the State. Can I use 0.62 as the binomial probability for a car being recovered?

I've calculated the rates per sub-region, then used Excel i.e. BINOMDIST(recovered, stolen, 0.62, TRUE), but it results in around half of them being significant (<0.05 or >0.95). This doesn't seem right...

Appreciate any help with this!
Hi Dason, I guess it just doesn't seem right because half the areas are significantly high/low, seems too many?

Would I be better off looking at the recovery rates per sub-region, and seeing if these results follow a normal distribution. Then checking for any significantly high/low values this way?


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I guess I just don't see any reason to believe that sub-regions should have the same probabilities of recovery as the state-wide average. It sounds like a beta-binomial model might be appropriate to model the recovery rates for the different sub-regions.
Yeah that was my original question - can I say the state average is the probability in each sub-region. I now realise I can't.

Thanks for the info on beta-binomial, it looks interesting. However a lecturer suggested I use a two-sample proportion test, and I think this has worked well.