Regression Contd., Important problems Correlation and Regression

Subject: Mathematics

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Overview

This note lets you know the product moment formula to find the Karl Pearson's coefficient and solves some important questions related to correlation and regresseion.
Regression Contd., Important problems Correlation and Regression

Regression Equations and regression coefficients:

Regression lines expressed in term of algebraic relations are known as the regression equations. Since there are two regression lines, So there are two regression equations.

  • The regression equation of y on x expresses the variation of y for a change in the value of x.
  • The regression equation of x on y expresses the variation of x for a change in the value of y.

Regression equation of y on x

Let the regression equation of y on x be

$$y=a+bx$$

$$\rightarrow\;\Sigma\;y=na+b\Sigma\;x$$

$$\rightarrow\;\frac{\Sigma}{n}\;y=a+\frac{b\Sigma\;x}{n}$$

$$\rightarrow\;\overline{y}=a+b\overline{x}$$

Subtracting two equations:

$$y-\overline{y}=b-(x-\overline{x}$$

This equation is the equation of line of regression of y on x. This equation shows that the line of regression of y on x passes through \(\;(\overline{x},\overline{y})\;\;\overline{x}\;and\overline{y}\) being the arithmetic averages of x and y series, b is known as the regression coefficient of y on x. In order to differentiate it from regression coefficient of x on y, b is written as b­yx­ where

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$$b_{yx}=\frac{n\Sigma\;xy-\Sigma\;x\Sigma\;y}{n\Sigma\;x^2-(\Sigma\;x)^2}=r\frac{\sigma_y}{\sigma_x}$$

Similarly the regression equation of x on y is,

$$y-\overline{x}=b_{xy}(y-\overline{y}$$

Where the line passes through \(\;(\overline{x},\overline{y})\) the regression coefficient of x on y denoted by bxy ­is given by

$$b_{xy}=\frac{n\Sigma\;xy-\Sigma\;x\Sigma\;y}{n\Sigma\;y^2-(\Sigma\;y)^2}=r\frac{\sigma_x}{\sigma_y}$$

­The regression equations can easily be obtained when the deviations of the items are taken from the assumed mean.

If u=x-a and v=y-b i.e. if the deviations of the items of x series and y-series be taken from the assumed means a and b respectively then,

$$\overline{x}=a+\frac{\Sigma\;u}{n},\;\;\;\;\overline{y}=b+\frac{\Sigma\;v}{n}$$

$$b_{yx}=\frac{n\Sigma\;uv-\Sigma\;u\Sigma\;v}{n\Sigma\;u^2-(\Sigma\;u)^2}$$

$$b_{xy}=\frac{n\Sigma\;uv-\Sigma\;u\Sigma\;v}{n\Sigma\;v^2-(\Sigma\;v)^2}$$

Relation between r and b:

$$r=\sqrt{b_{yx}b_{xy}}$$

Examples

Example 1: (Correlation)

Calculate the Karl Pearson’s coefficient of correlation from the following data using product moment formula.

X

12

9

8

10

11

13

7

Y

14

8

6

9

11

12

3

Solution:

Computation of correlation coefficient

X

Y

$$x=X-\overline{x}$$

$$y=Y-\overline{Y}$$

x2

y2

xy

12

9

8

10

11

13

7

14

8

6

9

11

12

3

2

-1

-2

=

1

3

-3

5

1

-3

0

2

3

6

4

1

4

0

1

9

9

25

1

9

0

4

9

36

10

1

6

0

1

9

18

$$\Sigma\;X=70$$

$$\Sigma\;Y=63$$

$$\Sigma\;x^2=28$$

$$\Sigma\;y^2=84$$

$$\Sigma\;xy=46$$

$$\overline{X}=\frac{\Sigma\;X}{n}=\frac{70}{7}=10\;\;\;\overline{Y}=\frac{\Sigma\;Y}{n}=\frac{63}{7}=9$$

$$r=\frac{\Sigma\;xy}{\sqrt{\Sigma\;x^2}\sqrt{\Sigma\;y^2}}=\frac{46}{\sqrt{28}\sqrt{84}}$$

$$\frac{46}{\sqrt{2352}}\;=\frac{46}{48.497}$$

$$=0.95$$

Example 2: (Regression)

Obtain the regression equations of X on Y and Y on X taking origin as 2 and 200 for X and Y respectively.

X

1

2

3

4

5

Y

166

184

142

180

338

Solution:

X

Y

u=X-A=X-2

v=Y-B=Y-200

uv

u2

v2

1

2

3

4

5

166

184

142

180

338

-1

0

1

2

3

-34

-16

-58

-20

138

34

0

-58

-40

414

1

0

1

4

9

1156

256

3364

400

19044

Total

5

10

350

15

24220

$$The\;regression\;equation\;of\;X\;on\;Y\;is\;X-\overline{X}\;=b_{xy}(Y-\overline{Y}$$

$$\overline{x}=a+\frac{\Sigma\;u}{n}=2+\frac{5}{5}=3$$

$$\overline{y}=b+\frac{\Sigma\;v}{n}=200=\frac{10}{5}=202$$

$$b_{xy}=\frac{n\Sigma\;uv-\Sigma\;u\Sigma\;v}{n\Sigma\;v^2-(\Sigma\;v)^2}$$

$$=\;\frac{5(350)-5(10)}{5(24220)-(10)^2}=\frac{1700}{121000}=0.014$$

The required equation is,

$$X-3=0.014(Y-202)$$

$$\rightarrow\;X=0.014Y+0.172$$

Again,

$$The\;regression\;equation\;of\;Y\;on\;X\;is\;Y-\overline{Y}\;=b_{yx}(X-\overline{X}$$

$$\overline{x}=3$$

$$\overline{y}=202$$

$$b_{yx}=\frac{n\Sigma\;uv-\Sigma\;u\Sigma\;v}{n\Sigma\;u^2-(\Sigma\;u)^2}$$

$$=\;\frac{5(350)-5(10)}{5(15)-(5)^2}=\frac{1700}{50}=34$$

The required equation is,

$$Y-202=34(X-3)$$

$$\rightarrow\;Y=34Y+100$$

Example 3: (Regression)

For a certain bivariate data

X

Y

Mean

10

18

Std

2.5

2.0

And the coefficient of correlation between X and Y is 0.8. Determine the following:

  • The regression of Y on X and the regression of X on Y
  • Estimated value of Y for X = 15

Solution:

We know,

$$b_{xy}=r\frac{\sigma_x}{\sigma_y}=0.8\frac{2.5}{2.0}=1$$

$$b_{yx}=r\frac{\sigma_y}{\sigma_x}=0.8\frac{2}{2.5}=0.64$$

the regression equation of Y on X is,

$$y-\overline{y}=b_{yx}(x-\overline{x}$$

$$Y-10=1(X-18)$$

the regression equation of X on Y is,

$$X-\overline{X}=b_{xy}(Y-\overline{Y}$$

$$X-18=0.64(Y-10)$$

Thus for X=15, Y=(15-18)+10=7

Example 4 : (Correlation)

Calculate the coefficient of correlation from the following data of price and demand:

Price (Rs)

14

16

19

22

24

30

Demand(kg)

24

22

20

24

23

26

Solution:

Computation of correlation coefficients:

Price(x)

u=x-19

u2

Demand(y)

v=y-23

v2

uv

14

16

19

22

24

30

-5

-3

0

3

5

11

25

9

0

9

25

121

24

22

20

24

23

26

1

-1

-3

1

0

3

1

1

9

1

0

9

-5

-3

0

3

0

33

$$\Sigma\;u=11$$

$$\Sigma\;u^2=189$$

$$\Sigma\;v=1$$

$$\Sigma\;v^2=21$$

$$\Sigma\;uv=34$$

$$r=\frac{n\Sigma\;uv-\Sigma\;u\Sigma\;v}{\sqrt{n\Sigma\;u^2-(\Sigma\;u)^2}\sqrt{n\Sigma\;v^2-(\Sigma\;v)^2}}$$

$$=\frac{6\times\;34-11\times\;1}{\sqrt{6\times\;189-11^2}\sqrt{6\times\;21-1^2}}$$

$$=\frac{204-11}{\sqrt{1134-221}\sqrt{126-1}}$$

$$=\frac{193}{\sqrt{1013\times\;125}}$$

$$=\frac{193}{\sqrt{126625}}$$

$$\frac{13}{355.84}$$

$$=0.542$$

Taken reference from

( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )

Things to remember
  • the regression equation of x on y is,

    $$y-\overline{x}=b_{xy}(y-\overline{y}$$

  • If u=x-a and v=y-b i.e. if the deviations of the items of x series and y-series be taken from the assumed means a and b respectively then,

    $$\overline{x}=a+\frac{\Sigma\;u}{n},\;\;\;\;\overline{y}=b+\frac{\Sigma\;v}{n}$$

    $$b_{yx}=\frac{n\Sigma\;uv-\Sigma\;u\Sigma\;v}{n\Sigma\;u^2-(\Sigma\;u)^2}$$

    $$b_{xy}=\frac{n\Sigma\;uv-\Sigma\;u\Sigma\;v}{n\Sigma\;v^2-(\Sigma\;v)^2}$$

  • $$r=\sqrt{b_{yx}b_{xy}}$$
  • It includes every relationship which established among the people.
  • There can be more than one community in a society. Community smaller than society.
  • It is a network of social relationships which cannot see or touched.
  • common interests and common objectives are not necessary for society.

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