I am going to examine the effect of educational achievement on rates of injury for 20 cities for people age 65+. So I have the number of people who was injured in each city, and the number of people with elementary school, diploma, and other certificates of each city. I have already tested The Fisher Exact Test!
Linear regression should work, provided that the assumptions for that analysis are met. However, you must control for population size of the city as larger cities have both more injured persons and more diplomas per capita. Otherwise, there will be autocorrelation between diplomas and injuries.
Fishers exact test is for analysis of contingency tables;as I understand it, that is not the kind of data you have...
There is no magic test for small sample sizes - you may have to live with the fact that the power of the test is relatively low.
I would suggest that you consult a basic statistical textbook, the information regarding these things is given there.
You definitely need to use some resamplig technique such as bootstrap or jacknife. There techniques consist of drawing random samples from your small sample and doing computations on them. This is often seen in life sciences where samples are small.
How small are your samples? What you describe doesn't really seem small to me, but then, if you only have a few from each city, it is possible. The main thing with small sample numbers, surely, is to not assume normality and use non-parametric statistics.
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