Higher Engineering Mathematics, Sixth Edition

576 Higher Engineering Mathematics

in Fig. 60.1 and the co-ordinate values (H 3 ,H 4 ,etc.)

are taken as the deviations. The equation of the regres-

sion line is of the form:X=b 0 +b 1 Yand the normal

equations become:

∑

X=b 0 N+b 1

∑

Y (3)

∑

(XY)=b 0

∑

Y+b 1

∑

Y^2 (4)

whereXandYare the co-ordinate values,b 0 andb 1

are the regression coefficients ofXonYandNis the

number of co-ordinates. These normal equations are of

the regression line ofXonY, which is slightlydifferent

to the regression line ofYonX. The regression line of

XonYis used to estimated values ofXfor given values

ofY. The regression line ofYonXis 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 corre-

sponding value ofXis calledlinear interpolation.If

it lies outside of the range ofY-values of the extreme

co-ordinates than the process is calledlinear extrapo-

lationand 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,valuesofX

corresponding 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 inductive

reactance of an electrical circuit, the following

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 fre-

quency is required, the frequency is the independent

variable,X, and the inductive reactance is the depen-

dent variable,Y. The equation of the regression line of

YonXis:

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