1 Coastal wildlife wardens are monitoring populations of herring gulls. Herring gulls usually lay 3 eggs per nest and the wardens wish to model the number of eggs per nest that hatch. They assume that the situation can be modelled by the binomial distribution \(\mathrm { B } ( 3 , p )\) where \(p\) is the probability that an egg hatches. A random sample of 80 nests each containing 3 eggs has been observed with the following results.

1. Initially it is assumed that the value of \(p\) is \(\frac { 1 } { 2 }\). Test at the \(5 \%\) level of significance whether it is reasonable to suppose that the model applies with \(p = \frac { 1 } { 2 }\).
2. The model is refined by estimating \(p\) from the data. Find the mean of the observed data and hence an estimate of \(p\).
3. Using the estimated value of \(p\), the value of the test statistic \(X ^ { 2 }\) turns out to be 2.3857 . Is it reasonable to suppose, at the \(5 \%\) level of significance, that this refined model applies?
4. Discuss the reasons for the different outcomes of the tests in parts (i) and (iii).