N1986 P2 Q10 [Modified]

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.