I took some stats in college, and this is a tip-of-the-tongue problem that I just can't think how to search. If there is an answer to this already, please point me in the right direction.
My problem fits well in an analogy of auto manufacturing:
There are hundreds of populations (different component) with varying population sizes (say only a few hundred for the wheels for a supercar, to hundreds of thousands for the gas tank that is used in every model). We would like to classify the parts into defective or not-defective. The defect rate will be low, varying maybe from 0%-10% depending on the part.
Knowing this, how many parts should you sample before you can conclude
1. x% probability that this component population is defect-free? or
2. x% probability that this component's population defect rate falls within some (small) range?
What distribution should be used?
Thanks for all your help!