Question on when to accept the null hypothesis?

Question

So to accept the null hypothesis $H_0: \theta = 0$, I know that the confidence interval has to contain 0 for any confidence level $1-\alpha$ but I am looking for a way to explain it more rigorously? Thanks!

Answer 1

Strictly speaking you should say not reject the null hypothesis, rather than accept the null hypothesis.

I suspect your definition of the confidence interval is something like saying that, on the assumption that the null hypothesis $H_0: \theta = 0$ is true, the probability that the confidence interval for $\theta$ contains $0$ is $1-a$.

So the probability of erroneously rejecting the null hypothesis $H_0: \theta = 0$ when it is true, on the grounds the confidence interval does not contain $0$, is $\alpha$.