Statistical distance and p-value
If σ is a known value, the calculation to find the p-value for testing the mean can be expressed in terms of the general formula for the statistical distance between a parameter and its estimate,
In the context of a test about means,
Since z has a standard normal(0, 1) distribution when the null hypothesis holds, it can be used as a test statistic and the p-value for the test can be determined from its tail areas.
For a two-tailed test, the p-value is the red tail area.
Example
The mean of a sample of n = 30 values is 16.8. Does the population have mean µ = 18.3 and standard deviation σ = 7.1, or is the mean now lower than 18.3?
H0 : µ = 18.3
HA : µ < 18.3
The p-value for the test is shown below:
The p-value can be evaluated using the statistical distance of 16.8 from 18.3 (a z statistic),
The p-value is reasonably large, meaning that a sample mean as low as 16.8 would not be unusual if µ = 18.3, so there is no evidence against µ = 18.3.