9 A cyclist has 3 bicycles, a road bike, a gravel bike and an electric bike. She wishes to know if the bicycle which she is riding makes any difference to whether she reaches a speed of 25 mph or greater on a journey. She selects a random sample of 120 journeys and notes the bicycle and whether or not her maximum speed was 25 mph or greater. She decides to carry out a chisquared test to investigate whether there is any association between bicycle type and whether her maximum speed is 25 mph or greater. Tables 9.1 and 9.2 show the data and some of the expected frequencies for the test.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Table 9.1}
| \multirow{2}{*}{} | Bicycle | |
| | Road | Gravel | Electric | Total |
| \multirow{2}{*}{Maximum speed} | Less than 25 mph | 2 | 21 | 19 | 42 |
| 25 mph or greater | 13 | 47 | 18 | 78 |
| Total | 15 | 68 | 37 | 120 |
\end{table}
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Table 9.2}
| \multirow{2}{*}{Expected frequency} | Bicycle |
| | Road | Gravel | Electric |
| \multirow{2}{*}{Maximum speed} | Less than 25 mph | | | 12.95 |
| 25 mph or greater | | | 24.05 |
\end{table}
- Complete the table of expected frequencies in the Printed Answer Booklet.
- Determine the contribution to the chi-squared test statistic for the Electric bicycle and maximum speed 25 mph or greater. Give your answer correct to 4 decimal places.
The contributions to the chi-squared test statistic for the remaining categories are shown in Table 9.3.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Table 9.3}
| \multirow{2}{*}{Contribution to the test statistic} | Bicycle |
| | Road | Gravel | Electric |
| \multirow{2}{*}{Maximum speed} | Less than 25 mph | 2.0119 | 0.3294 | 2.8264 |
| 25 mph or greater | 1.0833 | 0.1774 | |
- In this question you must show detailed reasoning.
Carry out the test at the 5\% significance level.
- For each type of bicycle, give a brief interpretation of what the data suggest about maximum speed.