2024 MI H1 Q13

A fruit juice company claims that the cups of juice dispensed from their vending machines have a mean volume of \(275\) ml. A quality control officer collects a random sample of \(40\) cups of juice from various vending machines. The volumes, \(x\), in ml, are summarised by

\[\sum (x-275) = -48, \quad \sum (x-275)^2 = 539.95\]

1. Suggest why, in this context, the data is summarised in terms of \((x-275)\) rather than \(x\).[1]
2. Find the unbiased estimates of the population mean and variance.[3]
3. Test at the \(5\)% level of significance whether the claim of the fruit juice company is valid.[4]

The company makes some improvement to their vending machines and now decides to take a new random sample of \(50\) cups of juice to investigate the situation. The volumes of juice in a cup are now assumed to have a normal distribution with standard deviation \(3.5\) ml. The mean volume of juice in this sample is \(k\) ml. A test at the \(4\)% significance level gives insufficient evidence that the population mean volume of juice is less than \(275\) ml.

4. Find the range of possible values of \(k\).[4]