2025 NYJC P2 Q10

The director of an art gallery wishes to find out the relationship between the time spent, \(t\) minutes, in the art gallery and the age, \(a\) (in years), of the visitors. Ten visitors were randomly selected and the data are summarised as follows:

1. A linear model is proposed for the relationship between \(t\) and \(a\). Given that the least squares regression line of \(t\) on \(a\) is \[t = 0.8957914a + 14.16013,\] show that \(k = 40\).[2]
2. Sketch a scatter diagram of \(t\) against \(a\) for the data given in the table and draw the line given in (a) on the same diagram.[2]
3. Use the least squares regression line of \(t\) on \(a\) in (a) to estimate the time spent in the art gallery by a \(60\)-year old visitor.[1]
4. A regression line \(t = ma + c\) where \(m\) and \(c\) are constants is used to model the above set of data. The residual for a data point \((a_i, t_i)\) where \(i = 1, 2, \dots, 10\) which is plotted on a scatter diagram is defined to be the number \[t_i - (ma_i + c).\]The sum of the squares of these residuals, denoted by \(S\), is given by\[S = \sum_{i=1}^{10} [t_i - (ma_i + c)]^2.\]Find an inequality that is satisfied by \(S\).[2]
5. The director wishes to explore the following two models which represent the relationship between \(t\) and \(a\), where \(p\) and \(q\) are constants.
   \[(A) \quad t = p + \frac{q}{a} ,\qquad (B) \quad t = pe^{qa}. \]
   For each model, find the product moment correlation coefficient and hence explain with a reason which model is more appropriate.[3]