Thanks for reviving this discussion Andreas and Bibani!
These can be quite difficult to follow, but here's what I feel I've learned since then:
- The smaller the p-value, the stronger the evidence we have to reject the null hypothesis. See the wiki and this link. A small p-value suggests a small probability of obtaining the observed results assuming the null hypothesis is true.
- The null hypothesis of the Jarque-Bera test is that the data is normally distributed.
- The conclusion of (1) and (2) must be that a small p-value for the Jarque-Bera test suggests the data is not normally distributed.
I confirmed this by running jarque_bera on two very large samples (10,000), the first of which is very obviously normal, and the second of which is very obviously bimodal. The test returns a p-value (second return value of jarque_bera, per the documentation) of 0.74 for the normally distributed data and a p-value of 0.0 for the bimodally distributed data. See my attached notebook if you'd like to review the results.
Note that part of the confusion stems from the value of the test statistic itself. Though the value is largely meaningless, it is true that if its value is far from zero, the data is likely not normally distributed. See the wiki for Jarque-Bera. For instance, in my test, the JB value (first value returned by the jarque_bera function) is 0.6 for the normally distributed data and 3331 for the bimodally distributed data. This is essentially "backwards" compared to its p-value, which could lead to some of the confusion here.
I think we can safely say this means the answers for Lecture 10 are incorrect and need adjusting.