2025 NJC P2 Q10

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The production manager of a food container manufacturing company wishes to take a random sample of a certain type of lunch box from the thousands produced one day at his factory, for quality control purposes. He wishes to check that the mean height of the lunch boxes is \(12.8\) cm.

1. State what it means for a sample to be random in this context.[1]
2. Explain why the manager should take a sample of at least \(30\) lunch boxes to carry out a hypothesis test.[1]

The heights, \(x\) cm, of a random sample of \(48\) lunch boxes are summarised as follows:\[\sum(x-13) = -3.6 \quad \text{and} \quad \sum(x-13)^2 = 12.02\]

3. Calculate the unbiased estimates of the population mean and variance of the height of the lunch boxes.[2]

The production manager claims that the mean height of the lunch boxes is \(12.8\) cm.

4. Test his claim at the \(10\%\) level of significance, stating the hypotheses clearly.[4]

In the hope of keeping the mean height of the lunch boxes not to exceed \(12.8\) cm, the production manager decides to replace the production line such that the population variance is reduced to \(0.04\) cm\(^2\). He then carries out a hypothesis test at \(2\%\) level of significance using another random sample of \(n\) lunch boxes.

5. Assuming that \(n\) is sufficiently large, find the critical region for this test in terms of \(n\).[4]