2026 EJC P2 Q9

A company manufactures protein bars. The manager claims that the mean mass of the protein bars is 50 grams. A random sample of 40 protein bars is taken, and their masses \(x\) grams, are summarised below.

\[\sum(x-50)=-28\quad\text{and}\quad\sum(x-50)^2=252.4\]

1. Calculate unbiased estimates of the population mean and variance of the mass of the protein bars.[2]
2. Test, at the 5% level of significance, whether the manager has overstated the mean mass of the protein bars.[4]
3. The manager now wants to determine at 5% level of significance whether the mean mass of a protein bar is different from 50 grams. Without doing a new test, explain how the conclusion will change.[1]

The manufacturer also produces yoghurt tubs and claims that the mean and variance of the yoghurt tubs are 500 grams and 8.24 grams\(^2\) respectively. It is assumed that the masses of these yoghurt tubs have a normal distribution. In a study, a random sample of \(n\) tubs was taken and the sample mean was 500.50 grams.

4. Find the least value of \(n\) such that the null hypothesis, \(\mathrm H_0:\mu=500\) would be rejected at the 5% level of significance against the alternative hypothesis, \(\mathrm H_1:\mu>500\).[3]
5. State the meaning of the \(p\)-value of this test in the context of this question.[1]