7 A dairy industry researcher, Robyn, decided to investigate the milk yield, classified as low, medium or high, obtained from four different breeds of cow, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D .
The milk yield of a sample of 105 cows was monitored and the results are summarised in contingency Table 1.
| \multirow{2}{*}{Table 1} | Yield |
| | Low | Medium | High | Total |
| \multirow{4}{*}{Breed} | A | 4 | 5 | 12 | 21 |
| B | 10 | 6 | 4 | 20 |
| C | 8 | 17 | 7 | 32 |
| D | 5 | 20 | 7 | 32 |
| Total | 27 | 48 | 30 | 105 |
The sample of cows may be regarded as random.
Robyn decides to carry out a \(\chi ^ { 2 }\)-test for association between milk yield and breed using the information given in Table 1.
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- Contingency Table 2 gives some of the expected frequencies for this test.
Complete Table 2 with the missing expected values.
| \multirow[t]{2}{*}{Table 2} | Yield |
| | Low | Medium | High |
| \multirow{4}{*}{Breed} | A | | | 6 |
| B | 5.14 | 9.14 | 5.71 |
| C | | | |
| D | 8.23 | 14.63 | 9.14 |
- For Robyn's test, the test statistic \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 19.4\) correct to three significant figures.
Use this information to carry out Robyn's test, using the \(1 \%\) level of significance.
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- (ii) By considering the observed frequencies given in Table 1 with the expected frequencies in Table 2, interpret, in context, the association, if any, between milk yield and breed.