AS June 2022 Q3

EdexcelAS paperCurrent spec10 marksCorrelation Linear Regression

3. Gabriela is investigating a particular type of fish, called bream. She wants to create a model to predict the weight, \(w\) grams, of bream based on their length, \(x\) cm.

For a sample of 27 bream, some summary statistics are given below.

\[\bar{x} = 31.07 \qquad \bar{w} = 628.59 \qquad \sum w^2 = 11\,386\,134\]\[\mathrm{S}_{xw} = 13\,082.3 \qquad \mathrm{S}_{xx} = 260.8\]

(a) Find the value of the product moment correlation coefficient between \(x\) and \(w\) (3)

(b) Explain whether the answer to part (a) is consistent with a linear model for these data. (1)

A residual plot for these data is shown below.

Residual plot: residual (from –80 to 100) against length (cm, from 26 to 38) for the 27 bream; the residual at length 32 cm is 80, and all five points with length above 33 cm have negative residuals

One of the bream in the sample has a length of 32 cm.

(d) Find its weight. (2)

(e) With reference to the residual plot, comment on the model for bream with lengths above 33 cm. (1)

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{S}_{ww} = 11\,386\,134 - 27(628.59)^2\ [= 717\,748.5213]\)	B1	1.1b
\(r = \dfrac{13\,082.3}{\sqrt{260.8 \times \text{‘}717\,748.5213\text{’}}}\)	M1	1.1b
\(r = 0.95618\ldots\) awrt 0.956	A1	1.1b
	(3)

Notes

B1: Correct expression for \(\mathrm{S}_{ww}\) (implied by a correct answer)

M1: Complete method to find \(r\) (Use of \(\mathrm{S}_{ww}\) = 11 386 134 is M0)

A1: awrt 0.956

Scheme	Marks	AO
Since \(r\) is close to 1, data is consistent with a linear model.	B1	2.4
	(1)

Mark scheme (c)

Scheme	Marks	AO
\(b = \dfrac{13\,082.3}{260.8}\ [= 50.162\ldots]\)	M1	3.3
\(a = 628.59 - \text{‘}b\text{’}(31.07)\)	M1	1.1b
\(w = -930 + 50.2x\)	A1	1.1b
	(3)

Notes

M1: Setting up linear model by finding gradient

M1: Attempting \(y\)-intercept of linear model

A1: Correct model with \(b\) = awrt 50.2 and \(a\) = awrt –930 (must use \(w\) and \(x\))

Scheme	Marks	AO
\(w = -930 + 50.2(32) + 80\)	M1	3.4
\(w = 756.4\)	A1	1.1b
	(2)

Mark scheme (e)

Scheme	Marks	AO
Negative residuals for all 5 observations with \(x \gt 33\) suggests the model systematically overestimates weights for the longest bream.	B1	3.5a
	(1)
	(10 marks)

Notes

B1: Evaluating the model for \(x \gt 33\) (must reference both the residuals and the model)
Negative correlation between residuals and length is B0.