Draw a logarithmic pattern and regress mass on time

(a) Sketch a scatter diagram that might be expected for the case when $x$ and $y$ are related approximately by $y=a+b\ln x$, where $a$ is positive and $b$ is negative. In this case, your diagram should include 6 points, approximately equally spaced with respect to $x$, and with all $x$- and $y$-values positive.1 mark

(b) The mass, $x$ grams, of a certain substance present in a chemical reaction varies with time, $t$ seconds. Observations at successive times give the data shown in the following table.

Time, $t$	5	15	20	25	30	35	40
Mass, $x$	4.5	7.9	8.7	9.6	10.8	11.3	11.9

(i) Draw a scatter diagram for these values, labelling the axes.1 mark

(ii) Calculate the product moment correlation coefficient between $t$ and $x$ and comment on its value in context of the data.2 marks

(iii) State, with a reason, the effect on the product moment correlation coefficient found in part (ii) if the mass was recorded in milligrams instead of grams.1 mark

(iv) State, with a reason, which least squares regression line should be used to express a possible linear relation between $x$ and $t$. Hence find the equation of a suitable regression line for this model.2 marks

(v) Use the equation of your regression line to estimate the mass of the certain substance present at 10 seconds and comment on the reliability of your estimate.2 marks