6 At a bird feeding station, birds are captured and ringed. If a bird is recaptured, the ring enables it to be identified. The table below shows the number of recaptures, \(x\), during a period of a month, for each bird of a particular species in a random sample of 40 birds.
| Number of | | recaptures, \(x\) |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Frequency | 2 | 5 | 5 | 9 | 10 | 4 | 3 | 1 | 0 | 1 | 0 |
- The sample mean of \(x\) is 3.4. Calculate the sample variance of \(x\).
- Briefly comment on whether the results of part (i) support a suggestion that a Poisson model might be a good fit to the data.
The screenshot below shows part of a spreadsheet for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean of 3.4 has been used as an estimate of the Poisson parameter. Some values in the spreadsheet have been deliberately omitted.
| A | B | C | D | E |
| 1 | Number of recaptures | Observed frequency | Poisson probability | Expected frequency | Chi-squared contribution |
| 2 | 0 or 1 | 7 | 0.1468 | 5.8737 | 0.2160 |
| 3 | 2 | 5 | | | 0.9560 |
| 4 | 3 | 9 | 0.2186 | 8.7447 | 0.0075 |
| 5 | 4 | 10 | 0.1858 | 7.4330 | 0.8865 |
| 6 | 5 | 4 | 0.1264 | 5.0544 | |
| 7 | \(\geq 6\) | 5 | 0.1295 | 5.1783 | 0.0061 |
- State the null and alternative hypotheses for the test.
- Calculate the missing values in cells
- C3,
- D3 and
- E6.
- Complete the test at the \(10 \%\) significance level.
- The screenshot below shows part of a spreadsheet for a \(\chi ^ { 2 }\) test for a different species of bird. Find the value of the Poisson parameter used.
| A | B | C | D | E |
| 1 | Number of recaptures | Observed frequency | Poisson probability | Expected frequency | Chi-squared contribution |
| 3 | 1 | 10 | 0.25716 | 12.8579 | 0.6352 |
| 4 | 2 | 7 | 0.27002 | 13.5008 | 3.1302 |
| 5 | 3 | 15 | 0.18901 | 9.4506 | 3.2587 |
| 6 | \(\geq 4\) | 11 | 0.16136 | 8.0679 | 1.0656 |