Amy is revising for a mathematics examination and takes a different practice paper each week. Her marks, \(y\%\) in week \(x\), are as follows.
| Week \(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) |
|---|
| Percentage mark \(y\) | \(38\) | \(63\) | \(67\) | \(75\) | \(71\) | \(82\) |
Draw a scatter diagram showing these marks.[1]
Suggest a possible reason why one of the marks does not seem to follow the trend.[1]
It is desired to predict Amy's marks on future papers. Explain why, in this context, neither a linear nor a quadratic model is likely to be appropriate.[2]
It is decided to fit a model of the form \(\ln(L - y) = a + bx\), where \(L\) is a suitable constant. The product moment correlation coefficient between \(x\) and \(\ln(L - y)\) is denoted by \(r\). The following table gives values of \(r\) for some possible values of \(L\).
| \(L\) | \(91\) | \(92\) | \(93\) |
|---|
| \(r\) | | \(-0.929944\) | \(-0.929918\) |
Calculate the value of \(r\) for \(L = 91\), giving your answer correct to 6 decimal places.[1]
Use the table and your answer to part (iv) to suggest with a reason which of 91, 92 or 93 is the most appropriate value for \(L\).[1]
Using this value for \(L\), calculate the values of \(a\) and \(b\), and use them to predict the week in which Amy will obtain her first mark of at least 90%.[4]
Give an interpretation, in context, of the value of \(L\).[1]