I am learning about rejection regions and tail tests. I am having trouble understanding one concept though, if we test a hypothesis and the test is a left tail test, then the rejection region would be in the left tail of the sampling distribution of the test statistic under the null hypothesis, right? Or would it be in the right tail of the sampling distribution of the test statistic under the alternative hypothesis, or null hypothesis?
Thank you.
You are correct: "If we test a hypothesis and the test is a left tail test, then the rejection region would be in the left tail...."
For example, if $X_1, X_2, ..., X_9$ is a random sample from a normal population with unknown mean $\mu$ and known standard deviation $\sigma=12.$ Then the null hypothesis for a 'z-test' might be $H_0: \mu = 100$ and the left-sided alternative would be $H_a: \mu < 100.$ Then the test statistic is
$$Z = \frac{\bar X - \mu_0}{\sigma/\sqrt{n}} = \frac{\bar X - 100}{12/\sqrt{9}},$$
where $\bar X$ is the sample mean and (assuming $H_0$ to be true) $Z \sim \mathsf{Norm}(0,1).$
You would reject at the 5% level of significance if $Z \le -1.645.$ If it turns out that the observed value of $Z$ is $z = -1.82$ (once you plug in the observed value of $\bar X$), then the P-value of the test is $P(Z < -1.82) = 0.0348,$ which you should be able to find from a printed table of the standard normal CDF (or using software).