A2 June 2023 Q1
1. 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.
\[\mathrm{S}_{xx} = 314.55 \qquad \mathrm{S}_{yy} = 9026 \qquad \mathrm{S}_{xy} = 1610\]\[\bar{x} = 19.65 \qquad \bar{y} = 108\]The child who completed the 100 m race in 20 seconds took 104 seconds to complete the 500 m race.
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 | + | + | + | + | – | – | + | – | – | – | – | – | – | – | – | + | + | + | + | + |
|---|
| Scheme | Marks | AO |
|---|---|---|
| \(r = \dfrac{1610}{\sqrt{314.55 \times 9026}}\) | M1 | 1.1b |
| \(= 0.9555\ldots\) awrt 0.956 | A1 | 1.1b |
| (2) |
Notes
M1: correct expression for \(r\)
A1: awrt 0.956
| Scheme | Marks | AO |
|---|---|---|
| \(b = \dfrac{1610}{314.55}\) | M1 | 2.1 |
| \(a = \bar{y} - b\bar{x} = 108 - \text{‘}b\text{’} \times 19.65\) | M1 | 1.1b |
| \(y = 5.12x + 7.42\)* | A1* | 1.1b |
| (3) |
Notes
M1: correct expression for gradient, or correct formula stated with \(b\) to more than 3.s.f
M1: correct expression, or correct formula stated with \(a\) to more than 3.s.f
follow through their \(b\)
*A1: correct linear model with \(\boldsymbol{a}\) = awrt 7.42 and \(\boldsymbol{b}\) = awrt 5.12
| Scheme | Marks | AO |
|---|---|---|
| Residual \(= 104 - (5.12(20) + 7.42) = \) awrt –5.8 | B1 | 3.4 |
| (1) |
Notes
B1: correct use of model awrt –5.8
| Scheme | Marks | AO |
|---|---|---|
| Positive residuals show that the model underestimates the 500m race times of the fastest and slowest children. | B1 | 3.5a |
| (1) | ||
| (7 marks) |
Notes
B1: correct contextual explanation mentioning positive residuals and underestimate o.e.