Test a minimum string length and interpret a strict boundary

A company sells balls of string. A manager claims average length is at least 300 m. A random sample of 100 balls gives $\sum(x-300)=-60$ and $\sum(x-300)^2=1240$.

(i) Find unbiased estimates of the population mean and variance.

(ii) Test at 5% whether the manager’s claim is valid.

The process is improved and the new population variance is known to be 12.1 m². A new random sample of 100 balls has mean $k$ m. A test at 10% indicates that the manager’s claim is valid for this process.

(iii) Find the least possible $k$, giving your answer to 2 decimal places.