Higher Engineering Mathematics, Sixth Edition

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.

1. X 14 18 23 30 50
   Y 900 1200 1600 2100 3800

[Y=− 256 + 80. 6 X]

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. 216 X]

In Problems 3 and 4, determine the equations of

the regression lines ofXonYfor the data stated,

correct to 3 significant figures.

3. The data given in Problem 1
   [X= 3. 20 + 0. 0124 Y]


4. The data given in Problem 2
   [X=− 0. 056 + 4. 56 Y]


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 signifi-

cant figures.

[Y= 1. 142 + 2. 268 X]

6. 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]


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 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]


8. 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