4 It is given that \(Y \sim \operatorname { Po } ( \lambda )\), where \(\lambda \neq 0\), and that \(\mathrm { P } ( Y = 4 ) = \mathrm { P } ( Y = 5 )\). Write down an equation for \(\lambda\). Hence find the value of \(\lambda\) and the corresponding value of \(\mathrm { P } ( Y = 5 )\).
\(555 \%\) of the pupils in a large school are girls. A member of the student council claims that the probability that a girl rather than a boy becomes Head Student is greater than 0.55 . As evidence for his claim he says that 6 of the last 8 Head Students have been girls.

1. Use an exact binomial distribution to test the claim at the \(10 \%\) significance level.
2. A statistics teacher says that considering only the last 8 Head Students may not be satisfactory. Explain what needs to be assumed about the data for the test to be valid.