Some private undergraduate schools meet the parental expected contribution, from what I think. Well is there any Medical Schools that really do this? Or are you stuck paying there tuition of $35,000 no matter what? This is not taking into consideration of scholarships.
Yes they do! They are usually more narrow in scope: maybe 20 - 20 students in a class of 100 will have some sort of package. About 5 may get some sort of almost-free ride. The other students have to apply all over the place for scholarships and grants.
Then some schools (I'm not sure how many) already build the package into your tuition. I know that Hopkins and Yale do this, and I believe Emory does it at some level also. Most state schools also have this form of tuition aid for their in-state matriculants: the Texas systems schools, California schools (which is why their tuition is so low in the first place), and a few other random state schools: OSU, MSU.
In the end, it will depend on where you want to go, and where you apply. I suggest researching each school personally.