Higher Engineering Mathematics, Sixth Edition

572 Higher Engineering Mathematics

Problem 2. The relationship between expenditure

on welfare services and absenteeism for similar

periods of time is shown below for a small company.

Expenditure

(£′000) 3.5 5.0 7.0 10 12 15 18

Days lost 241 318 174 110 147 122 86

Determine the coefficient of linear correlation for

this data.

LetXbe the expenditure in thousands of pounds andY

be the days lost.

The coefficient of correlation,

r=

∑

xy

√{(∑

x^2

)(∑

y^2

)}

wherex=(X−X)andy=(Y−Y),XandYbeing the

mean values ofXandYrespectively. Using a tabular

approach:

X Y x=(X−X) y=(Y−Y)

3.5 241 −6.57 69.9

5.0 318 −5.07 146.9

7.0 174 −3.07 2.9

10 110 −0.07 −61.1

12 147 1.93 −24.1

15 122 4.93 −49.1

18 86 7.93 −85.1

∑

X= 70. 5 , X=

70. 5

7

= 10. 07

∑

Y= 1198 , Y=

1198

7

= 171. 1

xy x^2 y^2

−459.2 43.2 4886

−744.8 25.7 21580

−8.9 9.4 8

4.3 0 3733

xy x^2 y^2

−46.5 3.7 581

−242.1 24.3 2411

−674.8 62.9 7242

∑

xy=− 2172

∑

x^2 = 169. 2

∑

y^2 = 40441

Thus

r=

− 2172

√

[169. 2 ×40441]

=− 0. 830

This shows that there isfairlygood inverse correlation

between the expenditure on welfare and days lost due

to absenteeism.

Problem 3. The relationship between monthly

car sales and income from the sale of petrol for a

garage is as shown:

Cars sold 25 31214 7 32814 7313

Income from

petrol sales 1291321172231471710911

(£′000)

Determine the linear coefficient of correlation

between these quantities.

Let Xrepresent the number of cars sold andYthe

income,inthousandsofpounds,frompetrol sales.Using

the tabular approach:

X Y x=(X−X) y=(Y−Y)

2 12 −7.25 −6.25

5 9 −4.25 −9.25

3 13 −6.25 −5.25

12 21 2.75 2.75

14 17 4.75 −1.25

7 22 −2.25 3.75

3 31 −6.25 12.75

28 47 18.75 28.75

14 17 4.75 −1.25

7 10 −2.25 −8.25

3 9 −6.25 −9.25