Chapter 2 What is a hypothesis test?

Recall our previous example where the cholesterol levels of 72 patients were measured, and we wanted to ascertain: Is the average cholesterol level of patients from this population different from 5.0 mmol/L? Hypothesis testing provides a formal framework from which questions such as this can be addressed using information from a collected sample.

When carrying out a formal hypothesis test, we normally consider two hypotheses:

  • The null hypothesis, denoted \(H_0\)
  • The alternative hypothesis, denoted \(H_1\).

In our example above, we would define the null and alternative hypotheses as follows:

\[H_0:\mu = 5\;\;\text{versus}\;\;H_1:\mu \neq 5,\] where:

  • \(\mu\) denotes the average cholesterol level of patients from this population
  • \(H_0\) denotes the null hypothesis that the average cholesterol level of patients from this population is equal to 5
  • \(H_1\) denotes the null hypothesis that the average cholesterol level of patients from this population is not equal to 5.

When we carry out a hypothesis test, to start out with, we assume the null hypothesis to be true. If our sample provides evidence that this was not a reasonable assumption to make, then we reject the null hypothesis and therefore have evidence in favour of the alternative hypothesis.

We will now consider an example.