Regression · 2016. 6. 24. · Y = + + ε i Population Linear Regression Model ... Y c =10X. Quantitative Aptitude & Business Statistics: Regression 32 ... 50 3.The coefficient of

Regression

Quantitative Aptitude & Business Statistics

Quantitative Aptitude & Business Statistics: Regression

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Regression Regression is the measure of

average relationship between two or more variables in terms of original units of the data.


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Regression analysis is a statistical tool to study the nature and extent of functional relationship between two or more variables and to estimate the unknown values of independent variable.


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Dependent variable :The Variable Which is predicted on the basis of another variable is called Dependent variable or explained variable .

Independent variable :The Variable Which is used to predict another variable is called independent variable or explanatory variable.


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Uses of Regression Analysis 1.Regression line facilitates to

predict the values of a dependent variable from the given value of independent variable.

2.Through Standard Error facilitates to obtain a measure of the error

involved in using the regression line as basis for estimation.


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3.Regression coefficients (bxy and byx) facilitates to calculate coefficient of determination (r2) and coefficient of correlation.

4.Regression Analysis is highly useful tool in economics and business.


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Distinction between Correlation and Regression

Correlation Regression 1. Correlation measures degree and direction of relationship between variables.

1. Regression measures nature and extent of average relationship between two or more variables.

2.It is a relative measure showing association between variables.

2.It is an absolute measure relationship.


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Correlation Regression 3. Correlation Coefficient is independent of both origin and scale.

3. Regression Coefficient is independent of origin but not scale.

4. Correlation Coefficient is independent of units of measurement.

4.Regression Coefficient is not independent of units of measurement.


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Correlation Regression 5.Correlation Coefficient is lies between -1 and +1.

5. Regression equation may be linear or non-linear .

6. It is not forecasting device.

6.It is a forecasting device.


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Regression lines

Regression line X on Y

Where X= Dependent Variable Y =Independent variable a=intercept and b= slope

bYaX +=


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( )YYbXX xy −=− Another way of regression line X on Y

( )YYrXXy

x −=−σσ


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Regression coefficients There are two regression coefficients byx and

bxy The regression coefficient Y on X is

x

yyx .rb

σ

σ=

The regression coefficient X on Y is

y

xxy .rb

σσ

=


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Regression coefficients

The regression coefficient X on Y is

y

xxy .rb

σσ

=


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Regression line Y on X

Where Y= Dependent Variable X =Independent variable a=intercept and b= slope

bXaY +=


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Another way of regression line Y on X ( )XXrYY

x

y −=−σσ

( )XXbyxYY −=−


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Properties of Linear Regression

Two Regression Equations. Product of regression coefficient. Signs of Regression Coefficient

and correlation coefficient. Intersection of means. Slopes .


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Angle between Regression lines

Value of r Angle between Regression Lines

a) If r=0

b) If r=+1 or -1

Regression lines are perpendicular to each other. Regression lines are coincide to become identical .


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Properties of regression coefficients

1.Same Sign. 2.Both cannot greater than one . 3.Independent of origin but not of scale . 4.Arithmetic mean of regression coefficients

are greater than Correlation coefficient. 5.r,bxy and byx have same sign. 6 .Correlation coefficient is the Geometric

Mean (GM) b/w regression coefficients.


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Independent of origin but not of scale.

This property states that if the original pairs of variables is (x,y) and if they are changed to the pair (u,v), where x=a + p u and y=c +q v

or

qcy

v

andp

axu

−=

−=

yxvu

xyuv

bpq

b

andbpq

b

×=

×=


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Normal Equations

Regression line Y on X

The two normal Equations are

bXaY +=

∑∑ += XbNaY

∑∑∑ += 2XbXaXY


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Calculate byx

( )∑ ∑

∑ ∑ ∑

−

−=

NX

X

NYX

XYb 2

2

yx

XbYa −=


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Normal Equations


The two normal Equations are

bYaX +=

∑∑ += YbNaX

∑∑∑ += 2YbYaXY


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Calculate bxy

( )∑ ∑

∑ ∑ ∑

−

−=

NY

Y

NYX

XYb 2

2

xy

YbXa −=


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Y i = + + ε

Population Linear Regression Model Relationship between variables is described

by a linear function The change of the independent variable

causes the change in the dependent variable

Dependent (Response) Variable

Independent (Explanatory) Variable

Slope Y-Intercept Random Error

a bx


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Sample Linear Regression Using Ordinary Least Squares (OLS), we can find the

values of a and b that minimize the sum of the squared residuals:

Partial Differentiate w.r.t parameters a and b then ,we will get the two normal equations

∑∑ += XbNaY

( )2 2

1 1

ˆn n

i i ii i

Y Y e= =

− =∑ ∑

∑∑∑ += 2XbXaXY


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From the following Data Calculate Coefficient of correlation

X

Advertisement Exp. (Rs. lakhs)

1 2 3 4 5

Y Sales

(Rs.lakhs)

10 20 30 50 40


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a .Find out Two Regression Equations

b. calculate coefficient of correlation c.Estimate the likely sales when

advertising expenditure is Rs.7 lakhs. d. What should be the advertising

expenditure if the firm wants to attain sales target of Rs.80 lakhs.


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X Y XY

1 2 3 4 5

10 20 30 40 50

1 4 9

16 25

100 400 900 1600 2500

10 40 90

160 250

=15 =150 =55 =5500 =550

2X 2Y


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Regression Equation of X on Y : X c=a + b Y Then the normal Equations are

Substituting the values in the above equations:

15=5a+150b 550=150a+5500b

∑∑ += YbNaX ∑∑∑ += 2YbYaXY

1

2


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Regression Equation of Y on X : Yc=a + b X Then the normal Equations are

Substituting the values in the above equations:

150=5a+15b 550=15a+55b

∑∑ += XbNaY ∑∑∑ += 2XbXaXY

1

2


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Regression line Yon X

Correlation coefficient r=1.0

Y01.0Xc =

X10Yc =


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c) Sales (Y) when the advertising 7 Expenditure (X) is Rs.7lakhs

Y=10x=10*7=70 d) Advertising Expenditure (X) to attain

sales (Y) target of 80lakhs. X=0.1Y=0.1*80=8.0


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Measure of Variation: The Sum of Squares

SST = SSR + SSE

Total Sample

Variability

= Explained Variability

+ Unexplained Variability


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Measure of Variation: The Sum of Squares

SST = Total Sum of Squares Measures the variation of the Yi values

around their mean Y SSR = Regression Sum of Squares Explained variation attributable to the

relationship between X and Y SSE = Error Sum of Squares Variation attributable to factors other

than the relationship between X and Y


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Coefficient of determination(r2)

The coefficient of determination is the square of the coefficient of correlation. It is equal to r2.

The maximum value of r2 is unity and in the case of all the variation in Y is explained by the variation in X ,it is defined as

Coefficient of determination( r2 )

nceTotalVariainacevarExplained

=


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Coefficient of non-determination(k2) Coefficient of non-determination(k2)=1-r2

nceTotalVariainacevarlainedexpUn

=


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Example In a partially destroyed record the

following data are available : Variance of x =25, Regression equation of X on Y : 5X-Y=22 Regression equation of Y on X : 64X-45Y=24 Find a) Mean values of X and Y ; b) Coefficient of correlation between x and Y c) Standard deviation of Y


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Solution

A) the mean values of X and Y lie on the regression lines and are obtained by solving the given regression equations.

Multiplying (1) by 45 ,we get

22yx5 =−24y45x64 =−

1

2

990y45x225 =− 3


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Subtracting (2) from (3)

Putting in (1) ,we get ;

6x =

6x

96x161

=

=

8y

22y30

=

=−


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B) the regression equation y on x is : 64x-45y=24

6564b

x6564

158y

2524x

4564y

yx =

+−=

−=


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Again regression equation x on y is 5x-y=22

+ve sign with r is taken as both the regression coefficients bxy and byx are positive

51b

x51

2522x

xy =

+=

158

51.

4564

b.br yxxy

=±=

±=


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Solution

Now it is given that

33.13340

5158

4564

.rb

4564b,

158

25)x(V

yy

x

yyx

yxx

2x

==σ⇒σ

×=

⇒∴σ

σ=

==σ

=σ=


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Example If the relationship between x and u

is u+3x=10 between two other variables y and v is 2y+5v=25 ,and the regression coefficient of y on x is known as 0.80,what would be the regression coefficient v on u ?


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Solution

Given u+3x=10 u=10-3x

2y+5v=25

−

−

=

313

10xu

−

−

=

25225y

v

vuyx bpq

b ×=

75880.0

152b

b3

125

80.0

vu

vu

=×=

×−−

=


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1.bxy and byx are (a) independent of both change of scale and

origin (b) independent of the change of scale and

not of origin (c)independent of the change of origin and

not of scale (d) neither independent of change of scale

nor of origin


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1.bxy and byx are (a) independent of both change of scale

and origin (b) independent of the change of scale

and not of origin (c) independent of the change of origin

and not of scale (d) neither independent of change of scale

nor of origin


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2.bxy measures (a) the changes in y corresponding to a

unit change in ‘x’ (b) the changes in x corresponding to a

unit change in ‘y’ (c) the changes in xy (d) the changes in yx


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2.bxy measures (a) the changes in y corresponding to

a unit change in ‘x’ (b) the changes in x corresponding to

a unit change in ‘y’ (c) the changes in x y (d) the changes in y x


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3.The coefficient of determination is defined by the formula

(a) r2=1– (b) r2= (c) both (d) none of these

iancetotaliancelainedun

varvarexp

iancetotaliancelained

varvarexp


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3.The coefficient of determination is defined by the formula

(a) r2= 1– (b) r2= ( c) both (d) none of these

iancetotaliancelainedun

varvarexp

iancetotaliancelained

varvarexp


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4.The method applied for driving the regression equations is known as

(a) least squares (b) concurrent deviation (c) product moment (d) normal equation


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4.The method applied for driving the regression equations is known as

(a) least squares (b) concurrent deviation (c) product moment (d) normal equation


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5.The two lines of regression become identical when

(a) r=1 (b) r=–1 (c) r=0 (d) (a) or (b)


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5.The two lines of regression become identical when

(a) r=1 (b) r=–1 (c) r=0 (d) (a) or (b)


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6.The term regression was first used in the year 1877 by _____

(a) Karl Pearson (b) A. L. Bowley (c) R. A. Fisher (d) Sir Francis Galton


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6.The term regression was first used in the year 1877 by _____

(a) Karl Pearson (b) A. L. Bowley (c) R. A. Fisher (d) Sir Francis Galton


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7.If regression lines are perpendicular to each other, the value of r will be __

(a) +1 (b) –1 (c) 0 (d) none of these


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7.If regression lines are perpendicular to each other, the value of r will be __

(a) +1 (b) –1 (c) 0 (d) none of these


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8.∑X=50; ∑Y=30; ∑XY=1000; ∑X2=3000; ∑Y2=180; n=10, the value

of byx will be (a) 0.6132 (b) 1.3636 (c) 0.3090 (d) none of these


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8.∑X=50; ∑Y=30; ∑XY=1000; ∑X2=3000; ∑Y2=180;n=12,the value of

byx will be (a) 0.6132 (b) 1.3636 (c) 0.3090 (d) none of these


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9.The standard error of an estimate is Zero ,r will be---

A) 1 B)+1 C)-1 D) none of these

±


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9.The standard error of an estimate is Zero ,r will be---

A) 1 B)+1 C)-1 D) none of these

±


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10.If there are two variables x and y,then the number of regression equations could be

A)1 B)2 C) Any number D)3


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10.If there are two variables x and y,then the number of regression equations could be.

A)1 B)2 C) Any number D)3


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11. The regression coefficients are Zero if r is equal to-----

A) 2 B) -1 C) 1 D) 0


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11 The regression coefficients are Zero if r is equal to-----

A)2 B)-1 C)1 D)0


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12 .When r =0 then Cov(x,y)----is equal to

A) +1 B) -1 C) 0 D) 3


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12 .When r =0 then Cov(x,y)----is equal to

A)+1 B)-1 C)0 D)3


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13.If r=1 ,then the standard error of estimate will be

A) Zero B)+1 C) -1 D) none of these


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13.If r =1 ,then the standard error of estimate will be

A) Zero B)+1 C) -1 D) none of these


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14.If bxy=+0.8,then the value of byx can be

A)+1.25 B)-1.25 C)+1.26 D)-1.24


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14.If bxy=+0.8,then the value of byx can be

A)+1.25 B)-1.25 C)+1.26 D)-1.24


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15 ____Gives the mathematical relationship between the variables.

A) Correlation B) Regression C) Both D) None


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15 ____Gives the mathematical relationship between the variables.

A) Correlation B) Regression C) Both D) None


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16. Equations of two lines of regression are 4x+3y+7 = 0 and 3x+ 4y + 8 = 0, the mean of x and y are

A) 5/7 and 6/7 B) – 4/7 and –11/7 C) 2 and 4 D) None of these


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16. Equations of two lines of regression are 4x+3y+7 = 0 and 3x+ 4y + 8 = 0, the mean of x and y are

A) 5/7 and 6/7 B) – 4/7 and –11/7 C) 2 and 4 c D) None of these


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17. Two lines of regression are given by 5x+7y–22=0 and 6x+2y–22=0. If the variance of y is 15, find the standard deviation of x?

A) B) C) D)

57

68


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17. Two lines of regression are given by 5x+7y–22=0 and 6x+2y–22=0. If the variance of y is 15, find the standard deviation of x?

A) B) C) D)

57

68


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18. If 2x + 5y – 9 = 0 and 3x – y – 5 = 0 are two regression equation, then find the value of mean of x and mean of y.

A) 2,1 B) 2,2 C) 1,2 D) 1,1


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18. If 2x + 5y – 9 = 0 and 3x – y – 5 = 0 are two regression equation, then find the value of mean of x and mean of y.

A) 2,1 B) 2,2 C) 1,2 D) 1,1


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19. If one of the regression coefficients is greater than unity, then other is less than unity.

A) True B) False C) Both D) None of these


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19. If one of the regression coefficients is greater than unity, then other is less than unity.

A) True B) False C) Both D) None of these


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20. The two regression lines obtained from certain data were y = x + 5 and 16x = 9y – 94. Find the variance of x if variance of y is 16.

A) 4/16 B) 9 C) 1 D) 5/16


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20. The two regression lines obtained from certain data were y = x + 5 and

16x = 9y – 94. Find the variance of x if variance of y is 16. A) 4/16 B) 9 C) 1 D) 5/16


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21. For a m×n two way or bivariate frequency table, the maximum number of marginal distributions is .

A) m B) n C) m +n D) m .n


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21. For a m×n two way or bivariate frequency table, the maximum number of marginal distributions is .

A) m B) n C) m +n D) m .n

THE END

Regression

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Regression · 2016. 6. 24. · Y = + + ε i Population Linear Regression Model ... Y c =10X. Quantitative Aptitude & Business Statistics: Regression 32 ... 50 3.The coefficient of