# p-Values

Posted by Beetle B. on Tue 18 July 2017

$$\newcommand{\Cov}{\mathrm{Cov}}$$ $$\newcommand{\Corr}{\mathrm{Corr}}$$ $$\newcommand{\Sample}{X_{1},\dots,X_{n}}$$

The p-value is the smallest significance level (i.e. $$\alpha$$) at which $$H_{0}$$ would be rejected. So if $$p\le\alpha'$$, you reject $$H_{0}$$. If $$p>\alpha'$$, you do not reject ($$\alpha'$$ is usually 0.01 or 0.05, etc).

When $$H_{0}$$ is rejected, we say the data is significant.

An equivalent definition: The p-value is the probability, calculated assuming $$H_{0}$$ is true, of obtaining a test statistic value at least as contradictory to $$H_{0}$$ as the value that actually resulted.

In other words, given $$H_{0}$$ is true, what is the probability of getting the observed value?

## The P-Value for a $$Z$$ Test

For an approximately normal distribution:

• For upper-tailed, $$p=1-\Phi(z)$$
• For lower-tailed, $$p=\Phi(z)$$
• For two-tailed, $$p=2(1-\Phi(|z|))$$

## P-Values for t Test

Just use the tables or software.