What’s the right version of these two?
Half of the students doesn’t bother to show up.
Half of the students don’t bother to show up.
Or are both right?
The related question Is two-thirds plural? discusses some relevant concepts, but the answer is very general, and it's not clear how to apply it here since "half" is a singular noun, while "two" is a plural noun.
Barrie England's answer there says that in "2/3 of the pizza was eaten," singular agreement is appropriate because "the emphasis is likely to be on the amount of pizza eaten, and not on the number of individual thirds. In contrast, in [2/3 of the visitors were men] the emphasis is on the number of visitors who were men, so plural concord ... is required."
What does it mean to put "emphasis" on "the amount" vs. "the number"? If "half of the students" is just an approximate or uncertain percentage (like "between 51% and 55%"), rather than an exact count, how do I know whether it's appropriate to emphasize the amount or the number of students?