This behavior is completely typical of the older child, and I can testify to this personally, being an older child. At school, you are struggling with being "cool" to your peers, you are trying to create your own persona, something different from how you are perceived at home by your parents and family, and, at least in my day, you are trying to impress your teachers. A younger sibling makes that quite difficult for any number of reasons. Younger siblings can say things about you or the family that you don't want your peers to know. They have a tendency to tease you in an embarrassing way. And a younger sibling who is perceived as a less than ideal student can harm your credibility with your teachers. I would say this happens to some degree with most siblings in the same school, although a few carry it to a greater extreme than others.