Confused about hypothesis testing, tails, and confidence intervals

Question

$Y_1, \ldots, Y_{10}$ are $N(\mu, \sigma)$ random variables. We observe $\overline{Y} = 20$ and a standard deviation of $1.265$. We are asked for a $95$% confidence interval. Use the t-table, we find $t(.975; 9) = 2.262$, so a $95$% confidence interval for $\mu$ is $20 \pm 2.262 \times 1.265$, giving a confidence interval of $$17.1 \le \mu \le 22.9.$$

Question 1

I would like to relate this to hypothesis testing. Am I correct in thinking that any test of the form $$H_0: \mu = c$$ $$H_a: \mu \ne c$$ (with $\alpha=.05$) will lead to us rejecting $H_0$ if and only if $c \notin [17.1 , 22.9]$?

I am even more confused about one-tail tests:

Question 2

Suppose a new hypothesis test was: $$H_0: \mu \le 18$$ $$H_a: \mu > 18$$ (with $\alpha=.05$). How does this relate to the confidence interval? Do we need to create a new interval to correspond to this test?

Question 3

What kind of one-tail tests are allowed? Do $H_0$ and $H_a$ need to cover all possibilities? Is a test of the form $$H_0: \mu \le 18$$ $$H_a: \mu > 21$$ legal?

Answer 1

1) In this particular example, yes. In some rare cases there can be exceptions.

2) The only difference is that if you do an one-sided 5% level test, you would use a two-sided 90% confidence interval and look at whether the lower end lies above 18. Many people would argue that you should do a one-sided 2.5% level test and look at the lower end of a 95% CI.

3) In practice it is hard to see why one would do this. What do you do when your confidence interval is from 19 to 30? Do we then reject the null hypothesis, but not in favor of the alternative? If the CI were from 19 to 20, would we reject both the null hypothesis and the alternative, too?