What the news people are talking about is anyone's guess and varies with the newscast. Perhaps most common is that they are giving a one sentence summary of research that requires several pages.
However, your last paragraph is mistaken. Statistically, each family does NOT have 2.4 children. The mean is 2.4 children. This is entirely possible. If you take a random sample of American families (tricky to do, but possible) then you would get an estimate of the mean. However, if you took a census of families, then, if the census really got every family (it doesn't) or, if the people it got are representative of the people it didn't get, with regard to number of children, then you would have proven the fact.
However, not only does the census miss people, the people it misses are different in many ways from the people it gets. The Census Bureau therefore tries to figure out how they are different; thus, again, giving an estimate of the number of kids per family.
But there are things you can prove; if you wanted to know, say, the average number of years that each professor in your department had been teaching, you could get accurate data and come up with an exact mean.
Your penultimate paragraph is also problematic as statistical tests are done precisely to prove hypotheses; more precisely, they are done (in the frequentist framework, anyway) to reject a null hypothesis at a given level of significance.