7 A biologist is investigating migrating butterflies. Fig. 7.1 shows the numbers of migrating butterflies passing her location in 100 randomly chosen one-minute periods.
\begin{table}[h]
| Number of butterflies | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | \(\geqslant 8\) |
| Frequency | 6 | 9 | 18 | 26 | 13 | 16 | 9 | 3 | 0 |
\captionsetup{labelformat=empty}
\caption{Fig. 7.1}
\end{table}
- Use the data to show that a suitable estimate for the mean number of butterflies passing her location per minute is 3.3.
- Explain how the value of the variance estimate calculated from the sample supports the suggestion that a Poisson distribution may be a suitable model for these data.
The biologist decides to carry out a test to investigate whether a Poisson distribution may be a suitable model for these data.
- In this question you must show detailed reasoning.
Complete the copy of Fig. 7.2 of expected frequencies and contributions for a chi-squared test in the Printed Answer Booklet.
\begin{table}[h]
| Number of butterflies | Frequency | Probability | Expected frequency | Chi-squared contribution |
| 0 | 6 | 0.0369 | 3.6883 | 1.4489 |
| 1 | 9 | 0.1217 | 12.1714 | 0.8264 |
| 2 | 18 | | | 0.2160 |
| 3 | 26 | | | 0.6916 |
| 4 | 13 | 0.1823 | 18.2252 | 1.4981 |
| 5 | 16 | 0.1203 | 12.0286 | |
| 6 | 9 | 0.0662 | 6.6158 | 0.8593 |
| \(\geqslant 7\) | 3 | 0.0510 | 5.0966 | 0.8625 |
\captionsetup{labelformat=empty}
\caption{Fig. 7.2}
- Complete the chi-squared test at the \(5 \%\) significance level.