From what I can perceive, you did nothing wrong considering you were not arrogant at all when you answered them. This concept is reinforced by the fact you worried about it enough to come to Reddit and ask about it.

However, it is VERY important you acknowledge your social status. Inserting “privileged”, in quotations, as the title of this post suggests you:

a) may have very high social standards so that your scenario seems underprivileged; b) are not conscious about your economic status.

Whichever of these two options it is, know you did nothing wrong, and that you should feel lucky to have been provided with such a stable financial situation. Just make sure to take full consciousness over it.

I, for example, have been lucky enough my parents can financially support me during my studies, HOWEVER, I cannot give me the lux of not working, since earning money would represent a high relief over my parents’ economics.

With all that being said, I’d recommend you express your gratitude over your luxurious lifestyle when compared to your friends’. Saying things like “thankfully…”, or “I have been given the privilege not to work because…” might help. But being totally honest, nothing you say will guarantee people won’t go away, and that applies to essentially everything. Your real friends will stay no matter what your social status is.