2025 TGM P2 Q8

At a canning factory, cans are filled with tomato puree. The machine which fills the cans is set so that the mass of the can has a mean \(822\text{g}\). After the machine is recalibrated, a quality control officer wishes to check whether the mean mass has changed. A random sample of \(45\) cans of tomato puree is selected and the mass of the puree in each can is recorded. The sample mean mass is \(\bar{x}\) grams and the sample variance is \(13.5\text{ grams}^2\).

1. Given that \(\bar{x} = 820.9\), carry out a test at the \(1\%\) level of significance to investigate whether the mean mass has changed. State, giving a reason, whether it is necessary for the masses to have a normal distribution for the test to be valid.[5]
2. Use an algebraic method to calculate the range of values of \(\bar{x}\), giving your answer correct to \(2\) decimal places, for which the results of the test at the \(1\%\) level of significance would reject the null hypothesis.[3]