S1 January 2010 Q6

EdexcelOld spec18 marksCorrelation & Regression

6. The blood pressures, \(p\) mmHg, and the ages, \(t\) years, of 7 hospital patients are shown in the table below.

Patient	A	B	C	D	E	F	G
\(t\)	42	74	48	35	56	26	60
\(p\)	98	130	120	88	182	80	135

[\(\sum t = 341,\ \sum p = 833,\ \sum t^2 = 18\,181,\ \sum p^2 = 106\,397,\ \sum tp = 42\,948\)]

(a) Find \(S_{pp}\), \(S_{tp}\) and \(S_{tt}\) for these data. (4)

(b) Calculate the product moment correlation coefficient for these data. (3)

(d) On the graph paper, draw the scatter diagram of blood pressure against age for these 7 patients. (2)

(e) Find the equation of the regression line of \(p\) on \(t\). (4)

(f) Plot your regression line on your scatter diagram. (2)

(g) Use your regression line to estimate the blood pressure of a 40 year old patient. (2)

Mark scheme (a)

Scheme	Marks
\(S_{pp} = 106397 - \dfrac{833^2}{7} = 7270\)	M1 A1
\(S_{tp} = 42948 - \dfrac{341 \times 833}{7} = 2369\), \(S_{tt} = 18181 - \dfrac{341^2}{7} = 1569.42857\ldots\) or \(\dfrac{10986}{7}\)	A1 A1
	(4)

Notes

M1 for at least one correct expression

1st A1 for \(S_{pp} = 7270\), 2nd A1 for \(S_{tp} = 2369\) or 2370, 3rd A1 for \(S_{tt} =\) awrt 1570

Mark scheme (b)

Scheme	Marks
\(r = \dfrac{2369}{\sqrt{7270 \times 1569.42857..}}\)	M1 A1ft
\(= 0.7013375\) awrt (0.701)	A1
	(3)

Notes

M1 for attempt at correct formula and at least one correct value (or correct ft) M0 for \(\dfrac{42948}{\sqrt{106397 \times 18181}}\)

A1ft All values correct or correct ft. Allow for an answer of 0.7 or 0.70
Answer only: awrt 0.701 is 3/3, answer of 0.7 or 0.70 is 2/3

Mark scheme (c)

Scheme	Marks
(Pmcc shows positive correlation.) Older patients have higher blood pressure	B1
	(1)

Notes

B1 for comment in context that interprets the fact that correlation is positive, as in scheme.
Must mention age and blood pressure in words, not just “\(t\)” and “\(p\)”.

Diagram for Q6 (d) + (f): scatter diagram of p against t with points (26, 80), (35, 88), (42, 98), (48, 120), (56, 182), (60, 135), (74, 130) and the line y = 1.5095x + 45.467 drawn — Mark scheme (d)

Scheme	Marks
(d) Points plotted correctly on graph: -1 each error or omission (within one square of correct position) * see diagram below for correct points	B2

	(2)

Mark scheme (e)

Scheme	Marks
\(b = \dfrac{2369}{1569.42857..} = 1.509466\ldots\)	M1 A1
\(a = \dfrac{833}{7} - b \times \dfrac{341}{7} = 45.467413\ldots\)	M1
\(p = 45.5 + 1.51t\)	A1
	(4)

Notes

1st M1 for use of the correct formula for \(b\), ft their values from (a)

1st A1 allow 1.5 or better

2nd M1 for use of \(\bar{y} - b\bar{x}\) with their values

2nd A1 for full equation with \(a\) = awrt 45.5 and \(b\) = awrt 1.51. Must be \(p\) in terms of \(t\), not \(x\) and \(y\).

Mark scheme (f)

Scheme	Marks
(f) Line drawn with correct intercept, and gradient (see the diagram for Q6 (d) + (f) in part (d))	B1ft B1
	(2)

Notes

1st B1ft ft their intercept (within one square). You may have to extend their line.

2nd B1 for correct gradient i.e. parallel to given line (Allow 1 square out when \(t\) = 80)

Mark scheme (g)

Scheme	Marks
\(t = 40,\ p = 105.84\ldots\) from equation or graph. awrt 106	M1 A1
	(2)
	(18 marks)

Notes

M1 for clear use of their equation with \(t\) = 40 or correct value from their graph.

A1 for awrt 106. Correct answer only (2/2) otherwise look for evidence on graph to award M1