4. The volume of milk, \(M\) litres, in cartons produced by a dairy, has distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(\mu\) and \(\sigma\) are unknown. A random sample of 12 cartons is taken and the volume of milk in each carton is measured ( \(M _ { 1 } , M _ { 2 } , \ldots , M _ { 12 }\) ). A statistic \(X\) is based on this sample.

1. Explain what is meant by "a random sample" in this case.
2. State the population in this case.
3. Write down the distribution of \(\frac { M _ { 12 } - \mu } { \sigma }\)
4. Explain what you understand by the sampling distribution of \(X\).
5. State, giving a reason, which of the following is not a statistic based on this sample.
   (I) \(3 M _ { 1 } + \frac { 2 M _ { 11 } } { 6 }\)
   (II) \(\sum _ { i = 1 } ^ { 12 } \left( \frac { M _ { i } - \mu } { \sigma } \right) ^ { 2 }\)
   (III) \(\sum _ { i = 1 } ^ { 12 } \left( 2 M _ { i } - 3 \right)\)