N2021 P2 Q9

In a chemical reaction the mass, \(x\) grams, of a particular product at time \(t\) minutes is given in this table.

The value of the product moment correlation coefficient is \(0.9803\) correct to \(4\) significant figures. The scatter diagram for the data is shown below.

1. Toby attempts to model the relationship between \(x\) and \(t\) with a straight line. Explain whether this is likely to provide a good model.[1]

Toby now tries a model in which \(x\) has been transformed to \(\ln x\).

2. 1. Sketch a scatter diagram of \(\ln x\) against \(t\) for the data given in the table.[1]
   2. Toby models the data with the equation \(\ln x=c+dt\). Find the values of the constants \(c\) and \(d\) and state the value of the product moment correlation coefficient for this model.[3]
3. Comment on Toby’s two models.[2]