I want to compute the average of a very long sequence $X=\{x_0,x_1,....\}$, where $x_i \in [0,a]$, with $a \in \mathbb{N}_1$. For simplicity let's assume $x_i \in [0,1]$.
Since values are constantly arriving, I'm trying to use Hoeffding's inequality to compute the minimum number of samples $n$ needed to build a confidence interval $\mu_X \pm \epsilon$. The following formula in [1] relates $n$ with $\epsilon$, such that: $$n \geq \frac{log(\frac{2}{\alpha})}{2\epsilon^2}$$
I know $n$, and the confidence level $\alpha$, so I can compute $\epsilon$. However, how I can determine that current $\epsilon$ is good or bad?
thanks!
[1] https://en.wikipedia.org/wiki/Hoeffding%27s_inequality#Confidence_intervals