CORRELATION

1
Session 4 Topic Regression Analysis Faculty
Ms Prathima Bhat K Department of Management
Studies Acharya Institute of
Technology Bangalore 90 Contact
prathimabhatk_at_gmail.com 9242187131
2
REGRESSION

• Regression Analysis, in general sense, means the
  estimation or prediction of the unknown value of
  one variable from the known value of the other
  variable.

3

• The Regression Analysis confined to the study of
  only two variables at a time is termed as Simple
  Regression. But quite often the values of a
  particular phenomenon may be affected by
  multiplicity of causes. The Regression analysis
  for studying more than two variables at a time is
  known as Multiple Regression.

4

• In Regression Analysis there are two types of
  variables. The variable whose value is influenced
  or is to be predicted is called dependent
  variable. The variable which influences the
  values or used for prediction is called
  independent variable. The Regression Analysis
  independent variable is known as regressor or
  predictor or explanator while the dependent
  variable is also known as regressed or explained
  variable.

5
LINEAR NON-LINEAR REGRESSION

• If the given bivariate data are plotted on a
  graph, the points so obtained on the diagram will
  more or less concentrate around a curve, called
  the Curve of Regression. The mathematical
  equation of the Regression curve, is called the
  Regression Equation. If the regression curve is a
  straight line, we say that there is linear
  regression between the variables under study. If
  the curve of regression is not a straight line,
  the regression is termed as curved or non-linear
  regression.

6
LINES OF REGRESSION

• Line of regression is the lines which gives the
  best estimate of one variable for any given value
  of the other variable. In case of two variable
  say x y, we shall have two regression
  equations x on y and the other is y on x.
• Line of regression of y on x is the line which
  gives the best estimate for the value of y for
  any specified value of x.
• Line of regression of x on y is the line which
  gives the best estimate for the value of x for
  any specified value of y.

7
LINES OF REGRESSION OF y on x

LINES OF REGRESSION OF x on y
8
REMEMBER

• When r0 i.e., when x y are uncorrelated, then
  the lines of regression of y on x, and x on y are
  given as y y 0 and x x 0. The lines are
  perpendicular to each other.
• When r1 then the two lines coincide.
• If the value of r is significant, we can use the
  lines of regression for estimation and
  prediction.
• If r is not significant, then the linear model is
  not a good fit and hence the line of regression
  should not be used for prediction.

9
COEFFICIENTS OF REGRESSION

• bxy is the Coefficient of regression of x on y.
• byx is the Coefficient of regression of y on x.

10
THEOREMS ON REGRESSION COEFFICIENTS

• The correlation coefficient is the Geometric Mean
  between the Regression Coefficients i.e., r2 bxy
  byx
• The sign to be taken before the square root is
  same as that of regression coefficients.
• If one of the regression coefficient is greater
  than one, then the other must be less than one.

11
THEOREMS ON REGRESSION COEFFICIENTS (Contd)

• The AM of the modulus value of regression
  coefficients is greater than the GM of the
  modulus value of the Correlation Coefficient.
• Regression coefficients are independent of change
  of origin but not of scale.

12
(No Transcript)
13
0.6132
1.361
(x-90) 1.361(y-70)
(y-70) 0.6132 (x-90)
x1.361y - 5.27
y0.6132x 14.812
14

• The data about the sales advertisement
  expenditure of a firm is given below
• Sales Advertmnt Expend.
• Means 40 6
• Standard Deviations 10 1.5
• Coefficient of Correlation is 0.9
• Estimate the likely sales for a proposed
  advertisement expenditure of Rs. 10 crores.
• What should be the advertisement expenditure if
  the firm proposes a sales target of 60 crores of
  rupees?

15
(x-40) (0.910/1.5) (y-6)
(y-6) (0.91.50/10) (x-40)
x 6y4
y 0.135x0.6
x 6104
y 0.135600.6
x 64
y 8.7
16

• Point out the consistency, if any, in the
  following statement
• The Regression Equation of y on x is 2y3x4 and
  the correlation coefficient between x y is 0.8
• By using the following data, find out the two
  lines of regression and from them compute the
  Karl-Pearsons coefficient of correlation.
• SX250 SY300 SXY7900 SX26500 SY210000
  n10

17
0.4
1.6
rxy2 bxy bxy
rxy2 1.6 0.4
rxy 0.8
18

• Find the two regression coefficients and hence
  the r .
• n5 X10 Y20 S(X-4)2100 S(Y-10)2160
  S(X-4)(Y-10)80

ANSWER
UX-4 UX-46 SU nU 30. Similarly SV50
byx
580 3050
byx
(11 17)
(11 4)
580 3050
5160 -(50)2
5100 -(30)2
r v(11/4)(11/17) 1.33 ( it is impossible)