AS June 2024 Q5

EdexcelAS paperCurrent spec8 marksCorrelation Linear Regression

5. A random sample of 24 adults is taken. The height, \(h\) metres, and the arm span, \(s\) metres, for each adult are recorded.

These data are summarised below.

\[\mathrm{S}_{hh} = 0.377 \qquad \mathrm{S}_{sh} = 0.352 \qquad \bar{s} = 1.70 \qquad \bar{h} = 1.68\]

The least squares regression line of \(h\) on \(s\) is

\[h = a + 0.919s\]

where \(a\) is a constant.

(a) Calculate the product moment correlation coefficient. (3)

A doctor uses the least squares regression line of \(h\) on \(s\) as a model to predict a person’s height based on their arm span.

(b) Use the model to predict the height of an adult with arm span 1.79 metres. (2)

Ewan has an arm span of 1.70 metres and a height of 1.75 metres. His information is added to the sample as the 25th adult.

(c) Explain how the gradient of the regression line for the sample of 25 adults compares with the gradient of the regression line for the original sample of 24 adults.
Give a reason for your answer. (3)

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{S}_{ss} = \dfrac{\mathrm{S}_{sh}}{b} = \dfrac{0.352}{0.919}\,[= 0.383\ldots]\) or \(\dfrac{352}{919}\)	M1	2.1
\(r = \dfrac{0.352}{\sqrt{0.377 \times \text{“}0.383\text{”}}}\)	M1	1.1b
\(r = 0.9263\ldots\) awrt 0.926	A1	1.1b
	(3)

Notes

1^st M1: realising the need to find \(\mathrm{S}_{ss}\) Must have \(0.919 = \dfrac{0.352}{\mathrm{S}_{ss}}\) or better

2^nd M1: attempt to find \(r\) ft their \(\mathrm{S}_{ss}\) provided \(\mathrm{S}_{ss} \gt 0.33\)

A1: awrt 0.926

Mark scheme (b)

Scheme	Marks	AO
\(h - 1.68 = 0.919(1.79 - 1.70)\)	M1	3.4
\(h = 1.76271\) awrt 1.76	A1	1.1b
	(2)

Notes

M1: using model with means and 1.79 to find \(h\). May see: \(0.1177 + 0.919 \times 1.79\)
Allow \(0.1177 + 0.919 \times 1.75\) as MR

A1: awrt 1.76

Mark scheme (c)

Scheme	Marks	AO
\(\mathrm{S}_{ss}\) would remain the same since \(s_{25} = \bar{s}\)	M1	2.4
\(\mathrm{S}_{sh}\) would remain the same additional \((s - \bar{s})(h - \bar{h}) = 0\)	M1	1.1b
So the gradient would be unchanged.	A1	2.2a
	(3)
	(8 marks)

Notes

1^st M1: explaining the effect on \(\mathrm{S}_{ss}\)

2^nd M1: considering the effect on \(\mathrm{S}_{sh}\)

A1: correct deduction

Alternative

1^st M1 explaining that \(s_{25} = \bar{s}\) so point lies on \(s = \bar{s}\) Must be explicit

2^nd M1 \(h_{25} \gt \bar{h}\) so the line would move upwards (parallel to original regression line)

A1 correct deduction