I need help explaining, and citing basic statistics texts, papers or other references, why it is generally incorrect to use the margin of error (MOE) statistic reported in polling to naively declare a statistical tie.
An example: Candidate A leads Candidate B in a poll, $39 - 31$ percent, $4.5 \%$ margin-of-error for $500$ surveyed voters.
My friend reasons like so:
Because of the intricacies of statistical modeling, the margin of error means that A's true support could be as low as 34.5 percent and B's could be as high as 35.5 percent. Therefore, A and B are actually in a statistical dead heat.
All help appreciated in clearly articulating the flaw my friend's reasoning. I've tried to explain that it is incorrect to naively reject the hypothesis "A leads B" if $p_A-p_B < 2MOE$.