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Say I have a sample of N measurements, n of them have characteristic C.

What are the density and distribution functions of C's presence in the population.

For example, out of N = 100 people randomly sampled in the city I spot n= 20 blond ones, what's the chance p that blonds are more than x = 25% of the city's population ?

Once I get it, how can I find the narrowest 90% CI around the mode ?

I may have low n so I can't do any normal approximation.

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    Maybe check out [these](https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval) to bound the proportion parameter and [this question](http://math.stackexchange.com/questions/117926/finding-mode-in-binomial-distribution?noredirect=1&lq=1) concerning estimating the mode, specifically. – 2017-02-03

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