BUSINESS STATISTICS PRESENTATION

ON REGRESSION ANALYSIS

PRESENTED BY-

HARI BHATTARAI

MBA 1ST YEAR

OBJECTIVES OF THE PRESENTATION-

What is regression analysis

Types and methods of regression analysis

Practical aspect of regression analysis with an

example

INTRODUCTION-

Regression analysis is the statistical tool which is

employed for the purpose of forecasting or making

estimates

Here we make use of various mathematical formulas

and assumptions to describe a real world situation.

In every situation, estimation becomes easy once it is

known that the variable to be estimated is related to and

dependent to some other variable.

For making estimates we first have to model the relationship

between the variable involved .

Models can me broadly be classified into –

Linear regression-

Linear regression analysis is a powerful technique used for

predicting the unknown value of a variable from the known

value of another variable.

More precisely, if X and Y are two related variables, then

linear regression analysis helps us to predict the value of Y

for a given value of X or vice verse.

For example age of a human being and maturity are related

variables. Then linear regression analyses can predict level

of maturity given age of a human being.

Multiple regression-

Multiple regression analysis is a powerful technique

used for predicting the unknown value of a variable from

the known value of two or more variables- also called the

predictors.

Multiple regression analysis helps us to predict the value

of Y for given values of X1, X2, …, Xk.

For example the yield of rice per acre depends upon

quality of seed, fertility of soil, fertilizer used, temperature,

rainfall. If one is interested to study the joint affect of all

these variables on rice yield, one can use this technique.

Dependent and Independent Variables-

By linear regression, we mean models with just one

independent and one dependent variable. The variable whose

value is to be predicted is known as the dependent variable

and the one whose known value is used for prediction is

known as the independent variable.

By multiple regression, we mean models with just one

dependent and two or more independent variables. The

variable whose value is to be predicted is known as the

dependent variable and the ones whose known values are

used for prediction are known independent variables.

Methods of solving regression models-

1) GRAPHICAL METHOD-

In this graphical method the average relationship

between the dependent variable and independent

variable is expressed by a line called “line of best fit”.

Example: Experience( in years) Income( in ‘000)

15 150

10 120

5 60

3 40

8 70

9 90

2 4 6 8 10 12 14 16

60

90

120

150

30

180

210

18

240

0

Line of best fit

income

experience

2) ALGEBRIC METHOD-In this method we make use of regression equation

and regression coefficients.

Regression equation(Linear).

The general equation is given by-y = a + bx a is the intercept

b is the slope of line

With the use of the above general equation we find the normal equations

Multiplying the general equation by N and taking the summatation of it

we find the first normal equation i.e.

∑Y = N.a + b∑X

And again to find the second normal equation we multiply the general

equation by x and then take the summatation i.e.

∑XY=a ∑X + b ∑X2

A statistical technique used to explain or predict thebehaviour of a dependent

variable

General equation => y = a + b1 x1 + b2x2 + .........+ bnxn

Regression equation(Multiple).

Normal equations for multiple regression are:

∑Y = N.a + b1∑X1 + b2∑X2

∑X1Y= a ∑X1 + b1 ∑ X1 2 + b2∑ X1 . X2

∑X2Y= a ∑X2 + b1 ∑ X1 . X2 + b2∑ X22

Lines of Regression

There are two lines of regression- that of Y on X and X on Y.

The line of regression of Y on X is given by Y = a + bX where a and b

are unknown constants known as intercept and slope of the equation.

This is used to predict the unknown value of variable Y when value of

variable X is known.

On the other hand, the line of regression of X on Y is given by X = c + dY

which is used to predict the unknown value of variable X using the

known value of variable Y.

Often, only one of these lines make sense.

Exactly which of these will be appropriate for the analysis in hand will

depend on labeling of dependent and independent variable in the

problem to be analyzed.

Regression coefficients-

The two regression co-efficient are byx and bxy .

The formula for the two regression co- efficient are given by –

or b y x = N .∑XY − ∑ X . ∑Y

N. ∑X2 − (∑X)2

b x y = N.∑ XY – ∑X . ∑Y

N. ∑Y2 – (∑Y)2

The coefficient of X in the line of regression of Y on X is called the

regression coefficient of Y on X and is denoted by b y x

It represents change in the value of dependent variable (Y)corresponding to

unit change in the value of independent variable (X).

And similarly the coefficient of Y in the line of regression of X on Y is

called coefficient of X on Y and is denoted by b x y .

How Good Is the Regression?

Once a regression equation has been constructed, we can

check how good it by examining the coefficient of

determination (R2).

R2 always lies between 0 and 1.

The closer R2 is to 1, the better is the model and its

prediction.

PRACTICAL ASPECT OF REGRESSION ANALYSIS-

Here we will show a linear regression analysis between two

variables X and Y.

Variable X is taken as “ driving experience” and variable Y is

taken as “number of road accidents(in a year)”.

Road accident is taken as the dependent variable and which

is related to independent variable X i.e. driving experience.

X (driving experience)

5 2 12 9 15 6 25 16

Y ( no. of road accidents)

64 87 50 71 44 56 42 60

From the date we will show-

The estimated regression line for the date.

Number of road accidents taking place when the

driving experience is 10 years and 30 years.

co efficient of determination(R2) and which will

help us to know that how much percentage of

dependent variable is explained by independent

variable.

X Y X.Y X2 Y2

5 64 320 25 4096

2 87 174 4 7569

12 50 600 144 2500

9 71 639 81 5041

15 44 660 225 1963

6 56 336 36 3136

25 42 1050 625 1764

16 60 960 256 3600

∑X=90 ∑Y=474 ∑X.Y=4739 ∑X2=1396 ∑Y2=29642

The following is the tabular representation of data related to

driving experience and number of road accidents.

Since the estimated regression line is given by Y = a + b.X , now

using the normal equations we calculate the value of a and b .

∑Y = N. a + b ∑X

474= 8.a + b.90

8a + 90b = 474 E .q - 1

∑XY=a ∑X + b ∑X2

4739 = a.90 + b.1396

90a + 1396 b = 4739 E.q-2

Now solving both the equation we get the value of a and b as-

Value of a = 76.66

Value of b = -1.5475

The estimated regression line is

Y = 76.66 – 1.5476 X

3 6 9 12 15 18 21 24 27

experience

80

70

60

50

40

30

20

10

No. Of accidents

Trend line forY = 76.66 – 1.5476 X

Since we all know that the road accidents are dependent upon the driving

experience and a new driver is considered to be inexperienced and for

him the risk of accident is more so there exist a negative relationship

between the two variables so the trend line is downward sloping in this

case.

From the above value of a and b we can see that value of a is 76.66 which

means if a driver has 0 experience then the no of road accidents that will

take place is 76.66

From the value of b we can say that for every extra year of driving

experience , the road accident is decreased by 1.5476

No of accidents with 10 yr experience No. of accidents with 30 yr experience

Y = 76.66 – 1.5476 X

Y = 76.66 – 1.5476 (10)

Y = 61. 184

Y = 76.66 – 1.5476 X

Y = 76.66 – 1.5476 (30)

Y= 30.232

Now we find coefficient of variation for the data

using regression coefficients.

b y x = N .∑XY − ∑ X . ∑Y

N. ∑X2 − (∑X)2

b x y = N.∑ XY – ∑X . ∑Y

N. ∑Y2 – (∑Y)2

= 8 (4739) − 90 . 474

8(1396) − (90)2

= − 1.547

= 8(4739) − 90. 474

8(29642)− (474)2

= − 0.381

Now R2 = b y x .b x y

= (- 1. 547) (- 0.381)

= 0.5894

From the above coefficient of determination we can say that almost 59 %

of variance of dependent variable is explained by the independent

variable.