dotzis16xd

2022-03-24

Let's say I have two samples of results of two bernoulli experiments.

$H}_{0}:{p}_{1}={p}_{2$

$H}_{1}:{p}_{1}\ne {p}_{2$

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?

And I want to try to reject

I already know a proper way to solve this, but I was wondering, if I have a confidence interval for

Korbin Ochoa

Beginner2022-03-25Added 11 answers

Step 1

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 $a\alpha -{\alpha}^{2}$ .

The true confidence level is presumably better than that, but without more analysis we don't know how much better.

Suppose the confidence intervals

Then under the null hypothesis,

Thus if they don't intersect, you can reject

The true confidence level is presumably better than that, but without more analysis we don't know how much better.

Abdullah Avery

Beginner2022-03-26Added 19 answers

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.

The best approach to take is the one I'm assuming you've already done: Construct a confidence interval for the difference