2025 HCI P2 Q8

A food producer claims that the mean mass of a can of beans it produces is \(425\) g. Following customer feedback, the production manager wishes to test if the mean mass of a can of beans is indeed \(425\) g.

The production manager took a random sample of size \(50\) and the mass of each can, in \(x\) g, is recorded and the results are shown below:

\[\sum x = 21209, \sum(x - 424.18)^2 = 522\]

1. State what it means for a sample to be random in this context.[1]
2. Find the unbiased estimates for the population mean and variance.[2]
3. State the hypotheses for the manager's test, defining any parameters you use. Carry out the test at the \(5\%\) level of significance, giving your conclusion in the context of the question.[5]

The production manager wishes to test whether the mean mass of a can of beans has increased using the alternative manufacturing process. He finds that the mean mass of \(55\) randomly chosen cans is \(426.5\) g. He carries out a hypothesis test at \(10\%\) level of significance.

4. Explain, with justification, how the population standard deviation of the mass of a can produced under the alternative process will affect the conclusion made by the production manager.[3]