6 A researcher is investigating whether there is any relationship between whether a cyclist wears a helmet and the distance, \(x \mathrm {~m}\), the cyclist is from the kerb (the edge of the road). Data are collected at a particular location for a random sample of 250 cyclists.
The researcher carries out a chi-squared test. Fig. 6 is a screenshot showing part of a spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted.
\begin{table}[h]
| A | B | C | D | E | F | G |
| 1 | \multirow{2}{*}{} | Observed frequency |
| 2 | | | \(\boldsymbol { x } \boldsymbol { \leq } \mathbf { 0 . 3 }\) | \(0.3 < x \leq 0.5\) | \(0.5 < x \leq 0.8\) | x > 0.8 | Totals |
| 3 | \multirow[t]{2}{*}{Wears helmet} | Yes | 26 | 27 | 23 | 46 | 122 |
| 4 | | No | 45 | 31 | 21 | 31 | 128 |
| 5 | \multirow{2}{*}{} | Totals | 71 | 58 | 44 | 77 | 250 |
| 6 | | | | | | | |
| 7 | | Expected frequency | |
| 8 | | | \(\boldsymbol { x } \boldsymbol { \leq } \mathbf { 0 . 3 }\) | \(0.3 < x \leq 0.5\) | \(0.5 < x \leq 0.8\) | \(\boldsymbol { x } \boldsymbol { > } \mathbf { 0 . 8 }\) | |
| 9 | \multirow[t]{2}{*}{Wears helmet} | Yes | 34.6480 | | | 37.5760 | |
| 10 | | No | 36.3520 | | | 39.4240 | |
| 11 | | | |
| 12 | \multirow{2}{*}{} | | Contribution to the test statistic | |
| 13 | | | \(\boldsymbol { x } \boldsymbol { \leq } \mathbf { 0 . 3 }\) | \(0.3 < x \leq 0.5\) | \(0.5 < x \leq 0.8\) | \(\boldsymbol { x } \boldsymbol { > } \mathbf { 0 . 8 }\) | |
| 14 | \multirow[t]{2}{*}{Wears helmet} | Yes | 2.1585 | 0.0601 | 0.1087 | 1.8885 | |
| 15 | | No | 2.0573 | 0.0573 | | 1.8000 | |
| 16 | |
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{table}
- Showing your calculations, find the missing values in each of the following cells.
- E10
- E15
- In this question you must show detailed reasoning.
Carry out a hypothesis test at the \(10 \%\) significance level to investigate whether there is any association between helmet wearing and distance from the kerb. - Discuss briefly what the data suggest about helmet wearing for different distances from the kerb.