Five intervals, jointly 90% confident

Question

You report confidence intervals for 5 quantities and want to be 90% confident that all five at once contain their true values.

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

Accepted Answer

Set the joint budget. This time the joint miss budget is \alpha = 0.10 (since you want 90% joint confidence, not 95%). Bonferroni splits that budget across the five intervals. Divide. Each interval is allowed a miss chance of \frac{\alpha}{m} = \frac{0.10}{5} = 0.02, so each must carry confidence 1 - 0.02 = 0.98 = 98\%. Check. Each interval misses at most 2% of the time; over five intervals the worst-case total is 5 	imes 0.02 = 0.10, matching the 90% joint target. Note the flexibility. The numerator is whatever joint miss budget you choose. Loosening the joint target from 95% to 90% lets ea…

Five intervals, jointly 90% confident

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