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.
(i)
Draw a scatter diagram for these values, labelling the axes.
[2]
(ii)
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 $m$.
(iii)
Calculate the value of $r$ for $m=65$, giving your answer correct to $7$ decimal places.
[1]
(iv)
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]
(v)
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]
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