2025 DHS P2 Q11

Mr Dough’s factory produces packets of flour for sale in local supermarkets packed in \(2\) sizes – Regular-sized packets and Large-sized packets. Automated machines weigh and pack the flour. A recent major power outage resulted in all his machines being needed to be recalibrated upon restarting.

Regular-sized packets are supposed to have a mean mass of \(1\) kg. Mr Dough suspects that his machines now pack Regular-sized packets that are underweight. He took a sample of his Regular-sized packets and measured their masses, \(x\) g (correct to nearest \(5\) g). The results are recorded as follows:

1. Determine the unbiased estimate of the population variance of the mass of Regular-sized packets.[1]
2. Explain whether Mr Dough should conduct a one-tail or a two-tail test.[1]
3. Carry out the test at \(5\)% level of significance and give your conclusion in context. Explain whether it is necessary to assume that the mass of Regular-sized packets of flour follow a normal distribution.[5]

Large-sized packets of flour are known to have a normal distribution with a mean mass of \(2\) kg. Mr Dough also suspects that his machines are now not packing the Large-sized packets according to the required mass correctly. A sample of \(15\) Large-sized packets of flour gave a total mass of \(30075\) g.

Assuming Large-sized packets of flour have a population variance of \(\sigma^2\) \(\mathrm{g}^2\), an appropriate test was carried out which gave a \(p\)-value of \(0.0529\).

4. Showing all necessary workings, prove that \(\sigma=10.0\) correct to \(3\) significant figures.[2]

Another packet of Large-sized packet of flour with mass \(h\) g is then added to the sample of \(15\). The combined sample of Large-sized packets resulted in a rejection of the claim that the mean mass of a Large-sized packet of flour is \(2\) kg at \(4\)% level of significance.

5. Using \(\sigma=10.0\), find the possible range of values of \(h\), giving your answer to the nearest gram.[3]