N2008 P2 Q8

A certain metal discolours when exposed to air. To protect the metal against discolouring, it is treated with a chemical. In an experiment, different quantities, \(x\) ml, of the chemical were applied to standard samples of the metal, and the times, \(t\) hours, for the metal to discolour were measured. The results are given in the table.

1. Calculate the product moment correlation coefficient between \(x\) and \(t\), and explain whether your answer suggests that a linear model is appropriate.[3]
2. Draw a scatter diagram for the data.[1]

One of the values of \(t\) appears to be incorrect.

3. Indicate the corresponding point on your diagram by labelling it \(P\), and explain why the scatter diagram for the remaining points may be consistent with a model of the form \(t=a+b\ln x\).[2]
4. Omitting \(P\), calculate least squares estimates of \(a\) and \(b\) for the model \(t=a+b\ln x\).[2]
5. Assume that the value of \(x\) at \(P\) is correct. Estimate the value of \(t\) for this value of \(x\).[1]
6. Comment on the use of the model in part (iv) in predicting the value of \(t\) when \(x=8.0\).[1]