2025 ACJC P2 Q10

Scientists are studying the growth of a particular species of bamboo. They want to understand how the height of the bamboo stalk changes over time. The table below shows the height, \(h\) inches, of a bamboo stalk at age, \(t\) weeks, after sprouting.

1. Draw a scatter diagram of these data, and explain using the diagram why the relationship between \(h\) and \(t\) is not well-modelled by an equation of the form \(h=a+bt\) or \(h=a+bt^2\), where \(b>0\).[3]

2. Consider the two models below:

   Model \(A\): \(h=c+d\sqrt{t}\),

   Model \(B\): \(h=c+d\ln t\),

   where \(c\) and \(d\) are constants. Explain which of Model \(A\) or Model \(B\) is a better fit to the data. State the equation of the least squares regression line in this case, giving your answer to \(3\) decimal places.

   [3]

3. Use the equation of your regression line in (b) to estimate the length of a \(6\)-week-old bamboo stalk. Comment on the reliability of this estimate.

   [2]
4. Comment on the suitability of using this model in the long run.[1]

5. Let \(H\) denote the height of the bamboo stalk calculated for each value of \(t\) in the table using the least squares regression line obtained in (b). Find the value of \(S\), where \(S=\sum(h-H)^2\) giving your answer to \(2\) decimal places.

   [1]
6. For each of the eight sample values of \(t\), \(H'\) is given by \(H'=p+q\ln t\), where \(p\) and \(q\) are constants. Would \(\sum(h-H')^2\) be greater or less than \(S\) and why?[1]