Required sample size to reduce the interval size by half I have a small doubt regarding a statistics problem and would like a confirmation.
If we want to reduce the length by half of the confidence interval (at $95\%$), we should:
Multiply the size of the sample by $2$ ? $4$ ? or by $8$ ?
As for me, I think that the sample size should be multiplied by $4$.
Why do I think so?
We are looking for $\frac{1}{4}E$, $E$ being one side of the interval:
$$\frac{1}{4}E= 1.96 * \frac{\sigma}{\sqrt{n}} $$
So,
$4n=(1.96)^2 * \frac{\sigma^2}{E^2}  $
So it should be multiplied by $4$, as said before.
Am I right ?
Thanks in advance.
Regards

Because this is a homework tag, I won't give you the explicit confirmation/explanation, but a simple way to check yourself is to simulate this with known $\sigma$ and known confidence intervals.
Simulate using a variety of sizes for $n$ and see if you can deduce the relationship explicitly. The main things to think about are
1- How the value of $\sqrt{n}$ changes as $n$ changes
2- How the symmetry of the confidence interval effects the answer, i.e, accounting for the $+/-$
