2025 SAJC P2 Q11

A beverage company produces bottled orange juice, claiming that the mean vitamin \(C\) content in each \(250\) ml bottle is \(50\) mg. It is known that the vitamin \(C\) content is normally distributed with a standard deviation of \(2\) mg

Due to feedback from consumers, the quality control manager wants to test, at the \(4\)% level of significance, whether the mean vitamin \(C\) content is, in fact, \(50\) mg. He randomly samples \(12\) bottles and measures the vitamin \(C\) content.

1. State the hypotheses and find the critical region for this test.[3]

Recently, the company launched a new "Vitamin Boost" orange juice line, advertised to contain an average of \(60\) mg of vitamin \(C\) content per \(250\) ml bottle. After some complaints that the juice is too sour, the manager suspects that the vitamin \(C\) content might be higher than advertised. He decides to conduct a hypothesis test to investigate. This time, he randomly samples \(30\) bottles from the new juice line.

(ii) Explain why the manager takes a sample of \(30\) bottles for this test when he only took a sample of \(12\) bottles in his earlier test.[2]

The vitamin \(C\) content, \(y\) mg, in the sample of \(30\) bottles is summarized as follows:

\[\sum y = 1865, \quad \sum y^2 = 116275\]

(iii) Calculate unbiased estimates of the population mean and variance for the vitamin \(C\) content in the bottles of the new juice line.[2]

(iv) At the \(5\)% level of significance, test whether the mean vitamin \(C\) content in the bottles of the new juice line is greater than \(60\) mg. Clearly state your hypotheses and define any symbols you use.[4]