What can we conclude from correlation? I just got my statistics test back and I am totally confused about one of the questions!  

A study was done that took a simple random sample of 40 people and measured whether
  the subjects were right-handed or
  left-handed, as well as their ages.
  The study showed that the proportion
  of left-handed people and the ages had
  a strong negative correlation. What
  can we conclude? Explain your answer. 

I know that we can't conclude that getting older causes people to become right-handed. Something else might be causing it, not the age. If two things are correlated, we can only conclude association, not causation. So I wrote: 

We can conclude that many people
  become right-handed as they grow
  older, but we cannot tell why.

That's exactly what association means, but my teacher marked me wrong! What mistake did I make? Is 40 too small of a sample size to make any conclusions?
 A: This is wrong: "We can conclude that many people become right-handed as they grow older." We cannot conclude this at all from the given data. 
For one, the study only takes a sample at one point in time, rather than selecting a sample and monitoring their progress through many decades. This is what would be needed for us to even entertain the possibility that aging causes a change in handedness. 
Other possible causes include that left handed people might have a shorter life expectancy, or perhaps there was a spike in the birth rate of right handed people in the past. There are many other possibilities that have been mentioned in others answers which would also account for the skewed proportions without requiring people to change handedness with age, which is what you falsely concluded in the test.
Also, just an observation, but it appears the "study" was conducted under false pretenses. Handedness is a false dichotomy, people can also be ambidextrous.
A: Here is one example of a plausible explanation that disagrees with your analysis:
Cultural expectations for left- and right-handedness have changed over time. Older people may have gone to school at a time where left-handedness was discouraged and students were forced to write with their right hands, training children never to use the left hand instead of the right. Younger participants in the study were in school more recently and learned to write at a time where left-handedness was not discouraged, creating a positive correlation between left-handedness and youth.
A: We can only conclude that the result is interesting and it deserves more research.
Without further study we can not say if right-handedness causes age (ie. being left handed causes biological alterations that shorten the life span), age causes right-handedness (ie. as people age they become right handed), they are correlated because they are caused by another variable (ie. older people became educated in a different system that discouraged left-handedness), the study had bad luck selecting its sample, the study sample was bad designed, the study was bad designed, etc.
A: Without getting much into conclusions that can be drawn from correlation coefficients, I wanted to point out a few things about sample size...
10% of people are left-handed.
https://www.huffingtonpost.com/2012/10/29/left-handed-facts-lefties_n_2005864.html
That means, unless you have specific data stating otherwise, it is likely that this hypothetical study sampled 36 right-handed people and 4 left-handed people.  The sampled ages of 4 left-handed people will tell you basically nothing about their mortality rate.  The difference in longevity between left-handed and right-handed people is probably not very high, so accurately approximating the difference would require quite a high sample size of each population (probably about 100-300 of each).  For further reading, see how to select sample size based on standard error (if it is known or can be reasonably estimated) and confidence level.
There is also an experimental design error.  Taking a survey of the ages of a few living people doesn't give a very good estimation of the life expectancy of these populations.  If you went to the morgue and surveyed the age of the last N people who died, you would be somewhat closer to having an accurate measure of life expectancy.
You also made a serious error in how you worded your conclusion, which others have pointed out.
Also, this is far beyond the scope of the pure math that you're being tested on, but it happens that the left-handed portion of the population declined to about 3% during the Victorian era, before rising back up to 10% in the present.  This could be largely because people of the time were pressured to become right-handed, and could also correlate to the design of industrial tools of the time.  Correlation doesn't necessarily imply causation, and the most common reason why is that there might be a third variable that is unnaturally forcing the correlation.  In this case, the life expectancy of people living in the present may not vary greatly between the left-handed and right-handed; but lack of knowledge about historical context might increase the likelihood of making a correlation = causation mistake, or historical data might imply a difference in life expectancy that once existed but now does not.
https://www.snopes.com/fact-check/left-hand-right-hand-death/
