How to interpret the margin of error in a poll? Recently the media reported on a political poll that stated that "46% of Republican voters in Mississippi think that interracial marriage should be illegal".  One example story (of many around the 'net) is this one from the NY Daily News: Interracial marriage should be illegal, say 46% of Mississippi Republicans in new poll.  From that article:

A survey conducted last month shows 46% of GOP voters in the state
  believe interracial marriage should be illegal - a plurality of the
  people questioned

Given that I know many people in interracial marriages (including in my own family) I was motivated to track down the full story rather than fall to the obvious hysteria.  I located the press release on the poll from Public Policy Polling here (PDF)  which has:

46% of these hardcore Republican voters believe interracial marriage
  should be illegal, while 40% think it should be legal. With Barbour
  included, Huckabee gets more support (22%) from the former than the
  latter (15%), as does Palin (13-6). The support for Bachmann (10-2),
  Gingrich (13-8), and Pawlenty (4-1) works the opposite way.

And at the end of the first page they state:

PPP surveyed 400 usual Mississippi Republican primary voters from
  March 24th to 27th. The survey’s margin of error is +/-4.9%.

Now what I don't understand is how to interpret the results especially the margin of error value.  
From the raw numbers, 46% (~50%) of 400 voters is 200 voters who responded who think that interracial marriages should be illegal.  But from stats of Mississippi in the 2008 Federal election, McCain polled about 725,000 Republican votes (to Obamas 556,000). My gut feeling is that you can't extrapolate that 46% number to the other 724,600 Republican voters and still retain a 5% margin of error.
So is the media misrepresenting the numbers (really? the media would do that?!?!?!) Or are there statistics at work that I don't understand?
Thanks for your help!
 A: The claim that the margin of error is $4.9$% follows from assuming that the poll was conducted as if a box had been filled with tickets--one for each member of the entire population (of "hardcore Republican voters")--thoroughly mixed, $400$ of those were blindly taken out, and each of the associated $400$ voters had written complete answers to all the poll questions on their tickets.  These $400$ poll results are the "sample."
The "as if" raises plenty of practical questions that go to whether the poll really can be viewed as arising in such a way.  (Can we really think of the population as represented by a definite set of tickets?  Is it fair to assume all tickets are completely filled out?  Was the sampling conducted in a manner akin to drawing from a thoroughly mixed box?  Etc.)  Other respondents have listed some of those questions.  Granting, however, that this is an adequate model of the poll leads us to the crux of the question: to what extent do these $400$ tickets represent the entire population?  We never know for sure, but we can develop some expectations by studying this process of sampling from a box of tickets.
To do this, we focus on one question at a time.  We might as well view each ticket as bearing either the "yes" or "no" answer for that question.  We now compare the true survey results (that is, the true proportions of yeses among all tickets in the box) to the results of the myriad possible samples of $400$ tickets.  (There are more than $1.9 \times 10^{1475}$ such samples.)  We have to make the comparison for any possible true proportion, but even so, it's merely a matter of mathematical calculation.  This calculation shows that the observed response in at least $95$% of all such samples lies within $\pm 4.9$% of the population value no matter what that population value might be.  For example, if exactly $50$% of the tickets in the box are "yes," then $95$% of the possible samples of $400$ tickets will contain between $50-4.9$% = $45.1$% and $50+4.9$% = $54.9$ yeses.
(That computed value of $95$% actually depends on the true proportion of yeses in the population: if that proportion is very small or very large, we find that quite a bit more than $95$% of all samples will give results accurate to within the margin of error.  A true proportion of $50$% is the worst case, which is used because we don't know the true proportion!)
This is all the margin of error means.  Because $95$% is a substantial fraction of all possible samples, we feel it's highly likely that the one sample that was actually obtained will be among these $95$%.  A doubter is allowed to suppose the sample could be one of the remaining $5$%: we cannot prove him wrong (based only on the poll results, anyway).  Yet, similar calculations show (for instance) that the proportion of yeses will differ from the true proportion by more than $12.2$% in only one of every million possible samples.  It's still possible the poll is among these one-in-a-million samples, but we have very shaky grounds to believe that.  Thus, there is usually a limit to what constitutes a "reasonable" amount of doubt about what the true proportion may be, and it's rarely as extreme as $\pm 100$%. 
The fundamental insight afforded by these calculations is that once the number of tickets in the box becomes moderately large (a few thousand in this case), the margin of error does not depend on how many tickets are in the box.  It should be intuitively clear that the only thing that really matters for a relatively small sample is the proportion of yeses in the box, because the proportion determines the chance of drawing a "yes" or "no" and that proportion doesn't appreciably change between drawing the first and drawing the last of the $400$ tickets.
In summary, assuming it's accurate to view the poll as acting like drawing tickets from a box, our right to "extrapolate" from the poll to the population (a process more formally known as statistical inference) is an uncertain one, because we can always be wrong; but when the sample is just a small fraction of the population, the amount by which we might be in error in making that extrapolation depends primarily on the size of the sample, not the size of the population.  This is why most credible polls, whether of local or international scope, use samples of a few hundred to a few thousand.  It is rare that larger samples are needed to achieve a high chance of getting reasonable accuracy.
A: I won't try to deliver my own answer, but I would refer you to the "What Is a Survey?" booklet compiled by the Survey Research Methods Section of the American Statistical Association. (Fritz Scheuren endorsing it on the title page is a former President of ASA from about five years ago. He used to be a high profile statistician in federal agencies such as the Social Security Administration and Internal Revenue Service, and now semi-retired from government to continue working as a VP of the National Opinion Research Center at University of Chicago.) The booklet delivers a clear and concise explanation of when and why you can, or can not, extrapolate the survey findings to the target population.
A: To answer your question:
It is possible to extrapolate from a sample of 400 to the views of all 700,000. This is contingent on the sample being random. Statistical Power is the topic you'd want to look into to confirm this. If I ask 400 of my closest friends, this doesn't work. To get a truly random sample, I'd have to get the list of all 700,000 people, and use a random number generator to pick 400 from it. Even so, there might be some selection biases. For example, if we're only calling landline telephones, then young people (who often only have cell phones) would be under represented in the sample. It's still possible to correct for these issues, but you have to be pretty careful. 
Nate Silverstein's blog has some really good posts on the reliability of different polling firms, problems with their techniques, and correct inference for US political polls. 
A: The short answer is yes, you can extrapolate.
Longer answer: The key question is whether the pollsters took a random sample of a population. They claim to have taken a random sample of Republican primary voters. But this is difficult. People refuse to answer polls, or they aren't home or other things can go wrong; even worse, the people who answer are not a random sample of the whole population (for instance, younger people are less likely to have land line telephones).  Most pollsters therefore try to weight the sample they get to match a known population. Exit polls of Republican primaries give good estimates of various traits of this population. 
Reputable pollsters (such as PPP) try hard to do this in a balanced way.
So, can you extrapolate from a relatively small sample to a large population? Yes you can, but there are some caveats.
