The market would take care of the standards I suppose. which of course is better than government standards anyway.
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The government does not set standards for private schools, colleges and universities AT ALL.
These are set by accrediting agancies, such as SACS (Southern Association of College and Schools) There are five of these. As a rule, if a school is not approved by SACS or one of the others (Western, North Central, New England),then the feds will not loan money to students to attend said school.
BoB Jones University is unaccredited by any of the regional accreditation bodies, but their graduates generally work in church-related positions. Whether Bob Jones is better or worse than Punxatawney State is anybody's guess. I don;t think I would hire a BJU grad for a pharmacy lab, but I don;t think they have degrees in hard sciences.
You are wrong to say that the government has standards. It has none, other than to rely on the regional bodies. It is typical of your minddset that you would think that "the market" would decide how qualified people are better than the government, because this is largely untrue. If you graduate from an unaccredited school, you are unlikely to be hired at all. At least by anyone who actually understands how education works in this country.
It is, of course, possible that an accounting major from Numbskull Private College of Numbers and Witchcraft might actually buy an accounting book and be the world's best accountant. Degrees are one thing and actual proficiency is another. But a degree in accounting would generally mean that the graduate had a better idea of what accounting is all about than someone who attended a place that merely called itself a college. The United States University of America has a very impressive-sounding name, but they will sell you any degree you might desire for a mere fraction of what a real university might charge, and you would not have to take a single class.