The question I would ask is: "Can Norman describe his belief as knowledge?" We can do so in this scenario, but only because of our position as an omniscient outsider. Norman does not have that sort of privileged information.

The relevant question for Norman is what he ought to believe on the basis of the evidence he has. He's got his clairvoyant feelings and some other conflicting external evidence. He can give credence to one or both of those and construct a belief appropriately, which he'll likely do on the basis of other beliefs. In this case, the most reasonable belief in absence of other supporting beliefs (supposing Norman values his clairvoyance) is perhaps a middle position: the President is either in New York or the president is in Florida.

On the other hand, if Norman possesses an evidence-based belief that clairvoyance is impossible, he might dismiss his clairvoyant feelings and conclude that the president is in Florida. Norman would possess a reasonable belief in this case, even if his manifest clairvoyance was in fact accurate. If Norman were to gain additional evidence that the president was in fact in New York (such as a first-hand sighting), he'd be reasonable in revising that belief and in giving more credence to further clairvoyant experiences.

We can categorize Norman's belief as knowledge or non-knowledge in all of these scenarios based on privileged information, but Norman cannot, and Norman's case represents the baseline we should consider when assessing our own beliefs. We can't say if our beliefs amount to knowledge since we aren't omniscient, but we can say if they're reasonable.