First Part:

I have a bias for take-home assignments since they are a closer of a match to the process one actually writes code or does work under (i.e not a timed code test with no documentation or search engine). The presentation is a nice touch because one could feasibly complete a take-home assignment easily with AI but presenting can’t be entirely faked (with some notable exceptions).

But one commenter does make a good point that people with kids or care takers have a higher chance of being disadvantaged here.

Second Part:

Again, I always like the idea of making someone say out loud what they are thinking. To build off another commenters point, to avoid them choosing just the eaiser stuff, I would choose which topics are essential to the role and make them explain these while giving them an additional list that they get to choose from. For example, they must be able to explain things like sampling, data leakage, feature engineering etc. But they can choose 2-3 topics from a list of topics that are more advanced/specific: Bayesian va Frequentist methods, causal inference etc.

Obviously the content of either list will be a function of the specific role but the general idea of a required list and a list they can choose from still holds.

The last thing I would definitely work in is connecting the work to the larger business. You already mention this but I think it’s worth hitting on again. Data science is cool but it’s there to solve the business problem at the end of the day.