Let's say I have two samples of results of two bernoulli experiments.
$H_0: p_1 = p_2$
$H_1: p_1 \neq p_2$
And I want to try to reject $H_0$ at a confidence level.
I already know a proper way to solve this, but I was wondering, if I have a confidence interval for $p_1$ and $p_2$, at the same level of significance. Can I just check if the intervals overlaps each other to test this ?
On
It is true that if the intervals don't overlap, then there is a statistically significant difference. However, the converse is not true. Overlapping intervals do not imply that you cannot reject the null hypothesis.
The best approach to take is the one I'm assuming you've already done: Construct a confidence interval for the difference $p_1 - p_2$ and check to see if it contains the point $0$ or not.
Suppose the confidence intervals $I_j$ have the property that $p \in I_j$ with probability $\ge 1-\alpha$. Then under the null hypothesis, $p \in I_1 \cap I_2$ with probability $\ge (1-\alpha)^2 = 1 - 2 \alpha + \alpha^2$, so this is a lower bound for the probability that $I_1$ and $I_2$ intersect. Thus if they don't intersect, you can reject H_0 with confidence level at most $2 \alpha - \alpha^2$. The true confidence level is presumably better than that, but without more analysis we don't know how much better.