11 Casey and Riley attend a large school. They are discussing the music preferences of the students at their school. Casey believes that the favourite band of 75\% of the students is Blue Rocking. Riley believes that the true figure is greater than 75\%. They plan to carry out a hypothesis test at the \(5 \%\) significance level, using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.75\) and \(\mathrm { H } _ { 1 } : p > 0.75\). They choose a random sample of 60 students from the school, and note the number, \(X\), who say that their favourite band is Blue Rocking. They find that \(X = 50\).

1. Assuming the null hypothesis to be true, Riley correctly calculates that \(\mathrm { P } ( X = 50 ) = 0.0407\), correct to 3 significant figures. Riley says that, because this value is less than 0.05 , the null hypothesis should be rejected.
   Explain why this statement is incorrect.
2. Carry out the test.
3. 1. State which mathematical model is used in the calculation in part (b), including the value(s) of any parameter(s).
   2. The random sample was chosen without replacement. Explain whether this invalidates the model used in part (b).