Do you believe Jack's apology to be sincere? Why or why not?

No, I do not believe Jack was sincere. He was merely putting on a show for his group and trying to redeem himself for his mistake. His apology was seen as courageous by the other hunters, he had put himself right.

The buzz from the hunters was one of admiration at this handsome behavior. Clearly they were of the opinion that Jack had done the decent thing, had put himself in the right by his generous apology and Ralph, obscurely, in the wrong. They waited for an appropriately decent answer.