I have a question as follows:
Assuming the null hypothesis is true for all cases. If we perform multiple pairwise comparisons at the $\alpha = 0.05$ level, what is the probability of making type I error with AT LEAST ONE of the tests?
The correct answer is greater than 0.05, and I don't know why. Thanks in advance!
I may have misinterpreted the above question--please let me know if so! I do not have much of a statistical background.
The probability of not making said error in each trial is $1-.05=.95$. So over $n$ trials, you have a probability of $.95^n$ of the error never occurring, and the probability of it occurring at least once (this is the complement of the situation where the error never occurs) is $1-.95^n$.
But since .95 is less than 1, when $n>1$, we have $.95^n<.95$, so $1-.95^n>1-.95=.05$ .