Jamie is training for National Archery Championship and she practices shooting each week. Her percentage score, \(y\%\) in week \(x\), is as follows.
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) |
|---|
| \(y\) | \(40\) | \(57\) | \(68\) | \(74\) | \(80\) | \(85\) | \(88\) |
Draw a scatter diagram to illustrate the data.[2]
Calculate the value of the product moment correlation coefficient.[1]
It is desired to predict Jamie's eventual score. 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 \(L\).
| \(L\) | \(93\) | \(94\) | \(95\) |
|---|
| \(r\) | \(-0.996\ 512\) | | \(-0.998\ 595\) |
Calculate the value of \(r\) for \(L = 94\), 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 \(93\), \(94\) or \(95\) is the most appropriate value of \(L\).[1]
Using the value for \(L\), calculate the values of \(a\) and \(b\), and use them to predict the week in which Jamie will obtain her first mark of at least \(92\%\).[3]
Give an interpretation, in context, of the value of \(L\). Explain the significance of \(L\) in relation to Jamie's training schedule.[2]