8 In excavating an archaeological site, Roman coins are found scattered throughout the site.
- State two assumptions needed to model the number of coins found per square metre of the site by a Poisson distribution.
Assume now that the number of coins found per square metre of the site can be modelled by a Poisson distribution with mean \(\lambda\).
- Given that \(\lambda = 0.75\), calculate the probability that exactly 3 coins are found in a region of the site of area \(7.20 \mathrm {~m} ^ { 2 }\).
A test is carried out, at the \(5 \%\) significance level, of the null hypothesis \(\lambda = 0.75\), against the alternative hypothesis \(\lambda > 0.75\), in Region LVI which has area \(4 \mathrm {~m} ^ { 2 }\).
- Determine the smallest number of coins that, if found in Region LVI, would lead to rejection of the null hypothesis, stating also the values of any relevant probabilities.
- Given that, in fact, \(\lambda = 1.2\) in Region LVI, find the probability that the test results in a Type II error.