2025 ASRJC P2 Q12

Flour is packed in bags of \(1.5\) kg, as stated on the packaging, by two separate machines \(A\) and \(B\) in a factory. During a quality control inspection, the production manager selects a random sample of \(40\) bags filled by machine \(A\) for checking. The masses, \(x\) kg, of these \(40\) bags are recorded as follows.

\[\sum x = 60.32 \quad \sum x^2 = 90.993\]

1. State what it means for a sample to be random in this context.[1]
2. Find unbiased estimates of the population mean and variance of the mass of the filled bags.[2]
3. Test, at the \(5\)% level of significance, whether machine \(A\) produces overweight bags. You should state your hypotheses and define any symbols you use.[5]
4. Explain in the context of the question, the meaning of '\(p\)-value'.[1]
5. The manager also wants to test whether the mean mass of the bags filled by machine \(B\) differs from \(1.5\) kg. It is known that the mass of the bags filled by machine \(B\) is normally distributed with variance \(0.00198\) kg\({}^2\). A sample of size \(n\) is randomly selected and its mean is found to be \(1.489\) kg. Find the range of values that \(n\) can take if there is sufficient evidence to conclude that the mean mass of the filled bags differs from \(1.5\) kg, at the \(5\)% level of significance.[3]