I have a typical dataset: sample A (method A) which has 200 samples and 25 failures. sample B (method B) which has 240 samples and 0-10 failures. Note that 0 failures is possible.
Null hypothesis is that the proportions are the same with 5 % significance level. Alternative hypothesis is that method B is the better and reduces failures. I want to reject the hypothesis. This is a one-tailed situation. I planned to use the basic z-test for comparing proportions.
Many Stats books say, however,
...there should be at least 10 failures and successes in both samples if one wants to use the z-test. This condition is obviously not fulfilled here because of sample B. So z-test can't be used, right?
Q1: How to bypass this restriction of minimum number of failures and successes?
Q2: What alternative test should I use to reject the null hypothesis?
(Using Numpy/Scipy or even MatLab is totally fine here. The data will be processed with a computer.)
Bonus question: What is the mathematical reason for the minimum requirement? If sample A has 10 failures and sample B has 10 failures we can reject the null hypothesis. But if sample B has even less failures we can't. This feels unintuitive.