(i)     Write down the Pearson’s product–moment correlation coefficient, $r$, for these data.

(ii)     Hence comment on the result.

Write down the equation of the regression line $y$ on $x$ for these data, in the form $y = ax + b$.

Estimate the total cost, to the nearest USD, of producing $13$ bicycles on a particular day.

All the bicycles that are produced are sold. The bicycles are sold for 304 USD each.

Explain why the factory does not make a profit when producing $13$ bicycles on a particular day.

All the bicycles that are produced are sold. The bicycles are sold for 304 USD each.

(i)     Write down an expression for the total selling price of $x$ bicycles.

(ii)     Write down an expression for the profit the factory makes when producing $x$ bicycles on a particular day.

(iii)     Find the least number of bicycles that the factory should produce, on a particular day, in order to make a profit.

(i)     $r = 0.985\;\;\;(0.984905 \ldots )$     (G2)

Notes: If unrounded answer is not seen, award (G1)(G0) for $0.99$ or $0.984$. Award (G2) for $0.98$.

(ii)     strong, positive     (A1)(A1)

$y = 259.909 \ldots x + 698.648 \ldots \;\;\;(y = 260x + 699)$     (G1)(G1)

Notes: Award (G1) for $260x$ and (G1) for $699$. If the answer is not an equation award a maximum of (G1)(G0).

$y = 259.909 \ldots  \times 13 + 698.648 \ldots$     (M1)

Note: Award (M1) for substitution of $13$ into their regression line equation from part (b).

$y = 4077.47 \ldots$     (A1)(ft)(G2)

$y = 4077{\text{ (USD)}}$     (A1)(ft)

Notes: Follow through from their answer to part (b). If rounded values from part (b) used, answer is $4079$. Award the final (A1)(ft) for a correct rounding to the nearest USD of their answer. The unrounded answer may not be seen.

If answer is $4077$ and no working is seen, award (G2).

$13 \times 304 - (4077.47) =  - 125.477 \ldots \;\;\;( - 125)\;\;\;$OR

$4077.47 - (13 \times 304) = 125.477 \ldots \;\;\;(125)$     (M1)

Notes: Award (M1) for calculating the difference between $13 \times 304$ and their answer to part (c).

If rounded values are used in equation, answer is $- 127$.

profit is negative$\;\;\;$OR$\;\;\;{\text{cost}} > {\text{sales}}$     (A1)

OR

$13 \times 304 = 3952$     (M1)

Note: Award (M1) for calculating the price of $13$ bikes.

$3952 < 4077.47$     (A1)(ft)

Note: Award (A1) for showing $3952$ is less than their part (c). This may be communicated in words. Follow through from part (c), but only if value is greater than $3952$.

OR

$\frac{{4077}}{{13}} = 313.62$     (M1)

Note: Award (M1) for calculating the cost of $1$ bicycle.

$313.62 > 304$     (A1)(ft)

Note: Award (A1) for showing $313.62$ is greater than $304$. This may be communicated in words. Follow through from part (c), but only if value is greater than $304$.

OR

$\frac{{4077}}{{304}} = 13.41$     (M1)

Note: Award (M1) for calculating the number of bicycles that should have been be sold to cover total cost.

$13.41 > 13$     (A1)(ft)

Note: Award (A1) for showing $13.41$ is greater than $13$. This may be communicated in words. Follow through from part (c), but only if value is greater than $13$.

(i)     $304x$     (A1)

(ii)     $304x - (259.909 \ldots x + 698.648 \ldots )$     (A1)(ft)(A1)(ft)

Note: Award (A1)(ft) for difference between their answers to parts (b) and (e)(i), (A1)(ft) for correct expression.

(iii)     $304x - (259.909 \ldots x + 698.648 \ldots ) > 0$     (M1)

Notes: Award (M1) for comparing their expression in part (e)(ii) to $0$. Accept an equation. Accept $3040x - y > 0$ or equivalent.

$x = 16{\text{ bicycles}}$     (A1)(ft)(G2)

Notes: Follow through from their answer to part (b). Answer must be a positive integer greater than $13$ for the (A1)(ft) to be awarded.

Award (G1) for an answer of $15.84$.