If the regression lines of X on y and y on X are respectively 2x-0.3y =0 and 4y-0.5x-5=0. Find the two regression coefficient and also the coefficient of correlation ,if the standard deviation of X is 3.Find the standard deviation of y.
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Business Mathematics and Statistics > Correlation and Regression > Regression Lines, Regression Equations and Regression Coefficients
Correlation and Regression
Regression Lines, Regression Equations and Regression Coefficients
When we make a distribution in which there is an involvement of more than one variable, then such an analysis is Regression Analysis. It generally focuses on finding or rather predicting the value of the variable that is dependent on the other. Let’s know more about regression.
Regression
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Annuity and Its types
Positive and Negative Correlation
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Questions
Regression Lines
Let there be two variables: x & y. If y depends on x, then the result comes in the form of simple regression. Furthermore, we name the variables x and y as:
y – Regression or Dependent Variable or Explained Variable
x – Independent Variable or Predictor or Explanator
Therefore, if we use a simple linear regression model where y depends on x, then the regression line of y on x is:
y = a + bx
Browse more Topics under Correlation And Regression
Scatter Diagram
Karl Pearson’s Coefficient of Correlation
Rank Correlation
Probable Error and Probable Limits
Regression Coefficient
The two constants a and b are regression parameters. Furthermore, we denote the variable b as byx and we term it as regression coefficient of y on x.
Also, we can have one more definition for the regression line of y on x. We can call it the best fit as the result comes from least squares. This method is the most suitable method for finding the value of y on x i.e. the value of a dependent variable on an independent variable.
Least Squares Method
∑ ei2 = ∑ (yi – y ^ i)2 = ∑ (yi – a – bxi)2
Here, variable yi is the actual value or the observed value. Further, y ^ i = a + bxi, denotes the estimated value of yi for a given random value of a variable of xi; ei = Difference between observed and estimated value and is the error or residue. The regression line of y or x along with the estimation errors are as follows:
Regression
On minimizing the least squares equation, here is what we get. We refer to these equations Normal Equations.
∑yi = na + b ∑xi
∑xiyi = a ∑xi2 + b ∑xi
We get the least squares estimate for a and b by solving the above two equations for both a and b.
b = Cov(x,y)/Sx2
= (r.SxSy)/Sx2
= (r.Sy)/Sx
The estimate of a, after the estimation of b is:
a =
¯
y
– b
¯
x
On substituting the estimates of a and b is:
[ y –
¯
y
]/Sy = r[ x –
¯
x
]/Sx
Sometimes, it might so happen that variable x depends on variable y. In such cases, the line of regression of x on y is:
x = a ^ + b^y
Regression Equation
The standard form of the regression equation of variable x on y is:
[ x –
¯
x
]/Sx = r[ y –
¯
y
]/Sy