7 A random sample of 100 adults with a chronic disease was chosen. Each adult was randomly assigned to one of three different treatments. After six months of treatment, each adult was then assessed and classified as 'much improved', 'improved', 'slightly improved' or 'no change'. The results are summarised in Table 1.
\begin{table}[h]
| Treatment \(A\) | Treatment \(B\) | Treatment \(C\) |
| Much improved | 12 | 16 | 4 |
| Improved | 13 | 12 | 6 |
| Slightly improved | 7 | 6 | 7 |
| No change | 5 | 3 | 9 |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
A \(\chi ^ { 2 }\) test, at the \(5 \%\) significance level, is to be carried out.
- State suitable hypotheses.
Combining the last two rows of Table 1 gives Table 2.
\begin{table}[h]
| Treatment \(A\) | Treatment \(B\) | Treatment \(C\) |
| Much improved | 12 | 16 | 4 |
| Improved | 13 | 12 | 6 |
| Slightly improved/ No change | 12 | 9 | 16 |
\captionsetup{labelformat=empty}
\caption{Table 2}
- By considering the expected frequencies for Treatment \(C\) in Table 1, explain why it was necessary to combine rows.
- Show that the contribution to the \(\chi ^ { 2 }\) value for the cell 'slightly improved/no change, Treatment \(C\) ' is 4.231 , correct to 3 decimal places.
You are given that the \(\chi ^ { 2 }\) test statistic is 10.51 , correct to 2 decimal places.
- Carry out the test.