June 2018 Paper 3 Q2

EdexcelCurrent spec7 marksIncludes hypothesis testingCorrelation & Regression

2. Tessa owns a small clothes shop in a seaside town. She records the weekly sales figures, £\(w\), and the average weekly temperature, \(t\)°C, for 8 weeks during the summer.

The product moment correlation coefficient for these data is \(-0.915\)

(a) Stating your hypotheses clearly and using a 5% level of significance, test whether or not the correlation between sales figures and average weekly temperature is negative. (3)

(b) Suggest a possible reason for this correlation. (1)

Tessa suggests that a linear regression model could be used to model these data.

(d) State, giving a reason, which variable would be the explanatory variable. (1)

Tessa calculated the linear regression equation as \(w = 10\,755 - 171t\)

(e) Give an interpretation of the gradient of this regression equation. (1)

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{H}_0: \rho = 0 \qquad \mathrm{H}_1: \rho \lt 0\)	B1	2.5
Critical value: \(-0.6215\) (Allow any cv in range \(0.5 \lt \|\text{cv}\| \lt 0.75\))	M1	1.1a
\(r \lt -0.6215\) so significant result and there is evidence of a negative correlation between \(w\) and \(t\)	A1	2.2b
	(3)

Notes

B1 for both hypotheses in terms of \(\rho\)

M1 for the critical value: sight of \(\pm 0.6215\) or any cv such that \(0.5 \lt |\text{cv}| \lt 0.75\)

A1 must reject \(\mathrm{H}_0\) on basis of comparing \(-0.915\) with \(-0.6215\) (if \(-0.915 \lt 0.6215\) is seen then A0 but may use \(|r|\) o.e. which is fine) and mention “negative”, “correlation/relationship” and at least “\(w\)” and “\(t\)”

Mark scheme (b)

Scheme	Marks	AO
e.g. As temperature increases people spend more time on the beach and less time shopping (o.e.)	B1	2.4
	(1)

Notes

B1 for a suitable reason to explain negative correlation using the context given.
e.g. “As temperature drops people are more likely to go shopping (than to the beach)”
e.g. “As temperature increases people will be outside rather than in shops”
A mere description in context of negative correlation is B0

SO e.g. “As temperature increases people don’t want to go shopping/buy clothes” is B0
e.g. “Less clothes needed as temp increases” is B0

Mark scheme (c)

Scheme	Marks	AO
Since \(r\) is close to \(-1\), it is consistent with the suggestion	B1	2.4
	(1)

Notes

B1 for a suitable reason e.g. “strong”/“significant”/“near perfect” “correlation”, \(|r|\) close to 1 and saying it is consistent with the suggestion. Allow “yes” followed by the reason.

Mark scheme (d)

Scheme	Marks	AO
\(t\) will be the explanatory variable since sales are likely to depend on the temperature	B1	2.4
	(1)

Notes

B1 For identifying \(t\) and giving a suitable reason.
Need idea that “\(w\) depends on \(t\)” or “\(w\) responds to \(t\)” or “\(t\) affects \(w\)” (o.e.)
Allow \(t\) (temperature) affects the other variable etc
Just saying “\(t\) is the independent variable” or “\(t\) explains change in \(w\)” is B0
N. B. Suggesting causation is B0 e.g. “\(t\) causes \(w\) to decrease”

Mark scheme (e)

Scheme	Marks	AO
Every degree rise in temperature leads to a drop in weekly earnings of £171	B1	3.4
	(1)
	(7 marks)

Notes

B1 for a description that conveys the idea of rate per degree Celsius.
Must have 171, condone missing “£” sign.