4 A student is investigating whether there is any association between the species of shellfish that occur on a rocky shore and where they are located. A random sample of 160 shellfish is selected and the numbers of shellfish in each category are summarised in the table below.
| Location |
| \cline { 3 - 5 }
\multicolumn{2}{|c|}{} | Exposed | Sheltered | Pool |
| \multirow{3}{*}{Species} | Limpet | 24 | 32 | 16 |
| \cline { 2 - 5 } | Mussel | 24 | 11 | 3 |
| \cline { 2 - 5 } | Other | 5 | 22 | 23 |
- Write down null and alternative hypotheses for a test to examine whether there is any association between species and location.
The contributions to the test statistic for the usual \(\chi ^ { 2 }\) test are shown in the table below.
| Contribution | Location |
| \cline { 3 - 5 } | Exposed | Sheltered | Pool | |
| \multirow{3}{*}{Species} | Limpet | 0.0009 | 0.2585 | 0.4450 |
| \cline { 2 - 5 } | Mussel | 10.3472 | 1.2756 | 4.8773 |
| \cline { 2 - 5 } | Other | 8.0719 | 0.1402 | 7.4298 |
- Calculate the expected frequency for mussels in pools. Verify the corresponding contribution 4.8773 to the test statistic.
- Carry out the test at the \(5 \%\) level of significance, stating your conclusion clearly.
- For each species, comment briefly on how its distribution compares with what would be expected if there were no association.
- If 3 of the 160 shellfish are selected at random, one from each of the 3 types of location, find the probability that all 3 of them are limpets.