S1 June 2013 (R) Q1

EdexcelOld spec10 marksCorrelation & Regression

1. Sammy is studying the number of units of gas, \(g\), and the number of units of electricity, \(e\), used in her house each week. A random sample of 10 weeks use was recorded and the data for each week were coded so that \(x = \dfrac{g - 60}{4}\) and \(y = \dfrac{e}{10}\). The results for the coded data are summarised below

\[\sum x = 48.0 \quad \sum y = 58.0 \quad \mathrm{S}_{xx} = 312.1 \quad \mathrm{S}_{yy} = 2.10 \quad \mathrm{S}_{xy} = 18.35\]

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

(b) Hence find the equation of the regression line of \(e\) on \(g\) in the form \(e = c + dg\).
Give the values of \(c\) and \(d\) correct to 2 significant figures. (4)

(c) Use your regression equation to estimate the number of units of electricity used in a week when 100 units of gas were used. (2)

Mark scheme (a)

Scheme	Marks
\(b = \dfrac{18.35}{312.1}\ [= 0.058795\ldots]\)	M1
\(a = 5.8 - \text{"}0.058795\ldots\text{"}\times 4.8\)	M1
\(\boldsymbol{a}\) = awrt 5.52	A1
So \(\boldsymbol{y = 5.52 + 0.0588x}\)	A1
	(4)

Notes

1st M1 for a correct expression for \(b\)

2nd M1 for a correct expression for \(a\) – ft their value of \(b\)

1st A1 for \(a\) = awrt 5.52

2nd A1 for a correct equation in \(y\) and \(x\) with \(a\) and \(b\) correct to awrt 3 sf

Mark scheme (b)

Scheme	Marks
\(\dfrac{e}{10} = \text{"}5.52\text{"} + \text{"}0.0588\text{"}\times\left(\dfrac{g - 60}{4}\right)\)	M1
\(4e = 220.71 + 0.588(g - 60)\)	dM1
\(\boldsymbol{e = 46 + 0.15g}\)	A1A1
	(4)

Notes

1st M1 for substitutions into their equation to get an equation in \(e\) and \(g\). Need \(y = \frac{e}{10}\) and \(x = \frac{g - 60}{4}\)

2nd dM1 Dep. on 1st M1 for an attempt to simplify (at least removing fractions). Allow one slip

1st A1 for an equation \(e\) = awrt 46 \(\pm\) …..

2nd A1 for an equation \(e = \)….+ awrt \(0.15g\)

ALT 1st M1 for use of \(d = \dfrac{10\times\text{"their } b\text{"}}{4}\) or sight of 0.15 used as gradient
2nd dM1 Dep. on 1st M1 for use of \(\bar{e} = 10\times\text{"their } \bar{y}\text{"}\) or sight of 58 and use of \(\bar{g} = 4\times\text{"their } \bar{x}\text{"} + 60\) or sight of 79.2 and use of these values to find \(c\) in \(c = \bar{e} - d\bar{g}\)

Mark scheme (c)

Scheme	Marks
\(e = \text{"}46\text{"} + \text{"}0.15\text{"}\times 100\)	M1
\(= \underline{\mathbf{61}}\)	A1
	(2)
	(10 marks)

Notes

M1 for substituting \(g = 100\) into their new equation (or \(x = 10\) and then attempting to \(\times\) ans.by 10)

A1 for awrt 61