Calculate theoretical test errors and supplied-tail boundaries

Observations are iid normal with known population variance \(\sigma^2=78\). Each sample has \(n=78\). The null mean is \(\mu_0=6\). Use the exact normal sampling model for \(\overline x\)

A two-tail decision rule prescribes null probability \(\frac{1}{10}\) below its lower boundary \(L\) and \(\frac{2}{25}\) above its upper boundary \(U\). Reject outside \([L,U]\). Find both boundaries, the total rejection probability and the probability of non-rejection under \(H_0\). These tail probabilities are given separately; do not halve either one. The actual rule uses unrounded boundaries