2024 NJC P2 Q9

A factory manufactures cans and bottles of iced tea. Machine A is used to fill the cans with iced tea and Machine B is used to fill the bottles with iced tea. Machine A is set to fill each can with \(300\) millilitres (ml) of iced tea. A random sample of \(60\) filled cans of iced tea was taken and the volume, \(x\) ml, of iced tea in each can was measured. The following summarised data was obtained.

\({\sum(x - 300) = -112.8} \qquad \sum(x - 300)^2 = 4532.87\)

1. Defining clearly any symbols you use, test at the \(8\%\) level of significance, whether the mean volume of iced tea per can is \(300\) ml.[6]
2. Explain in the context of the question, the meaning of ‘at the \(8\%\) level of significance’.[1]

3. The manager of the factory claims that the mean volume of iced tea that Machine B fills per bottle with is at least \(500\) ml. It is found that the volumes of iced tea in the bottles filled by Machine B follow a normal distribution with standard deviation \(5\) ml. A random sample of \(35\) filled bottles was taken and a test for the validity of the manager’s claim was carried out at the \(4\%\) level of significance. Find the critical region for this test, correct to 1 decimal place.

   [3]