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I have a dependent variable that is % of students graduating, but the explanatory variables are all absolute numbers, such as number of applicants and college fees. How would I interpret a coefficient of say 0.0575?

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For every extra applicant you predict 0.0575 percentage points increase in graduates, ie from 52 to 52.0575 percent. Similarly for every extra currency unit(euro, dollar, Swiss franc, ...) you predict the same increment in %.

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Adding to mdewey's answer, you do still have to check that the assumptions of regression are roughly met. When the DV is bounded, there can be problems with the normality of the residuals and there can be floor or ceiling effects.

Depending on the school you are looking at, either floor or ceiling problems could exist.

You might want to investigate beta regression, which works with bounded DVs (you'd have to convert to proportion, but that's easy).

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