Negotiate. There have been many studies about how women do not fight for a salary, they just accept the first offer.
15k is a big leeway, tho. I wonder if they had some wiggle room in the 5-10k range.
Always ask. They can say no, or work with you.
A few years ago, DH was looking for a new job and had decided NOT to go with company Q, so if they called with an offer, for me to say no. I, out of curiosity, asked them what the offer entailed when they did call, and it was higher and better than DH and I expected.
I did NOT say no, but I tabled it. DH then used that offer to counter a job that he DID want, and he got the same amount at a beter job.
His boss, who became a friend, asked months later if he had really been offered x amount at company Q, or was he bluffing. When boss found out that DH's wife (me) had done the deal, boss was duly impressed because he said women do not usually negotiate, let alone the wives!!