Full cost at these schools includes tuition, room, and board (meals.) NYU and Northwestern are around $100,000 in US dollars. On top of that you will have travel costs, incidentals, health insurance, easily another $10K per year. New York city is especially expensive to live in as well, so Northwestern might be a better choice. So, if you have your app ready to go for NU, I would stick with that.
You can run each school’s NPC to see what a cost estimate is, but often, NPCs aren’t accurate for internationals (look on each school’s website for details.) Work with your parents to make sure the NPC inputs are accurate and complete, and in US dollars.
Northwestern NPC: Net Price Calculator
NYU NPC: Net Price Calculator
This is the correct assumption.
Do you have affordable college options elsewhere? EU, UK, Australia, etc? Has your HS sent students to the US to highly selective schools like NYU or NU?
Are you interested in any of the other NYU locations?