I'm thinking of leaving my mom and my sister to go live out on my own.
Are you trying to change your life? Why do you want to move out?
By moving out I wouldn't have to tolerate the arrogance of my sister.
While it is expected that people live away from their parents when they grow up, leaving because of your poor relationship with your sister isn't a good reason. Rather than avoid your sister by moving out, try to examine what is causing so much strain in your relationship with her. Bringing up the issue and being more tolerant of your sister may mean that you won't have to move out unnecessarily.
In the end talking to you made me see that I'm acting like a spoiled brat, I'm going to apologize to her.
That sounds like a good idea. Conflict is okay, but it's good to talk things through and apologize when you need to.