4 The sexes and ages of a random sample of 300 runners taking part in marathons are classified as follows.
| Observed | Sex | \multirow{2}{*}{Row totals} |
| \cline { 3 - 4 } | Male | Female | |
| \multirow{3}{*}{} | Under 40 | 70 | 54 | 124 |
| \cline { 2 - 4 } | \(40 - 49\) | 76 | 36 | 112 |
| \cline { 2 - 5 } | 50 and over | 52 | 12 | 64 |
| Column totals | 198 | 102 | 300 |
- Carry out a test at the \(5 \%\) significance level to examine whether there is any association between age group and sex. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
- Does your analysis support the suggestion that women are less likely than men to enter marathons as they get older? Justify your answer.
For marathons in general, on average \(3 \%\) of runners are 'Female, 50 and over'. The random variable \(X\) represents the number of 'Female, 50 and over' runners in a random sample of size 300.
- Use a suitable approximating distribution to find \(\mathrm { P } ( X \geqslant 12 )\).