Likely Republican voters overwhelmingly believe the allegations against Mr. Moore are false, according to the CBS poll released Sunday. Forty three percent of those polled said they would vote for Jones while 49 percent will opt for his Republican rival.
The margin of error is 4.5%.
Among all likely voters, almost two out of ten (18%) think the allegations, whether true or false, are not serious. Moore vehemently denies the allegations.
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The Kentucky Republican backed Luther Strange, Moore's opponent in the GOP primary, then called for Moore to quit the general election race after several women last month accused him of sexual misconduct or behavior.
The poll shows seven out of 10 Republicans (71%) think the allegations against Moore are false.
The survey - published by CBS News over the weekend - shows Moore has a six point lead over Democrat Doug Jones ahead of the December 12th election. Less than half - 48 percent - said he's the best candidate for the job.
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Saturday's Washington Post poll shows the Alabama electorate is divided on the validity of the allegations against Moore, and that some voters still believe those claims are baseless. Change Research, in an online survey conducted Sunday and Monday, found that Moore led Jones 47 percent to 42 percent.
After McConnell said Moore should step aside, Moore tweeted that it was McConnell who should bow out of politics, saying he "has failed conservatives and must be replaced".
The sample of 739 likely voters upon which the poll was based included 38 percent who self-identified as Republicans, 31 percent who identified as Democrats, and 27 percent who identified as independents.
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The accusations against Moore are much more serious than those against Trump, and, because of his long history as a divisive figure in Alabama, he was already facing a possibly-competitive race before the allegations came out. No new allegations of sexual misconduct have emerged since around November 15; just this week Moore returned to the campaign trail, after laying low for 11 days.