Proving a bound with binomial coefficients

Question

I'm trying to prove the inequality below:

$$ \frac{\sum^{n/2 + \sqrt{n}}_{j=0} {n \choose j}}{2^n} \geq 0.95 $$

I have no idea where to start. I have tried to fill in the formula for small values of n and I see that it holds but I'm unable to proof this. Can anyone give a hint on how to prove this?

Any help would be greatly appreciated.

Answer 1

This expression can be interpreted as the probability that the number of heads obtained in $n$ tosses of a fair coin is less than or equal to $\mu+2\sigma$, where $\mu=\frac{n}{2}$ and $\sigma=\frac12 \sqrt{n}$ are the expected value and the standard deviation of the number of heads.