2025 TJC P2 Q10

Anand signed up for a privilege card with Temasek Airlines where cardholders earn more reward points as they fly further with the airline. From the airlines' website, Anand identified \(8\) different destinations and recorded the flight distance, \(x\) miles, of each destination from Singapore. He also recorded the corresponding reward points, \(y\), credited into his card after each flight. The data are shown below.

1. Sketch a scatter diagram of the data.[2]
2. Calculate the value of the product moment correlation coefficient, and explain why this value does not necessarily mean that a linear model is the best for the relationship between \(x\) and \(y\).[2]
3. Without calculating the product moment correlation coefficient, explain which of the following equations, where \(a\) and \(b\) are constants, and \(b > 0\), can be used to model the relationship between \(x\) and \(y\).
   (I) \(y = a + bx^2\)
   (II) \(y = a + b \ln x\)
   (III) \(y = a + be^{-x}\)[3]
4. Using the model identified in (c), find the equation of the corresponding regression line.[2]
5. Hence estimate the reward points that can be earned for a destination \(700\) miles away from Singapore. Comment on the reliability of your estimation.[2]
6. It is given that \(1\) mile is approximately \(1.609\) kilometres. State what will happen to the product moment correlation coefficient found in (b) if the flight distance is measured in kilometres instead of miles.[1]