"FordFairLane is a sniveling weasel, and I wish he would stop quoting me." - Jean-Jacques Rousseau
NOW we can agree on something.
That is not even how the margin of error works. The term "margin of error" is a term of art, meaning to specify a result within a certain range with a confidence interval. In this instance, where you take a population size of 243,000 and derive 478 answers, and with a claim that 61% believe Rivers was a better QB than Ben, with a margin of error of 5%, your actual outcome is:
- Margin of Error = z*(√p(1-p) / n)
- Margin of Error = 1.96*(√.61(1-.61) / 100) = 1.96*(√0.2379/100) = 1.96*(√0.002379) = 1.96*(0.04877) = 0.0956
- Margin of Error = .0956 or 9.56%
Therefore, the 95th percentile confidence interval for "Rivers is better than Ben" would be 51.4% to 70.6%.
But that means only that you can say within the 95th confidence interval that those who answered the poll favor Rivers over Ben between 51% to 71%. However, again your fundamental error is your presumption that the respondents to the poll are somehow representative of football fans in general or for an opinion poll - which this is - one taking responses from a proportionate share of football fans to generate an accurate model.
What we can accurately say about the poll you cited is, "Within the 95th percent confidence interval, I can say that between 51% and 71% of the fans who answered the poll believe Rivers is better than Ben." That's all you can say. That's it. And that statement has absolutely no relevance, at all, to anything on earth.