Hello and thank you so much in advance for your help!

My question has to do with hypothesis testing, specifically in regards to population and sample data:

1. Would you run this test if you had the entire population data for both Non-Smokers and Smokers?

https://onlinecourses.science.psu.edu/stat414/node/268

I understand that it is extremely rare that you ever have population data, which is why we sample. However, if we DID have population data, would it make sense to run this test? Can you explain your reasoning?

2. Does the explanation below make sense, in regards to the hypothesis test comparing two proportions from the link above? Can you explain your reasoning?

This test is not meant to infer on population properties based on sample data, it tests if two independent proportions are similar or not, be it from two random samples of the same population or two random samples of two different populations, or even two complete populations. If I have 2 respectively representative samples of 2 populations and I want to ensure the proportions are statistically similar, then what stops me (if I have all observations for both populationís) from increasing the size of my samples as long as they are representative of the population until I get the entire population? If H0: p1 = p2, and I know that p1 = P1 & p2 = P2 then I can also test if H0: P1 = P2.

Then you have to ask, why do you need to test the population proportions, maybe I donít think the populations are big enough and I may have a specification error in one or both measures. In this case I donít just want to take on face value that 5% of a population of 3,000 is different from 6% of a population of 400, so Iíd like to test the probability that these are in fact the same.

3. For an example, letís say I have two populations (not samples).

Population A has 50,000 Men and 30,000 Women

Population B has 2 Men and 1 Women

Given I have population data, would there be any statistical test whatsoever that should be performed in this scenario? Would it be accurate and statistically valid to say that Population B has a higher proportion of men than women? Would it be accurate and statistically valid to say that the proportion of men in Population A is not equal to the proportion of men in Population B? Even though the size of the two populations is so different?

Thank you so much i really appreciate it!