Finding a p-value from the t distribution

When testing the value of µ when σ is unknown, we use the test statistic

This has a t distribution (with n − 1 degrees of freedom) when H0 is true, so the p-value is found from a tail area of this distribution.

One-tailed test

H0 :   μ  =  μ0
HA :   μ  <  μ0

The steps for testing these hypotheses are shown in the diagram below.

Example

Consider a sample of n = 13 values with mean = 16.14 and standard deviation s = 2.15. A test for whether the population mean is more than 15.0 uses the hypotheses:

H0 :   µ  =  15

HA :   µ  >  15

Since the population standard deviation, σ, is unknown, the test must be based on a t statistic.