Let $_1,…,_$ be i.i.d. Bernoulli random variables with some unknown parameter $∈(0,1)$. Then which of the following is/are valid confidence interval(s) for $$ with nonasymptotic confidence level 95%?
$(0,1)$
$[0,\bar{X_ n}+\frac{.89}{n}]$
$[\bar{X_ n}-1.96\sqrt{\frac{p(1-p)}{n}},\bar{X_ n}+1.96\sqrt{\frac{p(1-p)}{n}}]$
I'm almost certain that the last one is correct since it's based on the central limit theorem (CLT) and is obtained after rearraning. Actually, the last one appears here but apparently it's not the only correct one and I'm not sure if
$$[0,\bar{X_ n}+\frac{.89}{n}]$$
Is also correct?
It doesn't really make sense to me so if someone could explain I would really appreciate it.
Another stackexchange post points out that, "if $̂ =0$ and $>30$, the 95% confidence interval is approximately $[0,3/]$ (Javanovic and Levy, 1997); the opposite holds for $̂ =1$". This article costs money which I cannot afford but it made me think whether there is another approach out there to confirm the second option is also valid. Or is it none of them?
I'm a little confused but got the spirit so cheers to anyone who is willing to clear my doubts.