sec d ch 12 regression part 2

8/10/2019 Sec d Ch 12 Regression Part 2

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CPT Section D Quantitative Aptitude Chapter12

CA.Dharmendra Gupta


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Regression is the measure of

average relationship betweentwo or more variables in termsof original units of the data


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Regression analysis is a statistical tool

to study the nature and extent offunctional relationship between two ormore variables and to estimate theunknown values of dependent variable.


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The Variable Which is predictedon the basis of another variableis called Dependent variable orexplained variable

Dependentvariable :

:The Variable Which is used topredict another variable is calledindependent variable orexplanatory variable

Independentvariable


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1.Regression line facilitates to predict thevalues 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 theregression 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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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.

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 line X on Y

Where X = Dependent Variable

Y = Independent variable

a = intercept andb = slope


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There are two regression coefficients byx and

bxy

The regression coefficient Y on X is

The regression coefficient X on Y is


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y

xxy .rb

=

The regression coefficient X on Y is


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Regression line Y on X

Where Y = Dependent Variable

X = Independent variable

a = intercept and

b = slope


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Two Regression Equations.

Product of regression coefficient.

Signs of Regression Coefficient and correlation coefficient.

Intersection of means.

Slopes .


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Value of r Angle between RegressionLines

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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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 Correlationcoefficient.

5.r,bxy and byx have same sign.

6 .Correlation coefficient is the Geometric Mean (GM) b/w regressioncoefficients.


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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


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Regression line Y on X

The two normal Equations are


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The two normal Equations are


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Yi

= + +

Relationship between variables is described by alinear function

The change of the independent variable causesthe change in the dependent variable

Dependent

(Response)

Variable

Independent

(Explanatory)

Variable

SlopeY-InterceptRandom Error

a bx


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Using Ordinary Least Squares (OLS), wecan find the values of a and b that minimizethe sum of the squared residuals:


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X

Advertisement Exp.

(Rs. lakhs)

1 2 3 4 5

Y

Sales

(Rs. lakhs)

10 20 30 40 50


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Find out Two Regression Equations

Calculate coefficient of correlation

Estimate the likely sales when advertisingexpenditure is Rs.7 lakhs

What should be the advertising expenditure if thefirm wants to attain sales target of Rs.80 lakhs


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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

1

2


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Regression Equation of Y on X :

Yc = a + bX

Then the normal Equations are

Substituting the values in the above equations:

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

1

2


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Xc=0.10Y

Regression line Yon X

Correlation coefficient r=1.0


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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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SST = SSR + SSE

Total Sample

Variability=

Explained

Variability +Unexplained

Variability


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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 non-determination(k2)=1-r2


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In a partially destroyed record the following dataare available :

Variance of x =25,

Regression equation of X on Y : 5X-Y=22Regression equation of Y on X64X-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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A) the mean values of X and Y lie on theregression lines and are obtained by solving the

given regression equations.

Multiplying (1) by 45 ,we get

1

2

3


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Subtracting (2) from (3)

Putting in (1), we get:


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


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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


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Now it is given that


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If the relationship between x and u is u+3x=10between 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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Given u+3x=10 u=10-3x

p = -3

2y+5v=25

5v = 25 -2y

v = 5 0.4 y

q = - 0.40

buv=( 3/0.40)0.8 =6


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MCQs


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(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

Answer:c


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(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

Answer:b


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(a) r2= 1

(b) r2=

( c) both

(d) none of these

Answer: c


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(a) least squares

(b) concurrent deviation

(c) product moment

(d) normal equation

Answer: a


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(a) r=1

(b) r =1

(c) r=0

(d) (a) or (b)

Answer:d


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(a)Karl Pearson

(b)A. L. Bowley

(c)R. A. Fisher

(d) Sir Francis Galton

Answer:d


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(a) +1

(b) 1

(c) 0

(d) none of these

Answer: c


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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

Answer:d


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A)1

B)2

C) Any number

D)3

Answer:B


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A)2

B)-1

C)1

D)0

Answer:D


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A)+1

B)-1

C)0

D)3

Answer:C


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A)+1.25

B)-1.25

C)+1.26

D)-1.24

Answer:(A)


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A) Correlation

B) Regression

C) Both

D) None

Answer:B


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17. Two lines of regression are given by 5x+7y22=0 and 6x+2y22=0. If the variance of y is 15,

find the standard deviation of x?

A) 2

B)

C)

D) None of these


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a) 2,1

b) 2,2

c) 1,2

d) 1,1Answer:A


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A) True

B) False

C) Both

D) None of these

Answer:a


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The two regression lines obtained from certain datawere 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

Answee;B


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sec d ch 12 regression part 2

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