As others pointed out, "Margin of Error" ignores many possible sources of error; it ONLY deals with the "sampling error" due to extrapolating from a smallish sample. If I was a rational voter in a red-neck district I might be tempted to lie — why bring danger on my family? — were a stranger with a white Southern drawl to ask me whom I'm voting for.
It is VERY easy for a pollster to get unwanted bias. Making afternoon phone calls? Maybe it is Rs rather than Ds who are most likely to answer by the 8th ring. I think that's one reason polling is expensive. Instead of just calling a different number, the pollster may spend time trying to contact the guy who didn't answer his phone.
The sampling error is easy to calculate with a simple formula:
. . . . . . Margin_of_Error = 1.96*sqrt(p*(1-p)/n)
or if we set p = 0.5, simply
. . . . . . Margin_of_Error = 0.98*sqrt(1/n)
Set n = 1067 and the latter formula yields Margin_of_Error = 0.03. This means that, ignoring all biases except sampling error, and supposing you got 50% of the pollees agreeing with "Yes, Pelosi is a crook," there is a 95% chance that the correct percent for the total population is in the range 47% to 53%.
0.98 is very close to 1 so Margin_of_Error = sqrt(1/n) is a good-enough formula. The margin of error is 10% for a n=100 sample, 1% for a n=10,000 sample, and 0.1% for a n=1,000,000 sample.
Pollsters usually set p = 0.50 and describe that as the Margin of Error for all the questions answered. But in fact, if the result on a certain question is 87%/13% instead of 50%/50%, then p = 0.13 (or p = 0.87) should be substituted in the formula. This would give a ±2% margin of error instead of ±3%. (Anyway the simple formula breaks down when the poll result is VERY lopsided.)
