How weak is the joint guarantee from six naive 95% intervals?

Question

You built 6 ordinary confidence intervals, each at the usual 95% level, and made no multiplicity correction. A colleague asks how confident you can be that all six at once contain their true values.

Using the Bonferroni idea, what joint confidence can you guarantee for all six holding together?

Accepted Answer

Run the logic backwards. Usually Bonferroni tells you how strong each interval must be for a target joint confidence. Here we do the reverse: given six plain 95% intervals, what joint guarantee falls out? Add up the miss budgets. Each 95% interval misses with probability 0.05. By the union bound, the chance that at least one of the six misses is at most the sum of the individual miss chances: P(	ext{any miss}) \le 6 	imes 0.05 = 0.30. So the chance that all six hold together is at least 1 - 0.30 = 0.70 = 70\%. Read it carefully. This is a guaranteed floor, not the exact number. The true joi…

How weak is the joint guarantee from six naive 95% intervals?

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