When you only care about one end of the interval

Question

A quality check finds a defect rate estimated at 1.0% with a standard error of 0.3%. The risk manager doesn't care how low the true rate might be, only how high it could plausibly be. They ask for a 95% upper confidence bound.

Construct the 95% one-sided upper bound, explain why it uses a different multiplier than a two-sided interval, and when one-sided bounds are the right tool.

Accepted Answer

The multiplier changes. A two-sided 95% interval splits the leftover 5% into two tails of 2.5% each, using z = 1.96. A one-sided 95% upper bound puts the entire 5% in one tail, using z = 1.645. Because you've stopped spending confidence protecting the lower side, the same 95% buys you a tighter bound on the side you care about. The bound. 	ext{upper bound} = \hat p + 1.645 	imes 	ext{SE} = 0.010 + 1.645 	imes 0.003 = 0.010 + 0.00494 \approx 0.0149. So with 95% confidence the true defect rate is no more than about 1.5%. Notice this is tighter than the upper end of the corresponding two-sid…

When you only care about one end of the interval

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