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Null hypothesis: The full model and the reduced model are statistically equivalent in terms of their predictive capability.
Alternative hypothesis: The full model and the reduced model are not statistically equivalent in terms of their predictive capability.
The lower the p-value, the higher the probability that the null hypothesis can be rejected in favor of the alternative hypothesis.
My understanding is such that the null hypothesis embodies the reduced model. If one cannot reject the null hypothesis, then one takes the more parsimonious model. However, if the two models differ, it is better to use the full model.
For hypotheses testing, the null hypotheses is always, there is no difference. Alternative is, there is difference.
As stated in the end of documentation, it is doing F-test (ANOVA I guess) on the regression models. It is a common way to compare two models, additionally I would also suggest to check the adj-R square in both models, it does something similar to tell you if the variables you dropped matters or not.