I have a statistics exercise, with a stochastic variable $X$ that has a $\text{Binomial}(64,p)$ distribution. P is unknown, and the goal is to find it.
In one of the sub-exercises, I have to approach the following:
$$P\left(\left|\frac{x}{64} - p\right| \le 0.1\right)$$
However, I simply have no idea where to even start. Can anyone attempt to explain how to approach a problem like this?
Lets approximate the $\dfrac{\text{Binomial}(64,p)}{64}$ with the normal distribution $N\bigl(p,\dfrac{p(1-p)}{64}\bigl)$, as the $\text{Binomial}(64,p)$ is just a repetition of identical expetiments. Then $\sqrt{\dfrac{64}{p(1-p)}}\bigl(\dfrac{\text{Binomial}(64,p)}{64}-p\bigr)$ is approximated as $N(0,1)$. And we come to
$P\bigl(\bigl|\frac{X}{64}-p\bigr|<0.1\bigr) = P\bigl( \sqrt{\frac{64}{p(1-p)}}\bigl|\frac{X}{64}-p\bigr|<\sqrt{\frac{64}{100p(1-p)}}\bigr) = Ф\bigl(\sqrt{\frac{64}{100p(1-p)}}\bigl)-Ф\bigl(-\sqrt{\frac{64}{100p(1-p)}}\bigl).$