Below are some schools that I would consider likely or extremely likely for someone with your profile. I suspect it’s fairly easy to transition between engineering to business to other fields at most of these schools, but that is something you would want to investigate further to be certain.
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Binghamton (NY): About 14k undergrads. ABET-accredited in multiple fields, including biomedical engineering. It’s a very strong school overall, with particularly popular bio and neuroscience majors.
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Duquesne (PA ): About 5100 undergrads in Pittsburgh…so if you’re visiting Pitt or CMU, this is an easy add-on. It’s ABET-accredited in biomedical engineering, has a lot of resources devoted to health fields.
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Fairfield (CT): About 4800 undergrads. ABET-accredited in multiple fields, including biomedical engineering. It offers a public health major and has historically had ties to investment banking.
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Lehigh (PA ): About 5600 undergrads. ABET-accredited in bioengineering as well as multiple other fields. Has all your other areas of interest covered.
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Loyola Chicago (IL): About 12k undergrads. It’s ABET-accredited only in general engineering, but one of its three specializations within that major is biomedical engineering. Offers prehealth and has a popular finance major.
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Rutgers - New Brunswick (NJ): About 36k undergrads and strong in many areas of your interest. I have no knowledge of how challenging it is to switch between departments here, but perhaps @DadOfJerseyGirl knows?
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Santa Clara (CA): About 6100 undergrads, ABET-accredited in multiple engineering fields, majors in public health and a popular finance major.
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Southern Methodist (TX): About 7100 undergrads, ABET-accredited in several fields, and its business school is very well respected. Transferring into business is probably harder here than at the other schools.
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U. of Miami (FL): About 13k undergrads, ABET-accredited in multiple fields, including biomedical engineering. Has all your other areas of interest covered.