Higher Engineering Mathematics

572 STATISTICS AND PROBABILITY

are of the regression line ofXonY, which is slightly

different to the regression line ofYonX. The regres-

sion line ofXonYis used to estimated values ofX

for given values ofY. The regression line ofYonX

is used to determine any value ofYcorresponding to

a given value ofX. If the value ofYlies within the

range ofY-values of the extreme co-ordinates, the

process of finding the corresponding value ofXis

calledlinear interpolation. If it lies outside of the

range ofY-values of the extreme co-ordinates than

the process is calledlinear extrapolationand the

assumption must be made that the line of best fit

extends outside of the range of the co-ordinate values

given.

By using the regression line ofXonY, values of

Xcorresponding to given values ofYmay be found

by either interpolation or extrapolation.

60.3 Worked problems on linear

regression

Problem 1. In an experiment to determine the

relationship between frequency and the induc-

tive reactance of an electrical circuit, the fol-

lowing results were obtained:

Frequency Inductive reactance

(Hz) (ohms)

50 30

100 65

150 90

200 130

250 150

300 190

350 200

Determine the equation of the regression line

of inductive reactance on frequency, assuming a

linear relationship.

Since the regression line of inductive reactance on

frequency is required, the frequency is the indepen-

dent variable,X, and the inductive reactance is the

dependent variable,Y. The equation of the regression

line ofYonXis:

Y=a 0 +a 1 X

and the regression coefficientsa 0 anda 1 are obtained

by using the normal equations

∑

Y=a 0 N+a 1

∑

X

and

∑

XY=a 0

∑

X+a 1

∑

X^2

(from equations (1) and (2))

A tabular approach is used to determine the summed

quantities.

Frequency,X Inductive X^2

reactance,Y

50 30 2500

100 65 10000

150 90 22500

200 130 40000

250 150 62500

300 190 90000

350 200 122500

∑

X= 1400

∑

Y= 855

∑

X^2 = 350000

XY Y^2

1500 900

6500 4225

13500 8100

26000 16900

37500 22500

57000 36100

70000 40000

∑

XY= 212000

∑

Y^2 = 128725

The number of co-ordinate values given,Nis 7.

Substituting in the normal equations gives:

855 = 7 a 0 + 1400 a 1 (1)

212000 = 1400 a 0 + 350000 a 1 (2)

1400 ×(1) gives:

1197000 = 9800 a 0 + 1960000 a 1 (3)

7 ×(2) gives:

1484000 = 9800 a 0 + 2450000 a 1 (4)

(4)−(3) gives:

287000 = 0 + 490000 a 1

from which,a 1 =

287000

490000

= 0. 586