The Z-Test › Computing P-Values from Z-Statistics

Once you have a z-statistic, you need to convert it to a p-value. The process depends on whether you're doing a one-sided or two-sided test.

For a two-sided test (testing whether the mean differs from $\mu_0$ in either direction):

$$\text{p-value} = 2 \times P(Z > |z|) = 2 \times \Phi(-|z|)$$

where $\Phi$ is the standard normal CDF. We double the probability because we're interested in extreme values in both tails.

For a right-tailed test (testing whether the mean is greater than $\mu_0$):

$$\text{p-value} = P(Z > z)$$

For a left-tailed test (testing whether the mean is less than $\mu_0$):

$$\text{p-value} = P(Z < z)$$