Higher Engineering Mathematics, Sixth Edition

Linear regression 577

The number of co-ordinate values given, N is 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

Substitutinga 1 = 0 .586 in equation (1) gives:

855 = 7 a 0 + 1400 ( 0. 586 )

i.e. a 0 =

855 − 820. 4

7

= 4. 94

Thus the equation of the regression line of inductive
reactance on frequency is:

Y=4.94+0.586X

Problem 2. For the data given in Problem 1,

determine the equation of the regression line of

frequency on inductive reactance, assuming a linear

relationship.

In this case, the inductive reactance is the independent
variableXand the frequency is the dependent variable
Y.Fromequations3and4,theequationoftheregression
line ofXonYis:

X=b 0 +b 1 Y

and the normal equations are

∑

X=b 0 N+b 1

∑

Y

and

∑

XY=b 0

∑

Y+b 1

∑

Y^2

From the table shown in Problem 1, the simultaneous
equations are:

1400 = 7 b 0 + 855 b 1

212000 = 855 b 0 + 128725 b 1

Solving these equations in a similar way to that in

Problem 1 gives:

b 0 =− 6. 15

and b 1 = 1. 69 ,correct to 3 significant figures.

Thus the equation of the regression line of frequency on

inductive reactance is:

X=−6.15+1.69Y

Problem 3. Use the regression equations

calculated in Problems 1 and 2 to find (a) the value

of inductive reactance when the frequency is 175Hz

and (b) the value of frequency when the inductive

reactance is 250ohms, assuming the line of best fit

extends outside of the given co-ordinate values.

Draw a graph showing the two regression lines.

(a) From Problem 1, the regression equation ofinduc-

tive reactance on frequency is

Y= 4. 94 + 0. 586 X. When the frequency, X,is

175Hz,Y= 4. 94 + 0. 586 ( 175 )= 107 .5, correct to

4 significant figures, i.e. the inductive reactance is

107.5ohmswhen the frequency is 175Hz.

(b) From Problem 2, the regression equation of fre-

quency on inductive reactance is

X=− 6. 15 + 1. 69 Y. When the inductive reac-

tance,Y, is 250ohms,

X=− 6. 15 + 1. 69 ( 250 )= 416 .4Hz, correct to 4

significant figures, i.e. the frequency is416.4Hz

when the inductive reactance is 250ohms.

The graph depicting the two regression lines is shown

in Fig. 60.2. To obtain the regression line of induc-

tive reactance on frequency the regression line equation

Y= 4. 94 + 0. 586 Xis used, andX(frequency) values of

100 and 300 have been selected in order to find the cor-

respondingYvalues. These values gave the co-ordinates

as (100, 63.5) and (300, 180.7), shown as points A

andBin Fig. 60.2. Two co-ordinates for the regression

line of frequency on inductive reactance are calculated

using the equationX=− 6. 15 + 1. 69 Y, the values of

inductive reactance of 50 and 150 being used to obtain

the co-ordinate values. These values gave co-ordinates

(78.4, 50) and (247.4, 150), shown as pointsCandD

in Fig. 60.2.

It can be seen from Fig. 60.2 that to the scale drawn, the

two regression lines coincide. Although it is not nec-

essary to do so, the co-ordinate values are also shown

to indicate that the regression lines do appear to be the