When we have a sample of the population, we calculate the mean $\bar X$ using the sample. The population mean $\mu_p$ is unknown. Hypothesis testing tells us whether the sample mean is a good representation of the population mean.

We have two hypothesis.

  1. Null hypothesis: the sample mean is a faithful representation of the population mean
  2. Alternative hypothesis: the sample mean is NOT a faithful representation of the population mean.

Hypothesis testing is about the conditional probability of the alternative hypothesis given the null hypothesis is true. However, the conclusion from hypothesis testing is only about whether the null hypothesis is accepted or rejected, nothing else.

Of course it doesn’t have to be mean.