S1 June 2015 Q4

EdexcelOld spec14 marksCorrelation & Regression

4. Statistical models can provide a cheap and quick way to describe a real world situation.

(a) Give two other reasons why statistical models are used. (2)

A scientist wants to develop a model to describe the relationship between the average daily temperature, \(x\) °C, and her household’s daily energy consumption, \(y\) kWh, in winter.

A random sample of the average daily temperature and her household’s daily energy consumption are taken from 10 winter days and shown in the table.

\(x\)	\(-0.4\)	\(-0.2\)	0.3	0.8	1.1	1.4	1.8	2.1	2.5	2.6
\(y\)	28	30	26	25	26	27	26	24	22	21

[You may use \(\sum x^2 = 24.76 \qquad \sum y = 255 \qquad \sum xy = 283.8 \qquad \mathrm{S}_{xx} = 10.36\)]

(b) Find \(\mathrm{S}_{xy}\) for these data. (3)

(c) Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + bx\)
Give the value of \(a\) and the value of \(b\) to 3 significant figures. (4)

(d) Give an interpretation of the value of \(a\) (1)

(e) Estimate her household’s daily energy consumption when the average daily temperature is 2 °C (2)

The scientist wants to use the linear regression model to predict her household’s energy consumption in the summer.

(f) Discuss the reliability of using this model to predict her household’s energy consumption in the summer. (2)

Mark scheme (a)

Scheme	Marks
To simplify (or represent) a real world problem (o.e.) To improve understanding (o.e.) To analyse a real world problem or can change variables/replicate easily (oe) To make predictions or find estimates (o.e.)	B1g B1h
	(2)

Notes

Make sure reasons refer to models and not tests

1st B1g (be fairly generous) for a sensible reason not using “quick”, “cheap” or “describe”

2nd B1h (be slightly harder) for two convincing reasons (both based on the list above)
Use professional judgement and mark as B0B0 or B1B0 or B1B1 do not use B0B1

Mark scheme (b)

Scheme	Marks
\(\sum x = 12\)	B1
\(\mathrm{S}_{xy} = 283.8 - \dfrac{\text{'}12\text{'} \times 255}{10},\ = -\)22.2	M1,A1cao
	(3)

Notes

B1 for \(\sum x = 12\) (May be by the table) (Can be implied by 3060 seen or the next line)

M1 for attempt at correct formula (ft their \(\sum x\) where \(10 \lt \sum x \lt 14\))

A1 for \(-22.2\) only

Mark scheme (c)

Scheme	Marks
\(b = \dfrac{\text{'}-22.2\text{'}}{10.36} =,\ -2.142857\ldots\) (A1 for awrt \(-2.1\))	M1A1
\(\left[a = \bar{y} - b\bar{x} \quad \Rightarrow\right]\ a = \dfrac{255}{10} - \text{'}b\text{'} \times \dfrac{\text{"}12\text{"}}{10} = 28.07143\)	M1
\(y = 28.1 - 2.14x\) \(\left[\text{Condone: } y = 28.1 + -2.14x\right]\)	A1
	(4)

Notes

M1 for a correct expression for \(b\) (ft their \(\mathrm{S}_{xy} \neq 283.8\))

A1 for awrt \(-2.1\) (allow \(-15/7\))

M1 for a correct expression for \(a\) and ft their 12 (allow use of a letter \(b\))

A1 for \(y = 28.1 - 2.14x\) (awrt 28.1 and awrt \(-2.14\)) Must be \(y\) and \(x\) and no fractions

Mark scheme (d)

Scheme	Marks
(28.1 kWh) of energy are used when the temperature is 0[°C]	B1
	(1)

Notes

B1 for a contextualised interpretation e.g. the amount of energy used when temperature is 0[°C] or [28.1] kWh used when temp. is 0[°C] [Can ft their 28.1]Need temp or °sign
[B0 for “value of \(y\) when \(x = 0\)” since no context in words]

Scheme	Marks
\(y = 28.1 - 2.14(2) = \)	M1
awrt 23.8	A1
	(2)

Mark scheme (f)

Scheme	Marks
The regression model is based on temperatures from the winter,	B1
so not reliable in the summer. Stating it is reliable (whatever the reason) is B0B0	dB1
	(2)
	(14 marks)

Notes

B1 for reasoning to suggest that temperatures are different in summer or the model was based only on data from the winter. Allow mention of extrapolation (o.e.)

dB1 so not reliable.