2024 RI H1 Q11

A company produces cooking oil in \(1\)-litre bottles. The manager suspects that too much oil is being dispensed into the \(1\)-litre bottles. To check the accuracy of the dispensing machine, the company manager selected a random sample of \(100\) bottles of cooking oil and their volumes, \(x\) litres, were measured. The results are summarised as follows.\n\[\sum x = 102.9 \text{ and}\hspace{0.5em} \sum x^2 = 110.08\]

1. Find unbiased estimates of the population mean and variance of the volume of cooking oil in a \(1\)-litre bottle.[3]
2. Test, at the \(5\)% level of significance, whether the manager's suspicion is justified.[5]

Independent of the results in part (b), the dispensing machine was re-calibrated. A random sample of \(80\) bottles of cooking oil dispensed from the newly calibrated machine was taken and the mean volume was found to be \(1.042\) litres. Using the unbiased estimate of the population variance found in part (a), a significance test of the null hypothesis \(\mu = 1\) against the alternative hypothesis \(\mu \neq 1\) was carried out at the \(\alpha\)% level of significance. The test showed that there is insufficient evidence to reject the null hypothesis.

3. State in context what \(\mu\) represents and find the set of values of \(\alpha\).[4]