My stats textbook is telling me that you can use a Pearson's r with a dichotomous variable and a continuous dependent variable, which is known as a biserial correlation (or point-biserial if the nominal variable is discrete).
However, you can also conduct a t-test with such data, which made me wonder what the decision-making process behind choosing a biserial correlation over a t-test actually involves.
Looking at it logically, my guess would be that you should use a t-test in the case of a discrete dichotomous variable, because measuring an association between two discrete categories and an outcome does not make intuitive sense (i.e. "is the degree of genderness correlated with height?" or "is there an association between gender and height?").
If it were non-discrete (i.e. there is a background trend between categories), my guess would be that a biserial correlation and t-test are approximately equivalent, and you could use either. For example, if the dichotomous variable is "pass/fail on an exam" and the dependent is "hours studied", it would make more sense than the above example to see whether pass/fail is associated with the number of hours studied. In other words, because there is a degree of performance between pass and fail, the concept of an association makes more sense than "is the degree of genderness correlated with height"?
My questions are: 1. Am I correct in my above assumptions? 2. I heard that Pearson's r and an independent t-test are more or less mathematically equivalent in this context. Is this true?