2 A headteacher models the number of children who bring a 'healthy' packed lunch to school on any day by the distribution $\mathrm { B } ( 150 , p )$. In the past, she has found that $p = \frac { 1 } { 3 }$. Following the opening of a fast food outlet near the school, she wishes to test, at the $1 \%$ significance level, whether the value of $p$ has decreased.\\
(i) State the null and alternative hypotheses for this test.\\

On a randomly chosen day she notes the number, $N$, of children who bring a 'healthy' packed lunch to school. She finds that $N = 36$ and then, assuming that the null hypothesis is true, she calculates that $\mathrm { P } ( N \leqslant 36 ) = 0.0084$.\\
(ii) State, with a reason, the conclusion that the headteacher should draw from the test.\\

(iii) According to the model, what is the largest number of children who might bring a packed lunch to school?\\

\hfill \mbox{\textit{CAIE S2 2018 Q2}}