Since she is only ten years old, she is probably jealous of you and how you may have certain privileges that because of her age she can't have yet. Is a way for her to show some individuality. At the same time, she is really missing you. The sister that used to play with her and had more time for her but now that you are 13 you have different interest.
Spend a little more time with her and doing things she wants to do no matter how silly they are. She will not accept it at first and may reject it but just keep at it.
My younger sister and I always hated each other when we were younger. As you get older your maturity levels and interests will start to be similar and you'll discover you are friends. Give her time to catch up with you and be patient with her.
It's simple,take a few days a week and pamper her,paint her nails,play dress up,it's guaranteed to work.If you really care about how she feels about you,then get on her level.
You are not giving much to work on. Like how old and if you live in the same house and what does she say's that gets her to hate you.