A2 June 2019 Q2

EdexcelCurrent spec10 marksCorrelation Linear Regression

2. A large field of wheat is split into 8 plots of equal area. Each plot is treated with a different amount of fertiliser, \(f\) grams/m². The yield of wheat, \(w\) tonnes, from each plot is recorded. The results are summarised below.

\[\sum f = 28 \qquad \sum w = 303 \qquad \sum w^2 = 13\,447 \qquad \mathrm{S}_{ff} = 42 \qquad \mathrm{S}_{fw} = 269.5\]

(a) Calculate the product moment correlation coefficient between \(f\) and \(w\) (2)

(b) Interpret the value of your product moment correlation coefficient. (1)

(d) Using your equation, estimate the decrease in yield when the amount of fertiliser decreases by 0.5 grams/m² (1)

The residuals of the data recorded are calculated and plotted on the graph below.

Graph of residuals (from –10 to 8) against f (from 0 to 8) on a grid: approximately (0, –9.4), (1, 0.2), (2, 3.7), (3, 6.3), (4, 5.9), (5, 2.5), (6, –1.9), (7, –7.3)

(e) With reference to this graph, comment on the suitability of the model you found in part (c). (2)

(f) Suggest how you might be able to refine your model. (1)

Mark scheme (a)

Scheme	Marks	AO
\(S_{ww} = 13447 - \dfrac{303^2}{8} = 1970.875\)
\(r = \dfrac{269.5}{\sqrt{42 \times 1970.875}}\)	M1	1.1b
\(r = 0.9367\ldots\) awrt 0.937	A1	1.1b
	(2)

Notes

M1: Complete correct method for finding \(r\)

A1: for awrt 0.937

Scheme	Marks	AO
As the amount of fertiliser increases the yield increases	B1	3.2a
	(1)

Mark scheme (c)

Scheme	Marks	AO
\(b = \dfrac{269.5}{42}\ [= 6.41666\ldots]\)	M1	3.3
\(a = \dfrac{303}{8} - \text{‘}b\text{’}\dfrac{28}{8}\ [= 15.41666\ldots]\)	M1	1.1b
\(w = 15.4 + 6.42f\)	A1	1.1b
	(3)

Notes

M1: For use of a correct model ie a correct expression for \(b\)

M1: For use of a correct model ie a correct expression (ft) for \(a\)

A1: For a correct model \(w = 15.4 + 6.42f\) with awrt 15.4 and awrt 6.42

Scheme	Marks	AO
3.21 tonnes	B1ft	1.1b
	(1)

Mark scheme (e)

Scheme	Marks	AO
The residual plot is close to an ’n’ shape or the residuals appear not to be randomly scattered	M1	2.4
The model in part (c) is unlikely to be suitable	A1	2.2b
	(2)

Notes

M1: Explaining a reason for their conclusion eg there is a pattern/trend in the residuals
Do not accept residuals not close to zero

A1: concluding it is not valid oe

Scheme	Marks	AO
Fit a curve rather than a line	B1	3.5c
	(1)
	(10 marks)