4 Birds are observed at feeding stations in three different places - woodland, farm and garden. The numbers of finches, thrushes and tits observed at each site are summarised in the table. The birds observed are regarded as a random sample from the population of birds of these species that use these feeding stations.
| \multirow{2}{*}{Observed Frequency} | Place |
| | Farm | Garden | Woodland | Totals |
| \multirow{4}{*}{Species} | Thrushes | 11 | 74 | 7 | 92 |
| Tits | 70 | 26 | 88 | 184 |
| Finches | 17 | 2 | 10 | 29 |
| Totals | 98 | 102 | 105 | 305 |
The expected frequencies under the null hypothesis for the usual \(\chi ^ { 2 }\) test are shown in the table below.
| \multirow{2}{*}{Expected Frequency} | Place |
| | Farm | Garden | Woodland |
| \multirow{3}{*}{Species} | Thrushes | 29.5607 | 30.7672 | 31.6721 |
| Tits | 59.1213 | 61.5344 | 63.3443 |
| Finches | 9.3180 | 9.6984 | 9.9836 |
- Verify that the entry 9.3180 is correct.
The corresponding contributions to the test statistic are shown in the table below.
| \multirow{2}{*}{Contribution} | Place |
| | Farm | Garden | Woodland |
| \multirow{3}{*}{Species} | Thrushes | 11.6539 | 60.7489 | 19.2192 |
| Tits | 2.0017 | 20.5201 | 9.5969 |
| Finches | 6.3332 | 6.1108 | 0.0000 |
- Verify that the entry 6.3332 is correct.
- Carry out the test at the \(1 \%\) level of significance.
- For each place, use the table of contributions to comment briefly on the differences between the observed and expected distributions of species.