# 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?