- Baako is investigating the times taken by children to run a 100 m race, \(x\) seconds, and a 500 m race, \(y\) seconds. For a sample of 20 children, Baako obtains the time taken by each child to run each race.
Here are Baako's summary statistics.
$$\begin{gathered}
\mathrm { S } _ { x x } = 314.55 \quad \mathrm {~S} _ { y y } = 9026 \quad \mathrm {~S} _ { x y } = 1610
\bar { x } = 19.65 \quad \bar { y } = 108
\end{gathered}$$
- Calculate the product moment correlation coefficient between the times taken to run the 100 m race and the times taken to run the 500 m race.
- Show that the equation of the regression line of \(y\) on \(x\) can be written as
$$y = 5.12 x + 7.42$$
where the gradient and \(y\) intercept are given to 3 significant figures.
The child who completed the 100 m race in 20 seconds took 104 seconds to complete the 500 m race.
- Find the residual for this child.
The table below shows the signs of the residuals for the 20 children in order of finishing time for the 100 m race.
| Sign of residual | + | + | + | + | - | - | + | - | - | - | - | - | - | - | - | + | + | + | + | + |
- Explain what the signs of the residuals show about the model's predictions of the 500 m race times for the children who are fastest and slowest over the 100 m race.