Two standard errors

Topic: Confidence intervals · more questions on this topic
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Let X_1,\ldots,X_n be independent and \operatorname{N}(\mu,1). Find the exact coverage of the interval \overline X\pm 2/\sqrt n, and the multiplier c for which \overline X\pm c/\sqrt n has coverage exactly 99%.

Worked answer (click to show)

\sqrt n(\overline X-\mu) is standard normal, so

\mathbb{P}\bigl(\overline X-2/\sqrt n<\mu<\overline X+2/\sqrt n\bigr)=\mathbb{P}(-2<Z<2)=2\Phi(2)-1=0.9545.

For exactly 99%, 2\Phi(c)-1=0.99, so \Phi(c)=0.995 and c=z_{0.995}=2.576.