2023 HCI P2 Q12

The table gives the record time, \(t\) seconds, for a particular \(100\)-metre freestyle swimming long course event for women for the number of years, \(y\), after Year \(2000\).

1. Draw the scatter diagram for these values, labelling the axes clearly. Circle the point on the scatter diagram that seems to have the value of \(t\) recorded wrongly and label it as \(P\).[3]

Omit point P identified in part (a) for the rest of this question.

2. It is desired to predict the record time in future years. Explain why, in this context, a linear model is likely to be inappropriate.[1]

The following models are proposed to describe the relationship between \(t\) and \(y\):

For \(a > 0\) and \(b > 0\),

Model A: \(t = a - b y^2\); \(\qquad\) Model B: \(t = a + \frac{b}{y}\).

3. Explain, without further calculation, why Model B provides a more accurate model of the relationship between \(t\) and \(y\) than Model A.[1]
4. For Model B, find the product moment correlation coefficient between \(t\) and \(\frac{1}{y}\). State the values of \(a\) and of \(b\) correct to \(5\) significant figures.[3]
5. Use your equation to estimate the record time in \(2025\), correct to \(3\) significant figures. Explain whether your estimate is reliable.[2]
6. It is required to estimate the value of \(y\) for which \(t = 52.45\). Use a suitable equation to find the estimated value of \(y\), correct to the nearest integer. Explain why neither the regression line of \(y^2\) on \(t\) nor the regression line of \(\frac{1}{y}\) on \(t\) should be used.[2]