= new reply since forum marked as read
However, even if technically a difference like this *can* be explained by statistical uncertainty caused by a smaller sample size and questions about how "random" the selection of participants was, that doesn't mean one can properly declare outright that this change is explained due to margin of error. It's more correct to declare that this apparent shift COULD be, rather that it IS. Uncertainty is uncertainty, and it works both ways. Just saying 
And you're essentially saying the same thing here, no offense 
Same point here. The observed data may turn out to be statistically insignificant once you test the difference, but you can’t conclude that solely because the margins of error overlap 
As an example, say you do two polls completely identically, same sample sizes, same methodology. With two 1,000‑person polls, a move from 60% to 65% is roughly a 2.4‑sigma shift. Under standard assumptions, if there were truly no change, seeing a swing that big in the positive direction would happen less than 1% of the time. So it is highly likely that the true change is positive; the main uncertainty is how big the increase really is.
A key point is that if the sample mean is 60% with a 3% margin of error, that does not mean the true approval rate is equally plausible anywhere from 57% to 63%. Under the usual polling assumptions, values closer to 60% are more plausible than values at the edges like 57% or 63%
