In the previous lectures, we practised the logic flow of hypothesis testing over and over again, so I hope you are now familiar with the concept of hypothesis testing. Basically, we have some data in our hands, and we come up with some hypotheses by incorporating the knowledge about certain population parameters based on the previous studies or experience. For example:

Hypotheses based on current data
  • The proportion of people who have type B may not be 9%.
  • The proportion of MCQs whose correct answers are Bs might be more than 25%.
  • The normal body temperature may not be 37 $^\circ$C.

How do we test them? As we mentioned repeatedly during previous lectures, there is no way we can directly test if those hypotheses are true or not. However, we can approach the problem in an indirect way. That is, we assume the opposite is true ($H_0$):

The null hypothesis
  • The proportion of people who have type B is 9%.
  • The proportion of MCQs whose correct answers are Bs is less or equal to 25%.
  • The normal body temperature is 37 $^\circ$C.

Based on that, we use what we learned from the section of probability theory to calculate the probability of observing the data we have or more extreme (the p-value). Then, we check:

Is the probability small enough or not ?

If it is not small, meaning the data we have is expected; if it is small, meaning that it is unlikely for us to observe the data we have given that our original assumption ($H_0$) is true. Yet, it happens, so there might be something wrong with our original assumption $H_0$. Therefore, we question our original assumption. In practice, we set a significance level $\alpha$. If the the p-value is smaller than $\alpha$, we reject our original assumption ($H_0$).

In this lecture, we continue practising NHST, using some unusual examples. The purpose of this lecture is to showcase how we should proceed with hypothesis testing when encountering some unusual cases. The basic idea is still the same. No matter what problems we have, we always need to figure out two things:

  1. What is the test statistic?
  2. What is the distribution of the test statistic given that $H_0$ is true?

We will elaborate those throughout the rest of this course.

References