# Hypothesis testing of test statistic, $\mu$ against $\mu <, >, \neq \mu_0$

In my notes, I am looking at a hypothesis testing of a test statistic distributed under the t-distribution because the population SD isn't known. The t score was computed for the test statistic, $$\mu$$, to be $$t = -1.31$$.

Then it says that if we want to test whether $$\mu < \mu_0$$ where $$\mu_0$$ is the value of our test statistic under the null hypothesis, we'd need to compute the area of the t distribution curve to the left of $$t = -1.31$$ to get our p value. This makes sense. Then it says if we want to test whether $$\mu \neq \mu_0$$, then we compute the area of the t distribution to the left of $$t = -1.31$$ and to the right of $$t = +1.31$$ (or just double the area to the left of $$t = -1.31$$). This also makes sense. Then it says that if we want to test whether $$\mu > \mu_0$$, we compute the area to the right of $$t = -1.31$$. My question is, should it be to the right of $$t = -1.31$$ or to the right of $$t = 1.31$$? The former makes sense mathematically, but the p-value is obviously going to be more than 0.5 in this case, so what is the point?

• If $T = -1.31,$ then obviously $\bar X < \mu_0$ so you have no evidence that $\mu > \mu_0$ and it makes sense to have a large P-value. May 5, 2021 at 17:58

Suppose I'm testing $$H_0: \mu = 55$$ vs. $$H_a: \mu > 55$$ and I happen to have $$n = 20$$ observations from $$\mathsf{Norm}(50, 7).$$ Then I might get $$\bar X = 52.76, T = -1.32,$$ and P-value $$0.898,$$ so I correctly fail to Reject $$H_0.$$

Example in R:

set.seed(2021)
t.test(rnorm(20, 50, 7), mu = 55, alt = "g")

One Sample t-test

data:  rnorm(20, 50, 7)
t = -1.3151, df = 19, p-value = 0.8979
alternative hypothesis: true mean is greater than 55
95 percent confidence interval:
49.81208      Inf
sample estimates:
mean of x
52.75886

1 - pt(-1.3151, 19)
[1] 0.897936
• For this particular example, on the alternative hypothesis, if you changed $\mu > 55$ to $\mu > c$ where $c > 55$, then you'd still get the same p value right? What if $c < 55$, it seems you'd still get the same p value? May 6, 2021 at 15:03
• Or in other words, I'm confused about the purpose of $c$ here. It seems moot? May 6, 2021 at 15:04