Robust estimation of the mean

Problem:

Suppose we want to estimate the mean µ of a random variable $$X$$ from a sample $$X_1 , \dots , X_N$$ drawn independently from the distribution of $$X$$. We want an $$\varepsilon$$-accurate estimate, i.e. one that falls in the interval $$(\mu − \varepsilon, \mu + \varepsilon)$$.

Show that a sample of size $$N = O( \log (\delta^{−1} )\, \sigma^2 / \varepsilon^2 )$$ is sufficient to compute an $$\varepsilon$$-accurate estimate with probability at least $$1 −\delta$$.

Hint: Use the median of $$O(log(\delta^{−1}))$$ weak estimates.

It is easy to use Chebyshev’s inequality to find a weak estimate of $$N = O( \sigma^2 / \delta \varepsilon^2 )$$.

However, I do not how to find inequality about their median. The wikipedia of median (https://en.wikipedia.org/wiki/Median#The_sample_median) says sample median asymptotically normal but this does not give a bound for specific $$N$$. Any suggestion is welcome.