 Solution

The sample mean $\overline{X}$ has normal distribution $N(\mu ,1/n)$ because the observations are independent with known variance 1. The pivotal quantity $Z=\sqrt{n}(\overline{X}-\mu )$ has the standard normal distribution. Consequentlyand inversion gives the confidence intervalReplacing 1.96 by the exact $0.975$ normal quantile gives exact 95% coverage; the printed numerical value gives the usual rounded interval. No variance estimation or Student distribution is needed.