2024 NYJC P2 Q11

Various studies suggest that plants exposed to music tend to grow taller compared to those not exposed to music. A junior biologist wants to investigate the effects of music on the growth of different plants. In one experiment, vines of a particular species were exposed to classical music. He recorded the average length of the vines, \(h\) centimetres, at age \(t\) days, in the table below.

1. Using the data, the junior biologist calculated the least squares regression line of \(h\) on \(t\) to be \(h=1.6393t+11.475\). Show that the value of \(k\) is \(48.1\), correct to 1 decimal place.[2]

The junior biologist realised that the relationship between \(h\) and \(t\) is not linear and can be modelled by either

\(h=a\sqrt[3]{t}+b\) or \(h=c\sqrt{t}+d\),

where \(a\), \(b\), \(c\) and \(d\) are constants.

2. By calculating the relevant product moment correlation coefficients, explain why \(h=a\sqrt[3]{t}+b\) is a better model than \(h=c\sqrt{t}+d\). Find the equation of the better model.[4]
3. An estimate for the average length of the vines when they are \(2\) months old is obtained using the better model. Explain whether the estimate is reliable.[1]
4. Sketch a scatter diagram for \(h\) against \(\sqrt[3]{t}\). Draw the line of best fit on your scatter diagram.[2]

For a line of best fit \(y=\mathrm{f}\left( x \right)\), the residual is the difference between the observed and predicted value. The difference between an observed value \(\left( p,q \right)\) plotted on the scatter diagram and the predicted value \(\left( p,\mathrm{f}\left( p \right) \right)\) is defined as \(q-\mathrm{f}\left( p \right)\).

5. Find the value of the residual corresponding to \(t=28\), for the line of best fit found in part (b).[1]
6. Explain why the line of best fit is referred to as the ‘least squares’ regression line in relation to the residuals.[2]