# confidence interval and rejection

Assume I have a data sample $$X_{1}, \dots X_{n}$$ from a normal distribution. Then, we consider the following hypothesis $$H_0: \quad \mu = \mu_{0}$$ and alternative $$H_1: \quad \mu \neq \mu_{0}$$ Assume, based on data and using $$t$$-test we reject the null. Then, we can obviously, report a p-value. The question, does it make any sense to provide the the confidence interval, given that we rejected the null based on that confidence interval?

While you are correct to recognize the relationship between hypothesis testing and confidence intervals, the confidence interval gives a range of "plausible" values. Sure, the exact interpretation is more nuanced than that of a credible interval, but at least there is a range of "plausible" values. This is particularly useful when the data set is large, leading to a tiny p-value yet a confidence interval that doesn't include zero but doesn't get far from it -- say $$(0.000003, 0.000005)$$. If you need to hit a value like $$0.1$$ for the work to be interesting, this allows you to qualify the statistical significance with added commentary about practical significance.