How can I prove that two samples belong to different populations (and whether they actually do)?

Question

I'm currently evaluating a factory and I'm trying to find out whether there is a connection between the valuation multiple and the number of cows (denominator). I've got a sample of similar new factories (nearly 40). I did a regression and R2 was somewhere near 0,4. However, in the graph I saw two different clusters - <1000 and >1000 cows. I did a regression on each and the results were R2 of nearly 0,02 and 0,04. Am I correct that these two samples belong to different populations (with different means and st.deviations) and that I can't analyse them as one? How can I prove it?

Thanks in advance!

Answer 1

Just based on the $R^2$, it's hard to say whether your groups are coming from two different populations.

One solution might be to run a regression with all of your observations, regressing your DV on your explanatory variable and a dummy variable for "big" factories. You can use a simple t-test, with the null hypothesis that the coefficient on "big" dummy is equal to zero. If the t-statistic is greater than the critical value for the level of significance you choose (e.g., greater than 1.96 for p=0.05), you can reject the null hypothesis--small groups and big groups have different intercepts.

If you think that small groups and big groups might have different slopes, you can add to your regression an interaction of your "big" dummy with the explanatory variable, and conduct an F-test with the null hypothesis that the coefficient on "big" dummy and your interaction term are both different from zero.