Let $n$ be a natural number, $p$ a real number between 0 and 1, and $X_n$ a random variable following a binomial distribution with parameters $n$ and $p$. We denote $F_n = \frac{X_n}{n}$ and $f$ a value taken by $F_n$. We recall that, for $n$ sufficiently large, the interval $\left[p - \frac{1}{\sqrt{n}} ; p + \frac{1}{\sqrt{n}}\right]$ contains the frequency $f$ with probability at least equal to 0.95. Deduce that the interval $\left[f - \frac{1}{\sqrt{n}} ; f + \frac{1}{\sqrt{n}}\right]$ contains $p$ with probability at least equal to 0.95.
Let $n$ be a natural number, $p$ a real number between 0 and 1, and $X_n$ a random variable following a binomial distribution with parameters $n$ and $p$. We denote $F_n = \frac{X_n}{n}$ and $f$ a value taken by $F_n$. We recall that, for $n$ sufficiently large, the interval $\left[p - \frac{1}{\sqrt{n}} ; p + \frac{1}{\sqrt{n}}\right]$ contains the frequency $f$ with probability at least equal to 0.95.
Deduce that the interval $\left[f - \frac{1}{\sqrt{n}} ; f + \frac{1}{\sqrt{n}}\right]$ contains $p$ with probability at least equal to 0.95.