Context: We have $X_i=1$ with probability $p$, $X_i=0$ with probability $1-p$, but $p$ is unknown. Given that the sample mean $Y_N=\frac{1}{N}\sum_{k=1}^N X_k$ is equal to $q$ after N observations, what is the probability that $q$ is not too far from $p$; say $q\in [p-0.05,p+0.05]$?
Is such a question answerable? I was thinking we could look at the probability that the sample mean is equal to $q$ given various values of $p$ and then see how probable it is that $q$ is in the desired interval, but we don't have any idea about the distribution of $p$ so we can't weight in this way.
Input greatly appreciated!
Expanding on my comment: by applying Chebyshev's inequality for every $\epsilon>0$ you get
\begin{equation*} \mathbb{P}(|p-q|>\epsilon) < \dfrac{pq}{N \epsilon^2} \end{equation*}