One sided hypothesis testing with two-sided interval

Question

So, I have a population which is believed to have $H_0:\mu = 4620$ and I should test for $H_1: \mu < 4620$. Given a sample of $n=81, \bar{X}=4450, S=900$, the p-value is: $pv=\Pr\left(\frac{\bar{X}-4620}{900/\sqrt{81}}\leq\frac{4450-4620}{900/\sqrt{81}}\right)\\=\Pr\left(Z\leq-\frac{170}{100}\right)\\\approx0.0445<\alpha=0.05$

So I reject the null hypothesis.

Then, I'm being asked to find a confidence interval for $\alpha=0.1$, which is: $CI_{0.9}\left(\mu\right)=\left[\bar{X}\pm\frac{S}{\sqrt{n}}\cdot t_{80,0.95}\right]=\left[4283.588,4616.412\right]$ and to infer from it, again, that the null should be rejected with $\alpha=0.05$.

Now, clearly $\mu_{H_{0}}>\sup CI_{0.9}$ so we can reject with $\alpha=0.1$. Furthermore, if I take the one-sided confidence interval with $\alpha =0.05$ it has the same upper bound as the two-sided interval. It seems like all we have done is to expand the non-rejection region to the left tail of the two-sided interval.

But what I'm having trouble with is how to formulate the argument - why does it make sense? Is it enough to state that the one-sided interval overlaps the two-sided one? That the values in the left tail, for the two-sided case, "don't matter"?

Why is this transformation of the two-sided interval into a suitable one-sided interval valid, in this case? Is there a general method/explaination of why these intervals are equivelant, if at all?

Edit: Found this answer Matching Confidence limits with One-Sided Hypothesis tests which explains it the other way around - why the two-sided interval should have been constructed with $\alpha=0.9$. But even after drawing the bell curve and coloring the expanded acceptance region, I can't understand the logic behind "merging" the left rejection area to the interval. It fits the desired $\alpha$, that's fine, but why would anyone test a one-sided hypothesis using a two-sided interval in the first place?

Answer 1

The thing which is confusing you is where to put the lower bound. In the two-sided 90% interval you give the lower bound is finite (4283.588). If you want a 95% one-sided interval it has the same upper bound (as you say) but the upper bound is infinite. So it is from $-\infty$ to 4616.412.

Answer 2

If you look at how you get to the confidence interval, there is nothing to do with the null or alternative. All you have is the sample statistics and the assumption that your sample follows a normal or t distribution.

You can use it as a tool to test different null hypothesis. Depends on whether you are doing a one-sided or two-sided test, you can construct one-sided or two-sided CI to tackle it. Given your null, it seems 95% 1-sided and 90% 2-sided yield the same upper bound and thus reject both. But if you are tasked for a null lower than 4283, the results will be different.

It's like having the same results doesn't guarantee they are the same thing.