Top 200 School to a Top 20 Grad school

Okay guys my ultimate goal is grad school and I can’t seem to find any answers to this question that aren’t over 10 years old. I’m an mech engineering major at a CC and transfer wise I’m looking more at USF honors college and UCF. These schools aren’t ranked crazy high or low but USF honors college has been noted for doing alot of research. These schools are rated on average to be about top 200. Will USF/UCF hurt my chances of getting into a grad program at a top 20 college compared to say UF? Or will I be treated the same as an applicant from UF given we both completed the same amount of research and internships? Thanks for the info.

For grad school, what you do at your top 200 college matters more than the fact that it’s “only” in the top 200. As long as it’s ABET accredited, of course.