- Stuart is investigating a treatment for a disease that affects fruit trees. He has 400 fruit trees and applies the treatment to a random sample of these trees. The remainder of the trees have no treatment. He records the number of years, \(y\), that each fruit tree remains free from this disease.
The results are summarised in the table below.
| \cline { 3 - 3 }
\multicolumn{2}{c|}{} | Treatment |
| \cline { 3 - 4 }
\multicolumn{2}{c|}{} | Applied | Not applied |
\multirow{3}{*}{| Number of years free | | from this disease |
} | \(y < 1\) | 15 | 25 |
| \cline { 2 - 4 } | \(1 \leqslant y < 2\) | 35 | 61 |
| \cline { 2 - 4 } | \(2 \leqslant y\) | 124 | 140 |
The data are to be used to determine whether or not there is an association between the application of the treatment and the number of years that a fruit tree remains free from this disease.
- Calculate the expected frequencies for
- Applied and \(y < 1\)
- Not applied and \(1 \leqslant y < 2\)
The value of \(\sum \frac { ( O - E ) ^ { 2 } } { E }\) for the other four classes is 2.642 to 3 decimal places.
- Test, at the \(5 \%\) level of significance, whether or not there is an association between the application of the treatment and the number of years a fruit tree remains free from this disease.
You should state your hypotheses, test statistic, critical value and conclusion clearly.