2025 TMJC P2 Q7

A coffee shop owner claims that its signature coffee has a mean caffeine content of \(120\) mg per cup. To investigate this claim, a random sample of \(60\) cups of coffee is selected and the caffeine content, \(x\) mg per cup, is summarised as follows.\[\sum x = 6960, \quad \sum x^2 = 822465\]

1. Calculate unbiased estimates of the population mean and variance for the caffeine content per cup.[2]

2. Test, at the \(5\%\) level of significance, whether the coffee shop owner’s claim is supported by the data. You should state your hypotheses and define any symbols that you use.

   [4]

The coffee shop also sells premium coffee, where its caffeine content is normally distributed with population variance \(200\) mg\({}^2\). The coffee shop owner now claims that its premium coffee has a mean caffeine content less than \(120\) mg per cup. Another large random sample of \(n\) cups of premium coffee is selected and the sample mean caffeine content per cup is found to be \(116.6\) mg. A test is carried out at the \(2\frac{1}{2}\%\) level of significance and the result supports the owner’s claim.

3. Find the set of values that \(n\) can take.[4]