# $p$-value,the significance level

Here in the definition 5.3.11 on the page 249 they write "The $$p$$-value associated with a test is the smallest significance level $$α$$ for which the null hypothesis is rejected."

My question is what is the order here: $$\bar{X}>\bar{x}$$ or $$\bar{X} <\bar{x}$$ and why? I.e. when it is the case that $$\alpha$$ is the smallest significabnce level, if

$$H_0:\mu=\mu_0$$ versus $$H_1:\mu>\mu_0$$

OR on the other hand

$$H_0:\mu=\mu_0$$ versus $$H_1:\mu<\mu_0$$ ?

• Is the PDF you link to a legitimate fair use? It would be better in any case to write the complete definition in the question here. – David K Oct 30 '18 at 13:06

A p-value less than or equal to $$\alpha$$ means you reject the null hypothesis and have evidence that the mean reflects the alternate hypothesis. The alternate hypothesis can be either $$H_a:\mu>\mu_0, \text{ or } H_a : \mu < \mu_0\text{ or simply } H_a : \mu\neq \mu_0$$ depending on the context of the test.
For example, you have not tested the mean of a population for several years so you want to assess whether it has changed. Because you don't know whether it is larger or smaller, it is appropriate to use $$H_a : \mu\neq \mu_0$$. In another situation, you may suspect the mean has decreased (test scores seem to be lower than normal) in which case you would test using $$H_a : \mu < \mu_0$$.
Edit: One can never prove an $$H_a$$ with a single test but only provide evidence to support it. One can never discount the possibility of a type I error. This is the random chance of obtaining a p-value less than $$\alpha$$. There also exists the possibility of another variable affecting the result. But with more testing and control of other variables, one can be more confident in the result.
For $$H_a:\mu>\mu_0$$, this is a result in the right tail to the right of a Z score corresponding to $$\alpha$$ of a normal distribution $$\text{ and } H_a : \mu < \mu_0$$, in the left tail $$\text{ and } H_a : \mu\neq \mu_0$$ which can be in either tail.
• Why we cannot prove $H_a$ and can only reject $H_0$? A geometric intuition using the normal distribution curve and at the same time the $p$-value would help. – user122424 Oct 30 '18 at 18:33