S1 January 2007 Q1

EdexcelOld spec15 marksCorrelation & Regression

1. As part of a statistics project, Gill collected data relating to the length of time, to the nearest minute, spent by shoppers in a supermarket and the amount of money they spent. Her data for a random sample of 10 shoppers are summarised in the table below, where \(t\) represents time and £\(m\) the amount spent over £20.

\(t\) (minutes)	£\(m\)
15	−3
23	17
5	−19
16	4
30	12
6	−9
32	27
23	6
35	20
27	6

(a) Write down the actual amount spent by the shopper who was in the supermarket for 15 minutes. (1)

(b) Calculate \(S_{tt}\), \(S_{mm}\) and \(S_{tm}\).

(You may use \(\Sigma t^2 = 5478 \quad \Sigma m^2 = 2101 \quad \Sigma tm = 2485\))

(6)

(d) Write down the value of the product moment correlation coefficient between \(t\) and the actual amount spent. Give a reason to justify your value. (2)

On another day Gill collected similar data. For these data the product moment correlation coefficient was 0.178

(e) Give an interpretation to both of these coefficients. (2)

(f) Suggest a practical reason why these two values are so different. (1)

Scheme	Marks
(£) 17	B1
	(1)

Mark scheme (b)

Scheme	Marks
\(\sum t = 212\) and \(\sum m = 61\)	B1, B1
\(S_{tm} = 2485 - \dfrac{61 \times 212}{10}, = 1191.8\)	M1, A1
\(S_{tt} = 983.6\) (awrt 984) and \(S_{mm} = 1728.9\) (awrt 1730)	A1, A1
	(6)

Notes

B1, B1 (Accept as totals under each column in qu.)

M1, A1 awrt 1190 or 119 (3sf)

A1, A1 (or 98.4 and 173)

M1 for one correct formula seen, f.t. their \(\sum t, \sum m\) [Use 1^st A1 for 1 correct, 2^nd A1 for 2 etc]

Mark scheme (c)

Scheme	Marks
\(r = \dfrac{1191.8}{\sqrt{983.6 \times 1728.9}}\)	M1, A1f.t.
\(= 0.913922\ldots\)	A1
	(3)

Notes

A1 awrt 0.914

M1 for attempt at correct formula, \(\dfrac{2485}{\sqrt{2101 \times 5478}}\) scores M1A0A0

A1ft f.t. their values for \(S_{tt}\) etc from (b) but don’t give for \(S_{tt} = 5478\) etc (see above)

Answer only (awrt 0.914) scores 3/3, 0.913 (i.e. truncation) can score M1A1ft by implication.

Mark scheme (d)

Scheme	Marks
0.914	B1f.t. (\(\|r\| \lt 1\))
e.g. linear transformation, coding does not affect coefficient (or recalculate)	dB1
	(2)

Notes

B1f.t. (Must be the same as (c) or awrt 0.914)

2^nd B1 dependent on 1^st B1 Accept \(\sum m = 261, \sum m^2 = 8541, \sum tm = 6725 \to 0.914\)

Mark scheme (e)

Scheme	Marks
0.914 suggests longer spent shopping the more spent. (Idea more time, more spent)	B1
0.178 different amounts spent for same time.	B1
	(2)

Notes

One mark for a sensible comment relating to each coefficient

For 0.178 allow “little or no link between time and amount spent”. Must be in context.

Just saying 0.914 is strong +ve correlation between amount spent and time shopping and 0.178 is weak correlation …scores B0B0.

Mark scheme (f)

Scheme	Marks
e.g. might spend short time buying 1 expensive item OR might spend a long time checking for bargains, talking, buying lots of cheap items.	B1g
	(1)
	(15 marks)

Notes

B1g for a sensible, practical suggestion showing that other factors might affect the amount spent.
E.g. different day (weekend vs weekday) or time of day (time spent queuing if busy)