Linear regression 579
∑
X =b 0 N+b 1
∑
Y
and
∑
XY=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 = 8 b 0 + 180 b 1
and 2045= 180 b 0 + 5100 b 1
Solving these simultaneous equations gives
b 0 = 44 .2andb 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 40cm, 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 40cm is 9.02N.
The radius,X, when the force is 32newtons 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 32Nis 7.08cm.
Now try the following exercise
Exercise 218 Further problemson linear
regression
In Problems 1 and 2, determine the equation of the
regression line ofYonX, correct to 3 significant
figures.
- X 14 18 23 30 50
Y 900 1200 1600 2100 3800
[Y=− 256 + 80. 6 X]
- 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. 216 X]
In Problems 3 and 4, determine the equations of
the regression lines ofXonYfor the data stated,
correct to 3 significant figures.
- The data given in Problem 1
[X= 3. 20 + 0. 0124 Y] - The data given in Problem 2
[X=− 0. 056 + 4. 56 Y] - 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 signifi-
cant figures.
[Y= 1. 142 + 2. 268 X]
- For the data given in Problem 5, determine the
equation of the regression line of current on
appliedvoltage,correct to3significant figures.
[X=− 0. 483 + 0. 440 Y] - 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 3mA and (b) the current flowing
when the applied voltage is 40volts,assuming
the regression lines are still true outside of the
range of values given.
[(a) 7.92V (b) 17.1mA] - In an experiment to determine the relationship
between force and momentum, a forceX,is
applied to a mass, by placing the mass on an
inclined plane, and the time,Y, for the velocity