3. A farmer is investigating the milk yields of two breeds of cow. He takes a random sample of 9 cows of breed \(A\) and an independent random sample of 12 cows of breed \(B\). For a 5 day period he measures the amount of milk, \(x\) gallons, produced by each cow. The results are summarised in the table below.
| Breed | Sample size | Mean \(( \overline { \boldsymbol { x } } )\) | Standard deviation \(\left( \boldsymbol { s } _ { \boldsymbol { x } } \right)\) |
| \(A\) | 9 | 6.23 | 2.98 |
| \(B\) | 12 | 7.13 | 2.33 |
The amount of milk produced by each cow can be assumed to follow a normal distribution.
- Use a two-tail test to show, at the \(10 \%\) level of significance, that the variances of the yields of the two breeds can be assumed to be equal. State your hypotheses clearly.
- Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is a difference in the mean yields of the two breeds of cow.
- Explain briefly the importance of the test in part (a) for the test in part (b).