2019 DHS P2 Q10

The speeds of an e-scooter (\(X\) km/h) and a pedestrian (\(Y\)km/h) measured on a particular stretch of footpath are normally distributed with mean and variance as follows

It is known that \(\mathrm{P}\left( Y<5.2 \right)=\mathrm{P}\left( Y\ge 7.0 \right)=0.379\).

1. State the value of \(\mu\) and find the value of \(\sigma\).[2]
2. Given that the speeds of half of the e-scooters measured are found to be within \(a\) km/h of the mean, find \(a\).[2]
3. A LTA officer stationed himself at the footpath and measured the speeds of 50 e-scooters at random. Find the probability that the 50th e-scooter is the 35th to exceed LTA’s legal speed limit of 10 km/h.[3]
4. On another day, the LTA officer randomly measured the speeds of 6 e-scooters and 15 pedestrians. Find the probability that the mean speed of the e-scooters is more than twice the mean speed of the pedestrians captured.[3]
5. Find the probability that the mean speed of \(n\) randomly chosen e-scooters is more than 10 km/h, if \(n\) is large.[2]