The problem is that it also depends on how random the sampling was. How many people were in the south? The midwest? In New England? And so forth.
Certain areas of America have different tendencies towards different attitudes.
5 people in Texas, for instance, are more likely to be conservative than 5 people in New York City.
They include that in the study, they list the people by census demographics at the bottom. This actually hurts your case. The south is under-represented (study 32%/Census 37.1), the west and north east being over-represented (Study 27 & 20%/Census 23.3 & 17.9, respectively). That aside, those deviations are not so egregious as to intentionally skew the data. Likewise, if you read their methodology, you would see that they called random landline phones.
Considering how large the country is, 863 results feels like it would have a much larger margin of error than 3.4%, simply because we're talking about proportions here.
I'm just gonna leave this here. Wikipedia can explain this far better than I care to.
The larger the number, the more likely it is to be representative of a group.
As it stands, though, a nationwide poll having a smaller sample size than a local poll reeks to me of bad planning anyways. And bad planning in sample size could easily be demonstrative of bad planning in other areas.
In short I have many problems with them, and the sample size is a big red flag marking it all. While I'm sure honest statisticians try their best to be honest, I have nothing showing evidence that the people who ran this poll are honest statisticians.
This is wharrgarbl. If you understood how confidence intervals and sampling worked you would understand that this is a baseless argument as well. If one asks a random number of people, they can extrapolate with confidence how the larger section of people feel. If they ask 1000 Virginians, they can say with a degree of certainty how Virginians feel about an issue, not necessarily Americans, but Virginians. If they ask 1000 Americans how they feel on an issue, they can do the same, and again, not necessarily say how Virginians feel on the issue. Just because some statewide polls ask 1000 people isn't a fair criticism, it just means the margin of error for that state is lower than it is for this study. I.E. you don't need to survey a certain proportion of the populace in polls to get an accurate answer.
Again, I'm focusing on the sample size criticism. If they asked 2400 people with their same methodology, question, etc, then they could've come to a far different result. That, however, is highly unlikely as there would only be about a 5% chance it would fall outside the margin of error, with most that did being closer to 25 & 33% than not. What realistically would've happened is the margin of error would drop & the 29% that they found would fluctuate between 25.6% & 32.4%. Thus, again, if you want to criticize the study, you're focusing on the wrong part.