Ten intervals, jointly 95% confident

Question

You report confidence intervals for 10 quantities and want to be 95% confident that all ten intervals at once contain their true values.

Using the Bonferroni idea, what confidence level must each individual interval carry?

Accepted Answer

Why each must be stronger. Ten separate 95% intervals each miss 5% of the time, so the chance that at least one misses is well above 5%. To make the joint "all ten hold" statement 95% confident, each interval must be individually tighter. Divide the miss budget. Bonferroni allots each interval only \alpha/m of the total miss budget: 	ext{per-interval miss} = \frac{0.05}{10} = 0.005, so each interval must carry confidence 1 - 0.005 = 0.995 = 99.5\%. Check the guarantee. Each interval misses at most 0.5% of the time; summing the worst case over ten gives at most 10 	imes 0.005 = 0.05, so all …

Ten intervals, jointly 95% confident

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