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Regression

Quantitative A Aptitude & & Business S Statistics

Quantitative Aptitude & Business Statistics: Regression 2

Regr gress ession

• n

 Regr

Regression

• n is t

the he meas measure of

• f

average r ge relat ations

• nshi

hip b p betwe ween t en two

• r
• r mor

more e var ariab ables i in n ter erms ms of

• f
• r
• rigi

ginal uni units of

• f the

he dat data. a.

Quantitative Aptitude & Business Statistics: Regression 3

 Regre

ression analysis is i is a a statistic ical tool

• ol to s
• stud

udy the nat he natur ure e and ex and extent

• f functional
• nal r

relations

• nshi

hip b p between een two or wo or mor more e var ariab ables and t and to

• es

estimate t the he unk unknown n val alues of

• f

independ pendent ent v variabl ble. e.

Quantitative Aptitude & Business Statistics: Regression 4

 Depend

endent ent v variabl ble e :The Var The Variab able Whi Which i is pr predi edicted on

• n the

he bas basis of

• f

another er v variabl able i e is c called ed Depe Depende dent v var ariab able or

• r ex

expl plained d variable . .

 Indepe

ndependen ent v var ariable : e :The Var The Variabl ble Whi Which i is us used ed t to

• pr

predi edict anot another variabl able is c called i ed indep epende endent nt var ariable or e or ex expl plan anatory v var ariab able.

Quantitative Aptitude & Business Statistics: Regression 5

Uses es of Regr

gres ession A sion Anal alysis ysis

1.Regr Regress ession

• n

li line fa facili lita tate tes to to pre redic ict the he valu lues of

• f a dep

dependen endent varia riable le fro rom th the gi given ven valu lue of

• f

independ pendent ent variable. 2.Throu

• ugh

gh St Stan anda dard Erro rror facili lita tate tes to to obt

• btai

ain a measu measure of

• f the

he er error

• r

in involv lved in in us using ng the he reg egress ession

• n

lin line as as bas basis fo for estimat mation

• n.

Quantitative Aptitude & Business Statistics: Regression 6

3.Regre ressio ion c coeffi ficients ts (bxy

xy and

and byx

yx) f

) facili litate tes t to calc lcula late te coef

• efficien

ent of

• f det

deter erminat ation ( n (r2) ) and c and coef

• efficient of
• f cor
• rrelation
• n.

4. 4.Re Regr gression Anal n Analysis i is hi highl ghly us usef eful tool

• ol i

in n ec econ

• nomics and

and business.

Quantitative Aptitude & Business Statistics: Regression 7

Distinc inction ion betwee ween C n Correlat latio ion a n and d Regre ressi ssion

Correl elation

• n

Regressi ession

• 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 relat ative ve meas asur ure e show

• wing a

ng assoc sociat ation

• n

between v een variabl ables es.

2.It is an absolute measure relationship.

Quantitative Aptitude & Business Statistics: Regression 8

Correl elation

• n

Regressi ession

• 3. Correlation

Coefficient is independent of both

• rigin 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.

Quantitative Aptitude & Business Statistics: Regression 9

Correl elation

• n

Regressi ession 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.

Quantitative Aptitude & Business Statistics: Regression 10

Regression lines

 Regr

egres ession

• n line

ne X on Y

• n Y

 Wher

here X= D Depe epend nden ent Var ariable e Y =Inde ndepe pend nden ent var ariab able a= a=intercept pt and and b= b= slope

• pe

bY a X + =

Quantitative Aptitude & Business Statistics: Regression 11

( )

Y Y b X X

xy

− = −

 Anot

nother her w way of ay of regr egres ession

• n l

line X e X on

• n Y

( )

Y Y r X X

y x

− = − σ σ

Quantitative Aptitude & Business Statistics: Regression 12

Regre ression c coeffic icie ients

 There are two regression coefficients byx and

bxy

 The regression coefficient Y on X is

x y yx

. r b σ σ =

The regression coefficient X on Y is

y x xy

. r b σ σ =

Quantitative Aptitude & Business Statistics: Regression 13

Regre ression c coeffic icie ients

The regression coefficient X on Y is

y x xy

. r b σ σ =

Quantitative Aptitude & Business Statistics: Regression 14

 Regr

egres ession

• n line

ne Y on X

• n X

 Wher

here Y= D Depe epend nden ent Var ariable e



X =Inde ndepe pend nden ent var ariab able

 a=

a=intercept pt and and b= b= slop

• pe

e

bX a Y + =

Quantitative Aptitude & Business Statistics: Regression 15

 Another way of regression line Y on X

( )

X X r Y Y

x y

− = − σ σ

( )

X X byx Y Y − = −

Quantitative Aptitude & Business Statistics: Regression 16

Properties of Linear Regression

 Two Regr

Two Regression Equa Equations.

 Pro

roduct ct of f re regre ression c coeffi fficient. t.

 Si

Sign gns of

• f Regr

Regression Coef n Coefficient and c correlation c coeff ffic icie ient. t.

 In

Inte ters rsecti tion o

• f

f means.

 Slopes .

pes .

Quantitative Aptitude & Business Statistics: Regression 17

 Angl

ngle e bet between een R Regr egression l n line nes Value of r Angle between Regression Lines

a) If r=0 b) If r=+1 or -1

Regression lines are perpendicular to each

• ther.

Regression lines are coincide to become identical .

Quantitative Aptitude & Business Statistics: Regression 18

Properties of regression coefficients

1. 1.Same S ame Sign. gn. 2. 2.Bot

• th c

h cannot annot gr great eater er t than han one

• ne .

3. 3.Independent ndependent of

• f or
• rigi

gin n but but not not of

• f scal

ale e . 4. 4.Arithm hmet etic mean mean of

• f regr

egressi ssion c

• n coef
• effici

cient ents ar are e gr great eater er t than han Cor

• rrel

elat ation c

• n coef
• effici

cien ent. 5. 5.r,bxy bxy and and by byx hav x have e same s ame sign. gn. 6 6 .Cor

• rrel

elat ation c

• n coef
• effici

cient ent i is the he Geomet eometric c Mean Mean (GM) b/ b/w regr egress ssion c n coef

• efficient

ents.

Quantitative Aptitude & Business Statistics: Regression 19

Independent of origin but not of scale.

 Thi

his pr proper

• perty st

stat ates t es that hat if the he or

• riginal pai

pairs of

• f

var variab ables es i is ( s (x, x,y) y) and and if t they hey ar are e change changed t d to

• the

he pai pair (u, u,v) v), w wher here x=a + x=a + p p u u and and y=c +q y=c +q v

• r
• r

q c y v and p a x u − = − =

yx vu xy uv

b p q b and b p q b × = × =

Quantitative Aptitude & Business Statistics: Regression 20

Nor Normal Equa Equation

• ns

 Regr

egres ession

• n line

ne Y on X

• n X

 The two normal Equations are

bX a Y + =

∑ ∑

+ = X b Na Y

∑ ∑ ∑

+ =

2

X b X a XY

Quantitative Aptitude & Business Statistics: Regression 21

Calcul culat ate b e byx

( )

∑ ∑ ∑ ∑ ∑

− − = N X X N Y X XY b

2 2 yx

X b Y a − =

Quantitative Aptitude & Business Statistics: Regression 22

Nor Normal Equa Equation

• ns

 Regression line X on Y  The t

The two

• nor

normal Equat quations ns ar are e

bY a X + =

∑ ∑

+ = Y b Na X

∑ ∑ ∑

+ =

2

Y b Y a XY

Quantitative Aptitude & Business Statistics: Regression 23

Calcul culat ate b e bxy

xy

( )

∑ ∑ ∑ ∑ ∑

− − = N Y Y N Y X XY b

2 2 xy

Y b X a − =

Quantitative Aptitude & Business Statistics: Regression 24

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

Dependen ependent (Respo pons nse) e) Va Varia iable le Indep ndepend nden ent (Expl planat nator

• ry)

Va Varia iable le

Slope

• pe

Y-Intercept ercept Random andom Error rror

a bx

Quantitative Aptitude & Business Statistics: Regression 25

Sam ample Li e Linea near Regr egression

 Using Ordi

dinary Leas Least Squa quares (OLS LS), 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

∑ ∑

+ = X b Na Y

( )

2 2 1 1

ˆ

n n i i i i i

Y Y e

= =

− =

∑ ∑

∑ ∑ ∑

+ =

2

X b X a XY

Quantitative Aptitude & Business Statistics: Regression 26

Fr From

• m the

he fol

• llowing

ng Dat ata C Cal alculate Coeffici cient o

• f correlation

X Adver dvertisem emen ent Exp.

• xp. (

(Rs. l lakh akhs) 1 2 3 4 5 Y Sal ales es (Rs.lakh akhs) s) 10 10 20 20 30 30 50 50 40 40

Quantitative Aptitude & Business Statistics: Regression 27

 a .Find o

nd out T Two R Regr gres ession

• n

Equat uation ions

 b. calculat

late c e coef effic icient ent o

• f correla

elation ion

 c.Estim

imat ate t e the l likely ely s sales les when en adver ertis isin ing e g expendi penditur ure i e is Rs.7 l 7 lakhs hs.

 d.

• d. W

What hat s sho hould be t be the he adv advertising expend pendit itur ure i e if the f firm want nts t to attai ain n sales les t target get o

• f Rs.80 l

80 lakhs hs.

Quantitative Aptitude & Business Statistics: Regression 28

X Y XY XY 1 2 3 4 5 10 10 20 20 30 30 40 40 50 50 1 4 9 16 16 25 25 100 100 400 400 900 900 1600 00 250 2500 10 10 40 40 90 90 160 160 250 250 =15 =15 =150 =150 =55 =55 =550 =5500 =550 =550

2

X

2

Y

Quantitative Aptitude & Business Statistics: Regression 29

 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

∑ ∑

+ = Y b Na X

∑ ∑ ∑

+ =

2

Y b Y a XY

1 2

Quantitative Aptitude & Business Statistics: Regression 30

 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

∑ ∑

+ = X b Na Y

∑ ∑ ∑

+ =

2

X b X a XY

1 2

Quantitative Aptitude & Business Statistics: Regression 31

 Regression line X on Y  Regression line Yon X  Correla

elation c ion coef effic icient ent r r=1.0

Y 01 . Xc =

X 10 Yc =

Quantitative Aptitude & Business Statistics: Regression 32

 c) S

Sal ales es ( (Y) w when hen the he adv adver ertisi sing ng 7 7 Expendi penditure ( e (X) i is Rs.7l 7lakh akhs

 Y=10x

10x=10* 10*7= 7=70 70

 d)

d) Adv dver ertisi sing E ng Expendi penditure ( e (X) t to

• at

attai ain n sal ales ( es (Y) t tar arget get

• f
• f 80l

80lak akhs. hs.

 X=0.

0.1Y 1Y=0. 0.1* 1*80= 80=8. 8.0

Quantitative Aptitude & Business Statistics: Regression 33

Meas easure of

• f Var

ariat ation: n: The S The Sum um of

• f Squar

quares

SST = SSR + SSE Total Sample Variability = Explained Variability + Unexplained Variability

Quantitative Aptitude & Business Statistics: Regression 34

Measure of Variation: The Sum of Squares

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

around their mean Y

 SSR =

= Regr egression S

• n Sum

um of

• f Squar

quares

 Exp

xplaine ned var d variat ation n at attribu butab able t to

• the

he rel elat ations nship bet betwee een n X and and Y

 SSE = E

= Error

• r Sum

um of

• f Squar

quares

 Var

ariation

• n at

attribu butab able t e to

• fact

actor

• rs ot
• ther

her than han the he rel elat ations nship bet betwee een n X and and Y

Quantitative Aptitude & Business Statistics: Regression 35

Coef Coefficien ent of

• f det

deter erminat ation( n(r2)

 The

he coef coefficien ent of

• f det

deter erminat ation i n is t s the he squar square e

• f
• f the

he coef coefficien ent of

• f cor

correlat ation.

• n. It is equal

s equal t to

• r2.

 The

he maxi aximum val value ue of

• f r2 is uni

s unity and and in t n the he case of case of al all t the he var variation

• n in

n Y is expl s explained ed by by the he var variation

• n in X

n X , ,it i is s def define ned d as as

 Coef

• efficient

ent of

• f det

deter erminat ation( n( r r2 )

nce TotalVaria inace var Explained =

Quantitative Aptitude & Business Statistics: Regression 36

Coef Coefficien ent of

• f non

non-det deter ermi mina nation

• n(k2)

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

nce TotalVaria inace var lained exp Un =

Quantitative Aptitude & Business Statistics: Regression 37

Example

 In a partially destroyed record the

follow ing 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 betw een x and Y c) Standard deviation of Y

Quantitative Aptitude & Business Statistics: Regression 38

Solut ution

• n

 A) the m

he mean ean val value ues of

• f X

X and and Y l lie e on t

• n the

he regr egres ession l n line nes and and ar are e obt

• btai

ained ed by by sol solving ng the he gi give ven r n regr egres ession

• n equat

equations

• ns.

 Mul

ultiplying ng (1) 1) by by 45 , 45 ,we get e get

22 y x 5 = − 24 y 45 x 64 = −

1 2

990 y 45 x 225 = −

3

Quantitative Aptitude & Business Statistics: Regression 39

 Subtracting (2) from (3)  Putting in (1) ,we get ;

6 x =

6 x 96 x 161 = =

8 y 22 y 30 = = −

Quantitative Aptitude & Business Statistics: Regression 40

 B) the regression equation y on x is :

64x-45y=24

65 64 b x 65 64 15 8 y 25 24 x 45 64 y

yx =

+ − = − =

Quantitative Aptitude & Business Statistics: Regression 41

 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

5 1 b x 5 1 25 22 x

xy =

+ =

15 8 5 1 . 45 64 b . b r

yx xy

= ± = ± =

Quantitative Aptitude & Business Statistics: Regression 42

Solu lutio tion

 Now it is given that

33 . 13 3 40 5 15 8 45 64 . r b 45 64 b , 15 8 25 ) x ( V

y y x y yx yx x 2 x

= = σ ⇒ σ × = ⇒∴ σ σ = = = σ = σ =

Quantitative Aptitude & Business Statistics: Regression 43

Exampl ample

 If the relationship betw een x and u

is u+3x=10 betw een tw o other variables y and v is 2y+5v=25 ,and the regression coefficient of y on x is know n as 0.80,w hat w ould be the regression coefficient v on u ?

Quantitative Aptitude & Business Statistics: Regression 44

Solut ution

• n

 Given

ven u+3x= u+3x=10

 u=10

u=10-3x 3x

 2y+5v

2y+5v=25

            −       − = 3 1 3 10 x u             −       − = 2 5 2 25 y v

vu yx

b p q b × =

75 8 80 . 15 2 b b 3 1 2 5 80 .

vu vu

= × = × − − =

Quantitative Aptitude & Business Statistics: Regression 45

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

• rigin

(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

Quantitative Aptitude & Business Statistics: Regression 46

1. 1.bx bxy and y and by byx ar are e (a) a) independent ndependent of

• f bot

both h change hange of

• f scal

cale e and or and origi gin (b) b) independent ndependent of

• f the

he change hange of

• f scal

cale e and and not not of

• f or
• rigi

gin (c) c) independent ndependent of

• f the

he change hange of

• f or
• rigi

gin n and and not not of

• f scal

cale e (d) d) nei neither her independent ndependent of

• f c

change hange of

• f s

scal ale e nor nor of

• f or
• rigi

gin n

Quantitative Aptitude & Business Statistics: Regression 47

2.bxy m xy measu sure res (a) t (a) the c he cha hanges in y n y cor

• rresponding t

to

• a

a unit it c chang ange e in ‘ ‘x’ (b) t (b) the c he cha hanges in x n x cor

• rresponding t

to

• a

a unit it c chang ange e in ‘ ‘y’ (c (c) t ) the c he cha hanges i in n xy (d) t (d) the c he cha hanges in y n yx

Quantitative Aptitude & Business Statistics: Regression 48

 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

Quantitative Aptitude & Business Statistics: Regression 49

3. 3.The coef he coefficient nt of

• f det

deter ermina nation

• n i

is def s define ned by d by the f he for

• rmul

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

iance total iance lained un var var exp

iance total iance lained var var exp

Quantitative Aptitude & Business Statistics: Regression 50

3. 3.The coef he coefficient nt of

• f det

deter ermina nation

• n i

is def s define ned by d by the f he for

• rmul

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

iance total iance lained un var var exp

iance total iance lained var var exp

Quantitative Aptitude & Business Statistics: Regression 51

4. 4.The m he met ethod hod appl applied f d for

• r dr

driving t g the he regr egress ession

• n

equat equation

• ns

s is know s known as as (a (a) leas east squar squares es (b (b) concur concurrent ent devi deviation

• n

(c (c) pr product

• duct m

mom

• ment

ent (d (d) nor normal al equat equation

Quantitative Aptitude & Business Statistics: Regression 52

4.The method applied for driving the regression equations is known as (a) least squares (b) concurrent deviation (c) product moment (d) normal equation

Quantitative Aptitude & Business Statistics: Regression 53

5. 5.The t two l

• lines of
• f regr

regression bec become ident entic ical w l when en (a) (a) r=1 =1 (b) (b) r= r=–1 (c) c) r=0 =0 (d) (d) (a) or (a) or (b) (b)

Quantitative Aptitude & Business Statistics: Regression 54

5. 5.The t two l

• lines of
• f regr

regression bec become ident entic ical w l when en (a) (a) r=1 =1 (b) (b) r= r=–1 (c) c) r=0 =0 (d) (d) (a) or (a) or (b) (b)

Quantitative Aptitude & Business Statistics: Regression 55

6. 6.The t term erm regr regression was as f firs rst us used i in n the he yea ear 187 1877 by by _ _____ (a) (a) Karl rl P Pearso rson (b) (b) A. L. B Bowley wley (c) c) R. A.

• A. Fish

sher (d) (d) Sir F r Fran rancis Gal alton

Quantitative Aptitude & Business Statistics: Regression 56

6. 6.The t term erm regr regression was as f firs rst us used i in n the he yea ear 187 1877 by by _ _____ (a) (a) Karl rl P Pearso rson (b) (b) A. L. B Bowley wley (c) c) R. A.

• A. Fish

sher (d) (d) Sir F r Fran rancis Gal alton

Quantitative Aptitude & Business Statistics: Regression 57

7.If r regr gres essio ion l n lines nes a are p perpen pendic dicular lar to eac

• each ot
• ther, t

the he v val alue of

• f r

r will be _ be __ (a) (a) +1 +1 (b) (b) –1 (c) c) 0 (d) (d) non none of

• f thes

hese

Quantitative Aptitude & Business Statistics: Regression 58

7.If r regr gres essio ion l n lines nes a are p perpen pendic dicular lar to eac

• each ot
• ther, t

the he v val alue of

• f r

r will be _ be __ (a) (a) +1 +1 (b) (b) –1 (c) c) 0 (d) (d) non none of

• f thes

hese

Quantitative Aptitude & Business Statistics: Regression 59

8. 8.∑X=5 =50; ; ∑Y=30; 0; ∑XY=10 1000; 0; ∑X2=30 3000; 0; ∑Y2=18 180; 0; n=10, 10, t the v value lue

• f
• f by

byx w will be be (a) (a) 0.6132 6132 (b) (b) 1.3636 3636

(c) (c) 0.3090

3090

(d) (d) non

none of

• f thes

hese

Quantitative Aptitude & Business Statistics: Regression 60

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

Quantitative Aptitude & Business Statistics: Regression 61

9.The s he stand ndar ard e d error

• r of an estim

imat ate i e is Zero ero , ,r r will be be---

• A)

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

• f t

thes hese

±

Quantitative Aptitude & Business Statistics: Regression 62

9.The s he stand ndar ard e d error

• r of an estim

imat ate i e is Zero ero , ,r r will be be---

• A)

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

• f t

thes hese

±

Quantitative Aptitude & Business Statistics: Regression 63

10. 10.If the here are t are two v

• var

ariables x x and and y,then t the num he number of

• f regr

regression equat uatio ions ns c could b uld be A) A)1 B) B)2 C) ) Any ny num number D)3 )3

Quantitative Aptitude & Business Statistics: Regression 64

10. 10.If the here are t are two v

• var

ariables x x and and y,then t the num he number of

• f regr

regression equat uatio ions ns c could b uld be. A) A)1 B) B)2 C) ) Any ny num number D)3 )3

Quantitative Aptitude & Business Statistics: Regression 65

11.

• 11. T

The reg he regression c coef

• efficients are

are Zero ero i if r r is equ equal t to-----

• A)

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

Quantitative Aptitude & Business Statistics: Regression 66

11 The regression coefficients are Zero if r is equal to----- A)2 B)-1 C)1 D)0

Quantitative Aptitude & Business Statistics: Regression 67

12 . 12 .Whe hen r r =0 t 0 the hen C Cov

• v(x,y)----
• ---is

is e equal ual to to A) A) +1 +1 B) B) -1 C) ) 0 D) ) 3

Quantitative Aptitude & Business Statistics: Regression 68

12 . 12 .Whe hen r r =0 t 0 the hen C Cov

• v(x,y)----
• ---is

is e equal ual to to A) A)+1 +1 B) B)-1 C)0 )0 D)3 )3

Quantitative Aptitude & Business Statistics: Regression 69

13.If r r=1 , 1 ,then en t the s e standar andard e error

• r of

estim imate w e will ll b be A) Z ) Zero ero B) B)+1 +1 C) ) -1 D) non ) none of

• f t

thes hese

Quantitative Aptitude & Business Statistics: Regression 70

13. 13.If r r =1 1 ,the hen t the s he stan andard err error of

• f

estim imate w e will ll b be A) Z ) Zero ero B) B)+1 +1 C) ) -1 D) non ) none of

• f t

thes hese

Quantitative Aptitude & Business Statistics: Regression 71

14. 14.If bx bxy=+0. 0.8, 8,then n the he val alue of

• f by

byx can be an be A) A)+1 +1.25 25 B) B)-1.2 .25 C)+1 +1.26 26 D) D)-1. 1.24 24

Quantitative Aptitude & Business Statistics: Regression 72

14. 14.If bx bxy=+0. 0.8, 8,then n the he val alue of

• f by

byx can be an be A) A)+1 +1.25 25 B) B)-1.2 .25 C)+1 +1.26 26 D) D)-1. 1.24 24

Quantitative Aptitude & Business Statistics: Regression 73

15 ____Gives the mathematical relationship between the variables. A) Correlation B) Regression C) Both D) None

Quantitative Aptitude & Business Statistics: Regression 74

15 ____Gives the mathematical relationship between the variables. A) Correlation B) Regression C) Both D) None

Quantitative Aptitude & Business Statistics: Regression 75

• 16. Equations of tw o 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

Quantitative Aptitude & Business Statistics: Regression 76

• 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/ 4/7 and 7 and –11/ 11/7 7 C) ) 2 and 2 and 4 c 4 c D) ) None of

• ne of these

hese

Quantitative Aptitude & Business Statistics: Regression 77

• 17. Two lines of regression are given by

5x+7y–22=0 and 6x+2y–22=0. If the variance

• f y is 15, find the standard deviation of x?

A) B) C) D)

5

7 6 8

Quantitative Aptitude & Business Statistics: Regression 78

• 17. Two lines of regression are given by

5x+7y–22=0 and 6x+2y–22=0. If the variance

• f y is 15, find the standard deviation of x?

A) B) C) D)

5

7 6 8

Quantitative Aptitude & Business Statistics: Regression 79

18.

• 18. I

If 2x 2x + 5y + 5y – 9 9 = 0 = 0 and 3x and 3x – y y – 5 = 0 ar 5 = 0 are t e two

• regr

egres ession equat n equation

• n, t

then hen find t nd the he val value e of

• f

mean of ean of x x and m and mean of ean of y. y. A) ) 2, 2,1 1 B) ) 2,2 ,2 C) ) 1,2 ,2 D) D) 1,1 ,1

Quantitative Aptitude & Business Statistics: Regression 80

18.

• 18. I

If 2x 2x + 5y + 5y – 9 9 = 0 = 0 and 3x and 3x – y y – 5 = 0 ar 5 = 0 are t e two

• regr

egres ession equat n equation

• n, t

then hen find t nd the he val value e of

• f

mean of ean of x x and m and mean of ean of y. y. A) ) 2,1 ,1 B) ) 2,2 ,2 C) ) 1,2 ,2 D) D) 1,1 ,1

Quantitative Aptitude & Business Statistics: Regression 81

19.

• 19. I

If one of

• ne of t

the r he regr egres ession

• n coef

coefficient ents is s gr great eater er t than han uni unity, t then hen ot

• ther

her is l s les ess t than han unity ty. . A) ) True ue B) ) False se C) ) Both th D) ) None of

• ne of these

hese

Quantitative Aptitude & Business Statistics: Regression 82

19.

• 19. I

If one of

• ne of t

the r he regr egres ession

• n coef

coefficient ents is s gr great eater er t than han uni unity, t then hen ot

• ther

her is l s les ess t than han unity ty. . A) ) Tru rue B) ) False se C) ) Both th D) ) None of

• ne of these

hese

Quantitative Aptitude & Business Statistics: Regression 83

20.

• 20. The

he two r

• regr

egress ession l n line nes obt

• btai

aine ned d from

• m

cer certain n dat data a wer ere e y = y = x + 5 and x + 5 and 16x 16x = 9y = 9y – 94. 94. Find nd t the he var varian ance of e of x x if var varianc nce of

• f y

y is 16. s 16. A) ) 4/ 4/16 16 B) B) 9 C) ) 1 D) ) 5/ 5/16 16

Quantitative Aptitude & Business Statistics: Regression 84

20.

• 20. The

he two r

• regr

egress ession l n line nes obt

• btai

aine ned d from

• m

cer certain n dat data a wer ere e y = y = x + 5 and x + 5 and 16x = 16x = 9y 9y – 94. 94. Find nd t the he var varian ance of e of x x if var varianc nce of

• f y

y is 16. s 16. A) ) 4/ 4/16 16 B) B) 9 C) ) 1 D) ) 5/ 5/16 16

Quantitative Aptitude & Business Statistics: Regression 85

21.

• 21. F

For

• r a m

a m×n n two w

• way

ay or

• r bi

bivar variat ate f e frequenc equency tabl able, e, the he maxi aximum num number ber of

• f mar

argi gina nal di dist stribu bution

• ns i

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

Quantitative Aptitude & Business Statistics: Regression 86

21.

• 21. F

For

• r a m

a m×n n two w

• way

ay or

• r bi

bivar variat ate f e frequenc equency tabl able, e, the he maxi aximum num number ber of

• f mar

argi gina nal di dist stribu bution

• ns i

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

THE END

Regressio ssion