S3 June 2005 Q3
3. A researcher carried out a survey of three treatments for a fruit tree disease. The contingency table below shows the results of a survey of a random sample of 60 diseased trees.
| No action | Remove diseased branches | Spray with chemicals | |
|---|---|---|---|
| Tree died within 1 year | 10 | 5 | 6 |
| Tree survived for 1–4 years | 5 | 9 | 7 |
| Tree survived beyond 4 years | 5 | 6 | 7 |
Test, at the 5% level of significance, whether or not there is any association between the treatment of the trees and their survival. State your hypotheses and conclusion clearly. (11)
| Scheme | Marks | |||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ||||||||||||||||||||||||||
| \(\dfrac{\text{RT} \times \text{CT}}{\text{GT}}\) | M1 | |||||||||||||||||||||||||
| \(6 \times 7\) | A1 | |||||||||||||||||||||||||
| \(3 \times 6\) | A1 | |||||||||||||||||||||||||
| \(\mathrm{H}_0\): Treatment & Survival are independent (not associated) \(\mathrm{H}_1\): Treatment & Survival are not independent (associated) | B1 both | |||||||||||||||||||||||||
| \(\alpha = 0.05\) \(\nu = (3 - 1) \times (3 - 1) = 4\) | B1 | |||||||||||||||||||||||||
| CR: \(\chi^2 \gt 9.488\) | B1ft | |||||||||||||||||||||||||
| \(\displaystyle\sum \frac{(O - E)^2}{E} = \frac{9}{7} + \frac{4}{7} + \frac{1}{7} + \frac{4}{7} + \frac{4}{7} + 0 + \frac{1}{6} + 0 + \frac{1}{6}\) Using \(\sum \frac{(O - E)^2}{E}\) Any 2 values | M1 A1 | |||||||||||||||||||||||||
| \(= 3.47619\ldots\) AWRT 3.48 | A1 | |||||||||||||||||||||||||
| Since 3.47619… is NOT in the critical region (i.e. \(\lt 9.488\)) there is insufficient evidence to reject \(\mathrm{H}_0\). Comparison | M1 | |||||||||||||||||||||||||
| There is no evidence of association between treatment and length of survival. Conclusion | A1ft | |||||||||||||||||||||||||
| (11) | ||||||||||||||||||||||||||
| (11 marks) |