2023 TJC P2 Q7

A vending machine is set to dispense cups of drink of \(\beta \) ml. It is known that the volume of drink dispensed per cup is distributed normally with standard deviation \(6.30\) ml. The company does a random check by taking a sample of \(8\) cups of drink to check if the volume of drink dispensed for each cup is less than \(\beta \) ml. He finds the volume of drink, in ml, are as follows:

\(190\), \(190\), \(192\), \(194\), \(195\), \(200\), \(203\), \(207\)

1. Find the mean of the sample of these \(8\) cups of drink and the unbiased estimate of the population variance for the volume of drink dispensed.[2]
2. State suitable hypotheses for the test, defining any symbols that you use.[2]
3. Given that the company concludes that the vending machine is dispensing less than \(\beta \) ml at \(5%\) level of significance, find the range of possible values of \(\beta \).[3]

The company repairs the machine and sets it to dispense a bigger cup of drink of \(210\) ml. He decides to perform a hypothesis test on a random sample of \(40\) cups of drink to find out if the machine is dispensing the correct volume of drink.

4. Explain why the company takes a sample of \(40\) cups of drink for this test when he only took a sample of \(8\) cups of drink in his earlier test.[2]