Higher Engineering Mathematics

LINEAR REGRESSION 573

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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 indepen-
dent variableXand the frequency is the dependent
variableY. From equations 3 and 4, the equation of
the regression 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 cal-

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

of inductive reactance when the frequency is

175 Hz and (b) the value of frequency when

the inductive reactance is 250 ohms, 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 of

inductive reactance on frequency is

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

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

to 4 significant figures, i.e. the inductive reac-

tance is 107.5 ohms when the frequency is

175 Hz.

(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 250 ohms,

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

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

when the inductive reactance is 250 ohms.

The graph depicting the two regression lines is

shown in Fig. 60.2. To obtain the regression line

of inductive reactance on frequency the regression

line equationY= 4. 94 + 0. 586 Xis used, andX(fre-

quency) values of 100 and 300 have been selected

in order to find the correspondingYvalues. These

values gave the co-ordinates as (100, 63.5) and (300,

180.7), shown as pointsAandBin Fig. 60.2. Two

co-ordinates for the regression line of frequency on

inductive reactance are calculated using the equa-

tion X=− 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

D

B

A

C

0 100 200 300 400 500

Frequency in hertz

50

100

150

200

250

300

Inductive reactance in ohms

Y

X

Figure 60.2