Linear regression 579
X = b 0 N + b 1
Y
and
X Y = b 0
Y + b 1
Y 2
(from equations (3) and (4))
The values of the summations have been obtained
in part (a) giving:
145 = 8b 0 + 180b 1
and 2045 = 180b 0 + 5100b 1
Solving these simultaneous equations gives
b 0 = 44.2 and b 1 =−1.16, correct to 3 significant
figures. Thus the equation of the regression line of
radius on force is:
X = 44.2 − 1.16Y
The force, Y , at a radius of 40 cm, is obtained
from the regression line of force on radius, i.e.
y = 33.7 −0.617(40) = 9.02,
i.e. the force at a radius of 40 cm is 9.02 N.
The radius, X , when the force is 32 newtons is
obtained from the regression line of radius on
force, i.e. X = 44.2 −1.16(32) = 7.08,
i.e. the radius when the force is 32 N is 7.08 cm.
Now try the following exercise
Exercise 218 Further problems on linear
regression
In Problems 1 and 2, determine the equation of the
regression line of Y on X , correct to 3 significant
figures.
1. X
14
18
23
30
50
Y
900 1200 1600 2100 3800
[Y =−256 +80.6X ]
2. X 6
3
9 15 2 14 21 13
Y 1.3 0.7 2.0 3.7 0.5 2.9 4.5 2.7
[Y = 0.0477 +0.216X ]
In Problems 3 and 4, determine the equations of
the regression lines of X on Y for the data stated,
correct to 3 significant figures.
3. The data given in Problem 1
[X = 3.20 + 0.0124Y ]
4. The data given in Problem 2
[X =−0.056 +4.56Y ]
5. The relationship between the voltage applied
to an electrical circuit and the current flowing
is as shown:
Current (mA) Applied voltage (V)
2
5
4
11
6
15
8
19
10
24
12
28
14
33
Assuming a linear relationship, determine
the equation of the regression line of applied
voltage, Y , on current, X , correct to 4 significant figures.
[Y = 1.142 + 2.268X ]
6. For the data given in Problem 5, determine the
equation of the regression line of current on
applied voltage, correct to 3 significant figures.
[X =−0.483 +0.440Y ]
7. Draw the scatter diagram for the data given
in Problem 5 and show the regression lines
of applied voltage on current and current on
applied voltage. Hence determine the values
of (a) the applied voltage needed to give a
current of 3 mA and (b) the current flowing
when the applied voltage is 40 volts, assuming
the regression lines are still true outside of the
range of values given.
[(a) 7.92 V (b) 17.1 mA]
8. In an experiment to determine the relationship
between force and momentum, a force X , is
applied to a mass, by placing the mass on an
inclined plane, and the time, Y , for the velocity
X = b 0 N + b 1
Y
and
X Y = b 0
Y + b 1
Y 2
(from equations (3) and (4))
The values of the summations have been obtained
in part (a) giving:
145 = 8b 0 + 180b 1
and 2045 = 180b 0 + 5100b 1
Solving these simultaneous equations gives
b 0 = 44.2 and b 1 =−1.16, correct to 3 significant
figures. Thus the equation of the regression line of
radius on force is:
X = 44.2 − 1.16Y
The force, Y , at a radius of 40 cm, is obtained
from the regression line of force on radius, i.e.
y = 33.7 −0.617(40) = 9.02,
i.e. the force at a radius of 40 cm is 9.02 N.
The radius, X , when the force is 32 newtons is
obtained from the regression line of radius on
force, i.e. X = 44.2 −1.16(32) = 7.08,
i.e. the radius when the force is 32 N is 7.08 cm.
Now try the following exercise
Exercise 218 Further problems on linear
regression
In Problems 1 and 2, determine the equation of the
regression line of Y on X , correct to 3 significant
figures.
1. X
14
18
23
30
50
Y
900 1200 1600 2100 3800
[Y =−256 +80.6X ]
2. X 6
3
9 15 2 14 21 13
Y 1.3 0.7 2.0 3.7 0.5 2.9 4.5 2.7
[Y = 0.0477 +0.216X ]
In Problems 3 and 4, determine the equations of
the regression lines of X on Y for the data stated,
correct to 3 significant figures.
3. The data given in Problem 1
[X = 3.20 + 0.0124Y ]
4. The data given in Problem 2
[X =−0.056 +4.56Y ]
5. The relationship between the voltage applied
to an electrical circuit and the current flowing
is as shown:
Current (mA) Applied voltage (V)
2
5
4
11
6
15
8
19
10
24
12
28
14
33
Assuming a linear relationship, determine
the equation of the regression line of applied
voltage, Y , on current, X , correct to 4 significant figures.
[Y = 1.142 + 2.268X ]
6. For the data given in Problem 5, determine the
equation of the regression line of current on
applied voltage, correct to 3 significant figures.
[X =−0.483 +0.440Y ]
7. Draw the scatter diagram for the data given
in Problem 5 and show the regression lines
of applied voltage on current and current on
applied voltage. Hence determine the values
of (a) the applied voltage needed to give a
current of 3 mA and (b) the current flowing
when the applied voltage is 40 volts, assuming
the regression lines are still true outside of the
range of values given.
[(a) 7.92 V (b) 17.1 mA]
8. In an experiment to determine the relationship
between force and momentum, a force X , is
applied to a mass, by placing the mass on an
inclined plane, and the time, Y , for the velocity