The correct answer if option (C).
Let us understand why.
Understanding the passage:1. Research:
a) indicates that college professors in general, were raised in economically advantaged households
b) Basis for above assertion: Overall, college professors grew up in communities with average household incomes higher than the national average household income
2. We need to find the flaw/weakness in the above argument
Thought Process:What would weaken the belief that college professors in general, were raised in economically advantaged households?
Given point (b).
1. (b) talks about average household incomes. (a) talks about economic advantage. The author is assuming a direct correlation between the two. i.e. higher the average household income, higher the economical advantage.
Assumption: Higher average household income translates to economic advantage (direct correlation).
2. Let us re-look (b). Does college professors growing up in a community whose average income is higher than the national average necessarily tell us that the college professors came from households with higher average income?
Say the community average is 50000$, and the national average is 45000$. What if professors come from households whose average is actually 40000$ (below the national average), but the community average is at 50000$ due to the other households (not having professors) with an average of say 55000$?
Assumption: A significant number of college professor households are at least average (average or above average) in their community.
Any statement that goes against the above assumptions will weaken the argument.
Let us look at the answer choices.
(A) inappropriately assumes a correlation between household income and economic advantageVery interesting choice. The researchers do seem to have assumed a correlation between household income and economic advantage. But is this an inappropriate assumption, or is it common sense, therefore appropriate

? I would park this option and check the other options for a more definite weakener.
(B) fails to note there are some communities with high average household incomes in which no college professors grew upIrrelevant. Other communities with high average incomes has no bearing given that college professors came from a community that is already known to have high average income. It is irrelevant to the argument.
(C) presumes without justification that college professors generally were raised in households with incomes that are average or above average for their communitiesCorrect. if college professors came from those households whose incomes were below the community average, it is highly probable that they do not really come from household with any real economic advantage.
(even though the community average is high, the professor's home's average is not so high => it may not really any better than the national average (may be lesser also) => no real economic advantage).
(D) does not take into account the fact that college professors generally have lower salaries than their counterparts in the private sectorIrrelevant. Has no impact on the argument
(E) fails to take into account the fact that many college professors live in rural communities which generally have low average household incomesThe argument is about professors in general - who come from communities with an average income higher than national average. This option talks about a group of professors who come from a different community (with a lower household income). This group is not part of the "general" - these are the professors who are not representative of professors in general. But our argument is concerned with professors in general.
Even if the option is true, it is simply talking about the many (not necessarily majority) profs who are exceptions to the general group of professors (about whom the argument is concerned).
Essentially, this option actually has no impact on the argument, because the group it talks about is not exactly the same.
Best option is (C). option (A) I guess is then common-sense.
Cheers!