2017 HCI P2 Q10

A large cohort of students sat for a mathematics examination. Based on selected data of the examination results, the following table shows \(y\), the proportions of students who scored \(x\) marks.

1. Draw a scatter diagram for these values, labelling the axes.[2]
2. Explain why, in this context, a linear model is not appropriate.[1]

It is decided to fit a model of the form \(\ln y=a{{(x-m)}^{2}}+b\), where and is a suitable constant, to the data. The product moment correlation coefficient between \({{\left( x-m \right)}^{2}}\) and \(\ln y\) is denoted by \(r\). The table below gives values of \(r\) for some possible values of \(m\).

3. Calculate the value of \(r\) for \(m=65\), giving your answer correct to \(7\) decimal places.[1]
4. Use the table and your answer in part (iii) to suggest with a reason which of \(62.5\), \(65\) or \(67.5\) is the most appropriate value for \(m\).[1]
5. Using the value of found in part (iv), calculate the values of \(a\) and \(b\), and use them to predict the proportion of students who scored \(45\) marks. Comment on the reliability of your prediction.[5]