Lower bound on tail of sum of independent variables.

Question

Let $n=2m$ be even and $X_1,\ldots, X_n$ Bernoulli independent random variables with $p=1/2$. Let $X=\sum_{i}X_i$. I wish to show that for any integer $t\in[0,n/8]$ $$ \operatorname{P}[X\geq n/2+t]\geq \frac{1}{15}e^{-16\frac{t^2}{n}} $$ I tried rewriting $$ \operatorname{P}[X\geq n/2+t]=\operatorname{P}[X\geq m+t] = \sum_{j=t}^m{2m\choose m+j}2^{-2m} $$ but I have no idea how to proceed, also, where should I use the fact that $t\leq n/8$. I tried looking for some lower bounds on binomial coefficients, but no luck. Could someone hint me what to do ?