June 2024 Paper 2 Q10

OCR ACurrent spec11 marksIncludes hypothesis testingCorrelation & Regression

10 Each month, the manager of a large store records the number, \(c\), of customers who visit the store, and the amount, £\(h\), spent on heating during that month. The manager wants to test whether there is linear correlation between \(c\) and \(h\).

For a randomly chosen year the value of Pearson’s product-moment correlation coefficient, \(r\), between \(c\) and \(h\) was \(-0.798\), correct to 3 significant figures.

(a) Using the table below, carry out the test at the 1% significance level. [5]

(b) Describe briefly two main features of a scatter diagram that could be drawn to illustrate the values of \(c\) and \(h\) for this year. There is no need to draw a diagram. [2]

(c) The manager makes the following statement.
“The value of \(r\) shows that when we spend more on heating, fewer customers visit the store. So we should spend less on heating.”
Comment briefly on this statement, making reference to the context. [2]

(d) Give a statement about a probability to explain the meaning of the value 0.7155 in the table below. [2]

Critical values of Pearson’s product-moment correlation coefficient

	1-tail test	5%	2.5%	1%	0.5%
	2-tail test	10%	5%	2%	1%
\(n\)	10	0.5494	0.6319	0.7155	0.7646
	11	0.5214	0.6021	0.6851	0.7348
	12	0.4973	0.5760	0.6581	0.7079
	13	0.4762	0.5529	0.6339	0.6835

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{H}_0: \rho = 0\)	B1	1.1
\(\mathrm{H}_1: \rho \neq 0\) where \(\rho\) is the correlation coefficient for the population or where \(\rho\) is the correlation coefficient between amount spent (\(h\)) and no. of customers (\(c\))	B1	2.5
\(0.798 \gt 0.7079\) oe	B1FT	1.1
Reject \(\mathrm{H}_0\)	M1	1.1
Sufficient evidence for a (linear) correlation between amount spent (\(h\)) and no. of customers (\(c\)) oe	A1	2.2b
	[5]

Notes

B1 B1: Subtract B1 for each error:

Undefined \(\rho\): B1B0
1-tail: B1B0
Allow other letters (but not \(r\), \(c\) or \(h\): B1B0)
Hypotheses in words (no parameter): B1B0
- \(\mathrm{H}_0\): There is no correlation
- \(\mathrm{H}_1\): There is correlation
- Do not allow “negative” or “positive” correlation for \(\mathrm{H}_1\): B0B0

Accept “pmcc” for correlation coefficient

B1FT: FT their setup/hypotheses

e.g. for \(\mathrm{H}_1: \rho \lt 0\), compare 0.798 with 0.6581

Must use \(n = 12\) and specify a corresponding value from the table

0.6581 or 0.7079 only (NB not 0.6851 from \(n = 11\))

Must compare this with 0.798 with the same sign

Condone \(-0.798 \lt -0.7079\) or \(|-0.798|\)
Do not accept \(-0.798 \lt 0.6581\)

NB this is the only mark that can be scored with no hypotheses

M1: This step must be seen, consistent with their hypotheses and their comparison. Condone Accept \(\mathrm{H}_1\)

A1: Conclusion must be in context, not definite and consistent with their hypotheses and comparison.

Disregard any mention of “negative” or “positive”
“Relationship” A0
“Prove(d)” A0
Condone “there is evidence of a linear correlation between \(h\) and \(c\)”
Condone “significant” for “sufficient”

Must conclude that there is evidence for a correlation.

Mark scheme (b)

Scheme	Marks	AO
Points (fairly) close to a (straight) line	B1	1.2
with negative gradient oe	B1	1.2
	[2]

Notes

B1: For a statement about the relative strength of the linear correlation:

Accept “points form a line” or
Accept “points lie (relatively) close to the line”
Not “points are close together” or “close to each other”

B1: For a statement about the appearance of the negative correlation:

Accept “line from 2^nd to 4^th quadrant”
Accept “two clusters in top left and bottom right”
Accept “line will be downwards sloping”
Not “negatively correlated” (must be a feature of the scatter diagram)

This mark only may be implied by a sketch (showing a scatter diagram with negative correlation, with or without a line of best fit).

Mark scheme (c)

Scheme	Marks	AO
Correlation does not imply causation	B1	2.3
A suggested 3rd factor affecting both \(c\) & \(h\) e.g. time of year, temperature, weather	B1	2.4
	[2]

Notes

B1: oe, may be implied (but do not allow “independent”)

B1: Any sensible comment about the statement but must be in context:

Accept “Some people may not visit if the store is too cold”
Accept “the shop being too warm may mean customers don’t want to go inside”
Not “There may be a third factor affecting both \(c\) and \(h\)” (a possible factor must be specified)
Accept “more people in the store may mean there is less need for heating” (so the implication might be the other way around) or equivalent statements about causation

Mark scheme (d)

Scheme	Marks	AO
If no (linear) correlation in the population, then for (a sample of) 10 (pairs)	B1	1.2
\(\mathrm{P}(r \gt 0.7155) = 0.01\) or \(\mathrm{P}(\|r\| \gt 0.7155) = 0.02\)	B1	2.5
	[2]

Notes

B1: For the setup, condone “n=10” but must reference ‘no correlation’

B1: Allow \(\geqslant\)
Accept an equivalent statement in words, but it must be about a probability:

e.g. “The probability that the pmcc is greater than 0.7155 is 1%”
e.g. “There is a 1% chance that there is actually no correlation when \(r\) is greater than 0.7155.”