Reverse a known-variance test for mean time and sample size

In a factory, the time in minutes for an employee to install an electronics component is a normally distributed continuous random variable $T$, with standard deviation 5.0 and ordinary expected value 38.0. After background music is introduced, a sample of $n$ components is taken and the mean time for randomly chosen employees is $\bar t$ minutes. A test at 5% determines whether the population mean installation time has been reduced.

(i) State appropriate hypotheses, defining any symbols.

(ii) Given $n=50$, state the set of values of $\bar t$ for which the test rejects $H_0$.

(iii) Suppose instead that $\bar t=37.1$ and $H_0$ is not rejected. Obtain an inequality involving $n$ and find the set of values that $n$ can take.