# Confused about hypothesis testing, tails, and confidence intervals

$Y_1, \ldots, Y_{10}$ 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?