 Solution

Independence and normality give $\overline{X}\sim N(\mu ,1/n)$. Consequently $Z=\sqrt{n}(\overline{X}-\mu )$ is standard normal, andRearranging the event gives $\mu >\overline{X}-z_{\alpha }/\sqrt{n}$. Thus the random interval $(\overline{X}-z_{\alpha }/\sqrt{n},\infty )$ covers the fixed parameter with probability $1-\alpha$. After observing the sample, its realized confidence interval isThe coverage refers to repeated samples, not to a posterior probability for the fixed parameter.