5 At a cinema there are three film sessions each Saturday, "early", "middle" and "late". The numbers of the audience, in different age groups, at the three showings on a randomly chosen Saturday are given in Table 1.
\begin{table}[h]
| \multirow{2}{*}{Observed frequencies} | Session |
| | Early | Middle | Late |
| \multirow{3}{*}{Age group} | < 25 | 24 | 20 | 40 |
| 25 to 60 | 4 | 2 | 10 |
| > 60 | 28 | 22 | 10 |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
The cinema manager carries out a test of whether there is any association between age group and session attended.
- Show that it is necessary to combine cells in order to carry out the test.
It is decided to combine the second and third rows of the table. Some of the expected frequencies for the table with rows combined, and the corresponding contributions to the \(\chi ^ { 2 }\) test statistic, are shown in the following incomplete tables.
\begin{table}[h]
| \multirow{2}{*}{Expected frequencies} | Session |
| | Early | Middle | Late |
| \multirow{2}{*}{Age group} | < 25 | 29.4 | 23.1 | |
| \(\geqslant 25\) | 26.6 | 20.9 | |
\captionsetup{labelformat=empty}
\caption{Table 2}
| \multirow{2}{*}{Contribution to \(\chi ^ { 2 }\)} | Session |
| | Early | Middle | Late |
| \multirow{2}{*}{Age group} | < 25 | 0.9918 | 0.4160 | |
| \(\geqslant 25\) | 1.0962 | 0.4598 | |
\captionsetup{labelformat=empty}
\caption{Table 3}
- In the Printed Answer Booklet, complete both tables.
- Carry out the test at the \(5 \%\) significance level.
- Use the figures in your completed Table 3 to comment on the numbers of the audience in different age groups.