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

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.