Theoretically, I was having a debate with my friend about confidence intervals and I got confused.
So, my understanding is the whole confidence interval is essentially the same as a two-tailed hypothesis test.
So, a 95% confidence interval is essentially the same as a two-tailed hypothesis test with significance level 10%.
So for example, if you have
H0: μ = x
and H1: μ ≠ x
with significance level 10%
You could turn this into a z-statistic and do a hypothesis test, or you could set up a 90% confidence interval.
The problem for me, theoretically comes in with one-tailed.
So for example, if you have H0: μ < x, and you see that it lies in the confidence interval, but above the sample mean. Do you reject H0? Do you split the hypothesis test essentially down the sample mean and use x̄ > the lower half of the interval when it's a one-tailed test, and do the opposite if you want to see if the mean has increased?