How do I convince my parents to let me re-do are attic and move in as a bed room?

So my sister is 12 and is sharing a room with my youngest sister (8) so obviously she wants her own bedroom. My parents do not share a room and refuse to share. So she has come up with the idea to move into the attic. Our attic is an almost finished attic, there is a floor and a roof, but there is not heating or air conditioning. However there is another unfinished section to are attic that is small, my sister would like to move all of the things in are attic into that small space. Are family is having a garage sale this summer so we have an opportunity to get rid of all the unwanted/needed things. The things that don't sell we can donate. She wants to completely re-do it, paint, finish, etc. But there are some more expensive things to do like heating and air. How should she try and convince are parents to let her re-do and move into are attic?