2022 JPJC P2 Q10

The Electronic Road Pricing (ERP) system is the primary method of regulating traffic in Singapore. ERP rates are determined based on traffic conditions. If the average traffic speed rises above \(45\)km/h on expressways, ERP charges at that gantry will be reduced. Conversely, ERP rates will be increased if traffic moves slower than the average speed of \(45\)km/h on expressways.

1. The authority reviewed the traffic conditions on a particular expressway by measuring the speeds of \(150\) randomly selected cars as they pass a speed camera. The speed, \(x\) km/h was recorded. The results are summarised by
   \(\sum{\left( x-40 \right)=\mathrm{585}}\),         \(\sum{{{\left( x-40 \right)}^{2}}=\mathrm{10998}}\).
   1. Find the unbiased estimates of the population mean and variance of the speed of a car on the expressway.[3]
   2. Test, at the \(5\%\) level of significance, whether the ERP rate needs to be increased for this expressway.[4]
   3. Explain the meaning of ‘\(p\)-value’ in the context of the question in (ii).[1]
2. A new speed camera is installed along the expressway and it is now known that the population standard deviation of the speed of a car along this expressway is \(7.79\) km/h. A large random sample of n cars is observed and the average speed of \(45.9\)km/h is recorded. A new test is carried out at \(5\%\) level of significance. Find the least value of \(n\) that results in the reduction of the ERP rate along this expressway.[4]