S1 June 2015 Q2
2. An estate agent recorded the price per square metre, \(p\) £/m\(^2\), for 7 two-bedroom houses.
He then coded the data using the coding \(q = \dfrac{p - a}{b}\), where \(a\) and \(b\) are positive constants.
His results are shown in the table below.
| \(p\) | 1840 | 1848 | 1830 | 1824 | 1819 | 1834 | 1850 |
|---|---|---|---|---|---|---|---|
| \(q\) | 4.0 | 4.8 | 3.0 | 2.4 | 1.9 | 3.4 | 5.0 |
The estate agent also recorded the distance, \(d\) km, of each house from the nearest train station. The results are summarised below.
\[\mathrm{S}_{dd} = 1.02 \qquad \mathrm{S}_{qq} = 8.22 \qquad \mathrm{S}_{dq} = -2.17\]The estate agent records the price and size of 2 additional two-bedroom houses, \(H\) and \(J\).
| House | Price (£) | Size (m\(^2\)) |
|---|---|---|
| \(H\) | 156 400 | 85 |
| \(J\) | 172 900 | 95 |
| Scheme | Marks |
|---|---|
| \(\dfrac{1840 - a}{b} = 4.0 \qquad \dfrac{1848 - a}{b} = 4.8\) | M1 |
| \(a = \)1800 \(b = \)10 | A1 |
| (2) |
Notes
M1 for setting up two suitable equations which could lead to \(a\) and \(b\) (may be implied by one correct answer)
A1 for \(a = 1800\) and \(b = 10\) (\(a = 10\) and \(b = 1800\) is A0) Correct answer only is 2/2
| Scheme | Marks |
|---|---|
| \(r = \dfrac{-2.17}{\sqrt{1.02 \times 8.22}} = -0.749417343\ldots\) awrt \(-\)0.749 | M1A1 |
| (2) |
Notes
M1 for a correct expression (condone missing \(-\))
A1 for awrt \(-0.749\)
(\(-0.75\) or awrt 0.749 with no working scores M1 A0).
| Scheme | Marks |
|---|---|
| \(-0.749\) | B1ft |
| (1) |
Notes
B1ft for \(-0.749\) or ft their answer to (b) to at least 2sf. Must be in the range \(-1 \lt \text{'}(b)\text{'} \lt 1\)
| Scheme | Marks |
|---|---|
| House H: 156 400/85 = [£1840/m\(^2\) or \(q = 4\)] House J: 172 900/95 = [£1820/m\(^2\) or \(q = 2\)] | M1 |
| Since (\(r = -0.749\),) there is negative correlation. or The higher the price (per square metre), the lower the distance from the train station. | dM1 |
| Therefore…..House H is likely to be closer. | A1 |
| (3) | |
| (8 marks) |
Notes
M1 for calculating price/square metre for both \(H\) and \(J\).
Can be implied by sight of 1840 and 1820 (so OK if not labelled or mis-labelled)
These may be seen in the table in the question.
Allow comment like “\(H\) is £20/square metre more than \(J\)”
dM1 dependent on 1st M1 for a statement that correlation is negative or a contextualised interpretation of the negative correlation.
\(r \gt 0\) If \(r \gt 0\) allow equivalent statements about positive correlation
A1 (dependent on both Ms) for House H is likely to be closer (No ft if \(r \gt 0\))