Final answer:
The minimum cardinality for 'students completing theses' at Penn State would be one for students in programs requiring a thesis, and zero for students in programs where a thesis is not required.
Step-by-step explanation:
The minimum cardinality for the relationship type 'students completing theses' for a student at Penn State in general implies the least number of theses a student might be associated with. In the context of an academic institution like Penn State, it's typically expected that a student will complete at least one thesis if they are in a program that requires it. Therefore, the minimum cardinality would be one, assuming that every student in such a program must complete a thesis to graduate. However, for students not in programs requiring a thesis, the minimum cardinality would be zero, as these students would not need to complete a thesis at all.