2024 VJC H1 Q12

A company bought ropes from rope Manufacturer \(A\). The manufacturer states that the mean length of rope produced is \(30\) cm. The company's manager claims that the mean length is greater than \(30\) cm. To test this claim, a random sample of \(50\) ropes are measured. The lengths of the rope, \(x\) cm, are summarised by

\[\sum x = 1505, \quad \sum x^2 = 45325.\]

1. Find the unbiased estimates of the population mean and variance.[3]
2. Test at the \(10\%\) significance level whether the manager's claim is supported by the data.[4]
3. State, with a reason, whether it is necessary to assume that the lengths of the rope are distributed normally for the test to be valid.[1]

The company also bought ropes from Manufacturer \(B\). The lengths of the rope from Manufacturer \(B\) are known to have a normal distribution with population variance \(1\) cm\({}^2\). A random sample of \(50\) ropes is obtained and the mean length is \(k\) cm. A test at the \(10\%\) significance level supports the manager's claim that the mean length of the ropes is greater than \(30\) cm.

4. Find the range of possible values of \(k\).[3]