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Simple Linear Regression
Meaning of Regression analysis: - โข Regression analysis means the estimation or prediction of the unknown value of one variable from the known values of one or more variables. It attempts to establish the โnature of relationshipโ between variables. โข Dependent/Explained variable - The variable whose value is to be predicted is called dependent or explained variable. โข Independent/Explanatory variable โ The variables which are used to predict the values of a dependent variable are called independent or explanatory variables. โข Simple regression โ study of only two variables, a dependent and an independent variable is called simple regression. โข Simple linear regression โ when the relationship between the dependent and independent variable is linear. Lines of Regression โ THE LEAST SQUARE APPROACH โข The least square line of regression of Y on X : This equation is used to estimate value of Y for a given value of X.
Y = a + bX
Where, a and b are constants The value of constant a and b can be find out with the help of two normal equations. The two normal equations are as follows: โY = na + b โX โXY = a โX + b โ๐^2 b = it is called the regression coefficient of Y on X and is denoted by bYX. It measures the change in Y corresponding to a unit change in X. Thus bYX = Slope of the line of regression of Y on X and given by
bYX=
๐โ๐๐โ(โ๐)(โ๐) ๐โ๐^2 โ(โ๐)^2
= b
The line of regression of Y on X passes through the point (๐,๐) and hence the equation of the line of regression of Y on X (Y = a + bX) can also be written as
Y - ๐ = bYX (X- ๐ )
โข The least square line of regression of X on Y : This equation is used to estimate a value of X for a given value of Y.
X = c + dY
where, c and d are constants The two normal equations for estimating c and d are given by โX = nc + dโY โXY = cโY + dโY^2 d = it is called the regression coefficient of X on Y and is denoted by bXY. It measures the change in X corresponding to a unit change in Y. Thus bXY= Slope of the line of regression of X on Y and given by
bXY =
๐โ๐๐โ(โ๐)(โ๐) ๐โ๐^2 โ(โ๐)^2 The line of regression of X on Y passes through the point (๐,๐) and hence the equation of the line of regression of X on Y (X = c + dY) can also be written as X- ๐ = bXY(Y - ๐) Remarks:
1. It may be remarked that there are always two lines of regression, one of Y on X and the other X on Y. Y on X = to predict value of Y from known values of X X on Y = to predict value of X from known values of Y 2. Since the two lines of regression of passes through the point (๐,๐), the mean values (๐,๐) can be obtained as the point of intersection of the two regression lines. Regression Coefficient โ Some Formulas
- Formulas for regression Coefficients in terms of Covariance and Variance:
bYX =
๐ถ๐๐ฃ(๐,๐) ๐๐^2
bXY =
๐ถ๐๐ฃ(๐,๐) ๐๐^2
(We can also assume X= maintenance cost and Y = age of cars and then solve the question accordingly) X (Age) Y(Cost) X^2 Y^2 XY 2 10 4 100 20 4 20 16 400 80 6 25 36 625 150 8 30 64 900 240 โX = 20 โY = 85 โX^2 = 120 โY^2 = 2025 โXY= 490 (a) Regression equation for Costs(Y) related to age(X): It means we have to find out equation Y on X, which is given as
Y - ๐ = bYX (X- ๐ )
And bYX =
๐โ๐๐โ(โ๐)(โ๐) ๐โ๐^2 โ(โ๐)^2
bYX =
4 ร 490 โ( 20 )( 85 ) 4 ร 120 โ( 20 )^2
1960 โ 1700 480 โ 400
260 80
bYX = 3.
๐ = โ๐ ๐
20 4 ๐ = 5 Regression equation Y on X:
Y - ๐ = bYX (X- ๐ )
Y โ 21.25 = 3.25 (X โ 5)
Y โ 21.25 = 3.25X โ 16.
Y = 5 + 3.25X, which is in the form of Y = a + bX
(b) Regression equation for Age (X) related to Cost(Y): It means we have to find out equation X on Y, which is given as
X- ๐ = bXY(Y - ๐)
And, bXY =
๐โ๐๐โ(โ๐)(โ๐) ๐โ๐^2 โ(โ๐)^2
bXY =
4 ร 490 โ( 20 )( 85 ) 4 ร 2025 โ( 85 )^2
1960 โ 1700 8100 โ 7225
260 875
bYX =.
โ๐ ๐
85 4 ๐ = 21. Regression equation X on Y:
X - ๐ = bXY (Y- ๐ )
X โ 5 = .297 (Y โ 21.25)
Y โ 5 = .297Y โ 6.
X = - 1.31125 + .297Y, which is in the form of X = c + dY
(c) Estimate the annual cost for a ten-year-old car: It means what will be the value of Y, if value of X = 10 (X = 10, Y =?) That is if the age of car(X) is 10, what is the cost (Y) for such a car. To find out value of Y for the given value of X, we will use regression equation Y on X.
Y = a + bX
Y = 5 + 3.25X, as obtained in part (a)
Substituting X = 10 in above equation, the estimated annual maintenance cost for a ten-year-old car is:
Y = 5 + 3.25ร
Y = 5 + 32.
Y = 37.5 (in Rs. hundred)
(d) Estimate the age(X) of a car whose maintenance cost(Y) is 50.
