2024 HCI P2 Q11

An experiment was carried out to investigate the growth rate of a particular species of plant. The following table gives the height of the plant specimen, \(h\) centimeters, at the start of the \(n^{\text{th}}\) month.

A possible model for the growth rate is given to be

\[h = \frac{an}{b + n},\]

where \(a\) and \(b\) are constants.

1. By writing the above equation in a form that is linear in \(\frac{1}{h}\) and \(\frac{1}{n}\), calculate the equation of the least squares regression line of \(\frac{1}{h}\) on \(\frac{1}{n}\). Hence find estimates for the values of \(a\) and \(b\), correct to \(3\) decimal places.[4]

2. Sketch a scatter diagram for \(\frac{1}{h}\) against \(\frac{1}{n}\) and include the least squares regression line found in part (a).

   [2]
3. Explain in the context of the question, the significance of the value of \(a\).[1]

4. Use the least squares regression line in part (a) to find the least integer value of \(n\) required for the plant to reach a height of \(18\) centimeters. Explain whether you would expect this estimate to be reliable.

   [3]

For a line of best fit \(y = \mathrm{f}(x)\), the residual for a point \((p, q)\) plotted on the scatter diagram is the vertical distance between \((p, \mathrm{f}(p))\) and \((p, q)\).

5. Mark the residual for each point on the scatter diagram in part (b).

   [1]
6. Find the sum of squares of the residuals for the least squares regression line of \(\frac{1}{h}\) on \(\frac{1}{n}\), giving your answer correct to \(5\) significant figures.[1]