regression · 2016. 6. 24. · y = + + ε i population linear regression model ... y c =10x....
TRANSCRIPT
Quantitative Aptitude & Business Statistics: Regression
2
Regression Regression is the measure of
average relationship between two or more variables in terms of original units of the data.
Quantitative Aptitude & Business Statistics: Regression
3
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.
Quantitative Aptitude & Business Statistics: Regression
4
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.
Quantitative Aptitude & Business Statistics: Regression
5
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.
Quantitative Aptitude & Business Statistics: Regression
6
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.
Quantitative Aptitude & Business Statistics: Regression
7
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.
Quantitative Aptitude & Business Statistics: Regression
8
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.
Quantitative Aptitude & Business Statistics: Regression
9
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.
Quantitative Aptitude & Business Statistics: Regression
10
Regression lines
Regression line X on Y
Where X= Dependent Variable Y =Independent variable a=intercept and b= slope
bYaX +=
Quantitative Aptitude & Business Statistics: Regression
11
( )YYbXX xy −=− Another way of regression line X on Y
( )YYrXXy
x −=−σσ
Quantitative Aptitude & Business Statistics: Regression
12
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
=
Quantitative Aptitude & Business Statistics: Regression
13
Regression coefficients
The regression coefficient X on Y is
y
xxy .rb
σσ
=
Quantitative Aptitude & Business Statistics: Regression
14
Regression line Y on X
Where Y= Dependent Variable X =Independent variable a=intercept and b= slope
bXaY +=
Quantitative Aptitude & Business Statistics: Regression
15
Another way of regression line Y on X ( )XXrYY
x
y −=−σσ
( )XXbyxYY −=−
Quantitative Aptitude & Business Statistics: Regression
16
Properties of Linear Regression
Two Regression Equations. Product of regression coefficient. Signs of Regression Coefficient
and correlation coefficient. Intersection of means. Slopes .
Quantitative Aptitude & Business Statistics: Regression
17
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 .
Quantitative Aptitude & Business Statistics: Regression
18
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.
Quantitative Aptitude & Business Statistics: Regression
19
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
×=
×=
Quantitative Aptitude & Business Statistics: Regression
20
Normal Equations
Regression line Y on X
The two normal Equations are
bXaY +=
∑∑ += XbNaY
∑∑∑ += 2XbXaXY
Quantitative Aptitude & Business Statistics: Regression
21
Calculate byx
( )∑ ∑
∑ ∑ ∑
−
−=
NX
X
NYX
XYb 2
2
yx
XbYa −=
Quantitative Aptitude & Business Statistics: Regression
22
Normal Equations
Regression line X on Y
The two normal Equations are
bYaX +=
∑∑ += YbNaX
∑∑∑ += 2YbYaXY
Quantitative Aptitude & Business Statistics: Regression
23
Calculate bxy
( )∑ ∑
∑ ∑ ∑
−
−=
NY
Y
NYX
XYb 2
2
xy
YbXa −=
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
Dependent (Response) Variable
Independent (Explanatory) Variable
Slope Y-Intercept Random Error
a bx
Quantitative Aptitude & Business Statistics: Regression
25
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
Quantitative Aptitude & Business Statistics: Regression
26
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
Quantitative Aptitude & Business Statistics: Regression
27
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.
Quantitative Aptitude & Business Statistics: Regression
28
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
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
∑∑ += YbNaX ∑∑∑ += 2YbYaXY
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
∑∑ += XbNaY ∑∑∑ += 2XbXaXY
1
2
Quantitative Aptitude & Business Statistics: Regression
31
Regression line X on Y
Regression line Yon X
Correlation coefficient r=1.0
Y01.0Xc =
X10Yc =
Quantitative Aptitude & Business Statistics: Regression
32
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
Quantitative Aptitude & Business Statistics: Regression
33
Measure of Variation: The Sum of Squares
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 = 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
Quantitative Aptitude & Business Statistics: Regression
35
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
=
Quantitative Aptitude & Business Statistics: Regression
36
Coefficient of non-determination(k2) Coefficient of non-determination(k2)=1-r2
nceTotalVariainacevarlainedexpUn
=
Quantitative Aptitude & Business Statistics: Regression
37
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
Quantitative Aptitude & Business Statistics: Regression
38
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
Quantitative Aptitude & Business Statistics: Regression
39
Subtracting (2) from (3)
Putting in (1) ,we get ;
6x =
6x
96x161
=
=
8y
22y30
=
=−
Quantitative Aptitude & Business Statistics: Regression
40
B) the regression equation y on x is : 64x-45y=24
6564b
x6564
158y
2524x
4564y
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
51b
x51
2522x
xy =
+=
158
51.
4564
b.br yxxy
=±=
±=
Quantitative Aptitude & Business Statistics: Regression
42
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=
Quantitative Aptitude & Business Statistics: Regression
43
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 ?
Quantitative Aptitude & Business Statistics: Regression
44
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
=×=
×−−
=
Quantitative Aptitude & Business Statistics: Regression
45
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
Quantitative Aptitude & Business Statistics: Regression
46
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
Quantitative Aptitude & Business Statistics: Regression
47
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
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.The coefficient of determination is defined by the formula
(a) r2=1– (b) r2= (c) both (d) none of these
iancetotaliancelainedun
varvarexp
iancetotaliancelained
varvarexp
Quantitative Aptitude & Business Statistics: Regression
50
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
Quantitative Aptitude & Business Statistics: Regression
51
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
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.The two lines of regression become identical when
(a) r=1 (b) r=–1 (c) r=0 (d) (a) or (b)
Quantitative Aptitude & Business Statistics: Regression
54
5.The two lines of regression become identical when
(a) r=1 (b) r=–1 (c) r=0 (d) (a) or (b)
Quantitative Aptitude & Business Statistics: Regression
55
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
Quantitative Aptitude & Business Statistics: Regression
56
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
Quantitative Aptitude & Business Statistics: Regression
57
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
Quantitative Aptitude & Business Statistics: Regression
58
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
Quantitative Aptitude & Business Statistics: Regression
59
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
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 standard error of an estimate is Zero ,r will be---
A) 1 B)+1 C)-1 D) none of these
±
Quantitative Aptitude & Business Statistics: Regression
62
9.The standard error of an estimate is Zero ,r will be---
A) 1 B)+1 C)-1 D) none of these
±
Quantitative Aptitude & Business Statistics: Regression
63
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
Quantitative Aptitude & Business Statistics: Regression
64
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
Quantitative Aptitude & Business Statistics: Regression
65
11. The regression coefficients are Zero if r is equal to-----
A) 2 B) -1 C) 1 D) 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 .When r =0 then Cov(x,y)----is equal to
A) +1 B) -1 C) 0 D) 3
Quantitative Aptitude & Business Statistics: Regression
68
12 .When r =0 then Cov(x,y)----is equal to
A)+1 B)-1 C)0 D)3
Quantitative Aptitude & Business Statistics: Regression
69
13.If r=1 ,then the standard error of estimate will be
A) Zero B)+1 C) -1 D) none of these
Quantitative Aptitude & Business Statistics: Regression
70
13.If r =1 ,then the standard error of estimate will be
A) Zero B)+1 C) -1 D) none of these
Quantitative Aptitude & Business Statistics: Regression
71
14.If bxy=+0.8,then the value of byx can be
A)+1.25 B)-1.25 C)+1.26 D)-1.24
Quantitative Aptitude & Business Statistics: Regression
72
14.If bxy=+0.8,then the value of byx can be
A)+1.25 B)-1.25 C)+1.26 D)-1.24
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 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
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) – 4/7 and –11/7 C) 2 and 4 c D) None of these
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 of y is 15, find the standard deviation of x?
A) B) C) D)
57
68
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 of y is 15, find the standard deviation of x?
A) B) C) D)
57
68
Quantitative Aptitude & Business Statistics: Regression
79
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
Quantitative Aptitude & Business Statistics: Regression
80
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
Quantitative Aptitude & Business Statistics: Regression
81
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
Quantitative Aptitude & Business Statistics: Regression
82
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
Quantitative Aptitude & Business Statistics: Regression
83
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
Quantitative Aptitude & Business Statistics: Regression
84
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
Quantitative Aptitude & Business Statistics: Regression
85
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
Quantitative Aptitude & Business Statistics: Regression
86
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