Linear regression 577
The number of co-ordinate values given, N is 7.
Substituting in the normal equations gives:
855 = 7a 0 + 1400a 1
(1)
212000 = 1400a 0 + 350000a 1
(2)
1400 ×(1) gives:
1197000 = 9800a 0 + 1960000a 1
(3)
7 × (2) gives:
1484000 = 9800a 0 + 2450000a 1
(4)
(4) − (3) gives:
287000 = 0 + 490000a 1
from which, a 1 =
287000
490000
= 0.586
Substituting a 1 = 0.586 in equation (1) gives:
855 = 7a 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 + 0.586 X
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
variable X and the frequency is the dependent variable
Y . From equations 3 and 4, the equation of the regression
line of X on Y is:
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 = 7b 0 + 855b 1
212000 = 855b 0 + 128725b 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 + 1.69 Y
Problem 3. Use the regression equations
calculated 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 reactance is
107.5 ohms when the frequency is 175 Hz.
(b) From Problem 2, the regression equation of frequency on inductive reactance is
X =−6.15 +1.69 Y . When the inductive reactance, Y , is 250 ohms,
X =−6.15 +1.69(250) = 416.4 Hz, correct to 4
significant figures, i.e. the frequency is 416.4 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 equation
Y = 4.94 +0.586X is used, and X (frequency) values of
100 and 300 have been selected in order to find the corresponding Y values. These values gave the co-ordinates
as (100, 63.5) and (300, 180.7), shown as points A
and B in Fig. 60.2. Two co-ordinates for the regression
line of frequency on inductive reactance are calculated
using the equation X =−6.15 +1.69Y , 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 points C and D
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 necessary to do so, the co-ordinate values are also shown
to indicate that the regression lines do appear to be the
The number of co-ordinate values given, N is 7.
Substituting in the normal equations gives:
855 = 7a 0 + 1400a 1
(1)
212000 = 1400a 0 + 350000a 1
(2)
1400 ×(1) gives:
1197000 = 9800a 0 + 1960000a 1
(3)
7 × (2) gives:
1484000 = 9800a 0 + 2450000a 1
(4)
(4) − (3) gives:
287000 = 0 + 490000a 1
from which, a 1 =
287000
490000
= 0.586
Substituting a 1 = 0.586 in equation (1) gives:
855 = 7a 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 + 0.586 X
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
variable X and the frequency is the dependent variable
Y . From equations 3 and 4, the equation of the regression
line of X on Y is:
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 = 7b 0 + 855b 1
212000 = 855b 0 + 128725b 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 + 1.69 Y
Problem 3. Use the regression equations
calculated 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 reactance is
107.5 ohms when the frequency is 175 Hz.
(b) From Problem 2, the regression equation of frequency on inductive reactance is
X =−6.15 +1.69 Y . When the inductive reactance, Y , is 250 ohms,
X =−6.15 +1.69(250) = 416.4 Hz, correct to 4
significant figures, i.e. the frequency is 416.4 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 equation
Y = 4.94 +0.586X is used, and X (frequency) values of
100 and 300 have been selected in order to find the corresponding Y values. These values gave the co-ordinates
as (100, 63.5) and (300, 180.7), shown as points A
and B in Fig. 60.2. Two co-ordinates for the regression
line of frequency on inductive reactance are calculated
using the equation X =−6.15 +1.69Y , 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 points C and D
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 necessary to do so, the co-ordinate values are also shown
to indicate that the regression lines do appear to be the