2024 ACJC H1 Q13

The manager of a firm that produces tyres claims that the mean lifetime of the tyres (thousand kilometres) is \(18.5\). He selects a random sample of \(200\) tyres and the lifetimes of these tyres, \(x\) (thousand kilometres) are summarised as follows:

\[\sum(x - 18.5) = 75 \quad \sum(x - 18.5)^2 = 1968.04\]

1. Test at the \(5\%\) significance level whether the manager's claim is supported by the data.[6]
2. Explain what is meant by 'at the \(5\%\) significance level' in the context of the question.[1]

A new manufacturing process is developed to increase the lifetime of tyres (in thousands of kilometres). Over a period of time, it was found that the mean lifetime of tyres produced by the new process is \(\mu_0\). The lifetime of tyres is known to follow a normal distribution with variance \(9.5\). A random sample of \(400\) tyres made by the new process is examined and found to have a mean lifetime of \(20\).

3. Find the set of values of \(\mu_0\) in order that the firm may claim that the new process is effective at \(1\%\) level of significance.[4]