1986 TYS

The length of string in the balls of string made by a particular manufacturer has mean $\mu $m and variance $27.4$ m$^{2}$.

The manufacturer claims that $\mu =300$. A random sample of $100$ balls of string is taken and the sample sum (total length) is found to be $S$m.

1. If $S=29920$, test whether this provides significant evidence, at the $3%$ level, that the manufacturer’s claim overstates the value of $\mu $.
2. Find the set of values of $S$ (to one decimal place) for which the test concludes that the manufacturer’s claim overstates the value of $\mu $ at the $5%$ significance level.

State, giving a reason, whether it is necessary for the length of string in the balls of string to have a normal distribution for the test to be valid.