This feels incredibly basic but I can't seem to find the answer!
Let's say I knew the pass rate for an exam in a huge population of students is 50% (let's say millions of students).
A new class of N=30 students take the test and 60% of the class pass. Another new class of N=5000 students take the test and 55% pass.
The test is binary, only pass or fail with no scores.
Is there a statistically sound way of "adjusting" both these classes towards the population average?
For instance, for the first class - although 60% passed there's only N=30, so this is adjusted towards the population mean by a lot (let's say 51%?)
The second class has a 55% pass rate but given there's N=5000 there will be only a small adjustment (maybe 54.9%)?
This makes sense in my head but I can't seem to find the right statistical approach!
The reason I want to do this is, if we found out there was an unmarked test from one student from both these new classes - what would we predict their chances of passing the test to be?