# Calculate fraction of multiple tests for which the null hypothesis was rejected

Consider a test (e.g., a permutation test) that tests, at the individual level, if a binary event occured or not.

• A paper that I was recently reading, for example, tested whether or not a pair of two students cheated in a multiple-choice exam (plagiarism).
• They then run such a test for every pair of students, and reject the null hypothesis for $$10\%$$ of all pairs.
• They then concluded that $$10\%$$ of all pairs cheated.

Now, I am wondering if this conclusion makes sense. For each single test, there is a certain probability that the authors identified a false positive (they chose a $$5\%$$ significance level). Shouldn't this imply that $$5\%$$ of the pairs for which the null hypothesis (no cheating) was rejected should be false positives? Can I then conclude that the share of cheaters should be by $$5\%$$ lower.

More generally, I wonder whether it ever makes sense to calculate the fraction for which the $$H_0$$ is rejected. Shouldn't I, at least, account for multiple testing?