- Liam and Simone are studying the distribution of oak trees in some woodland. They divided the woodland into 80 equal squares and recorded the number of oak trees in each square. The results are summarised in Table 1 below.
\begin{table}[h]
| Number of oak trees in a square | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 or more |
| Frequency | 1 | 4 | 21 | 23 | 13 | 11 | 7 | 0 |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
Liam believes that the oak trees were deliberately planted, with 6 oak trees per square and that a constant proportion \(p\) of the oak trees survived.
- Suggest the model Liam should use to describe the number of oak trees per square.
Liam decides to test whether or not his model is suitable and calculates the expected frequencies given in Table 2.
\begin{table}[h]
| Number of oak trees in a square | 0 or 1 | 2 | 3 | 4 | 5 | 6 |
| Expected frequency | 5.53 | 14.89 | 24.26 | 22.24 | 10.87 | 2.21 |
\captionsetup{labelformat=empty}
\caption{Table 2}
- Showing your working clearly, complete the test using a \(5 \%\) level of significance. You should state your critical value and conclusion clearly.
Simone believes that a Poisson distribution could be used to model the number of oak trees per square. She calculates the expected frequencies given in Table 3.
\begin{table}[h]
| Number of oak trees in a square | 0 or 1 | 2 | 3 | 4 | 5 | 6 or more |
| Expected frequency | 12.69 | 16.07 | \(s\) | 14.58 | \(t\) | 9.37 |
\captionsetup{labelformat=empty}
\caption{Table 3}
- Find the value of \(s\) and the value of \(t\), giving your answers to 2 decimal places.
- Write down hypotheses to test the suitability of Simone's model.
The test statistic for this test is 8.749
- Complete the test. Use a \(5 \%\) level of significance and state your critical value and conclusion clearly.
- Using the results of these tests, explain whether the origin of this woodland is likely to be cultivated or wild.