Regression Equation Setup

[mathjax] Given, Sample size (=mathrm{n}=)
Let, Final score (=Y)
Mid-term score (=X_{1})
Home work score (=X_{2})
So, multiple regression Equation (Y) on (X_{1}) and (X_{2})
Where, a = Sample (Y)-intercept (value of (Y) when (X_{1}=0) and (X_{2}=0) )
(b_{1}=) Sample regression coefficient of (Y) on (X_{1}) and keeping (X_{2}) constant.
(b_{2}=) Sample regression coefficient of (Y) on (X_{2}) and keeping (X_{1}) constant.
The constant can be solved by the following equation
(Sigma Y=) na (+b_{1} Sigma X_{1}+b_{2} Sigma X_{2})
(Sigma X_{1} Y=mathrm{a} Sigma X_{1}+b_{1} Sigma X_{1}^{2}+b_{2} Sigma X_{1} X_{2})
(Sigma X_{2} Y=) na (Sigma X_{2}+b_{1} Sigma X_{1} X_{2}+b_{2} Sigma X_{2}^{2})

begin{array}{|l|l|l|l|l|l|l|l|l|}
hline mathrm{Y} & mathrm{X}_{1} & mathrm{X}_{2} & mathrm{Y}^{2} & X_{1}^{2} & X_{2}^{2} & mathrm{Y} times mathrm{X}_{1} & mathrm{Y} times mathrm{X}_{2} & mathrm{X}_{1} times mathrm{X}_{2} \
hline & & & & & & & & \
hline & & & & & & & & \
hline mathrm{Y} & Sigma mathrm{X}_{1} & Sigma mathrm{X}_{2} & Sigma mathrm{Y}^{2} & Sigma X_{1}^{2} & Sigma X_{2}^{2} & Sigma mathrm{Y} times mathrm{X}_{1} & Sigma mathrm{Y} times mathrm{X}_{2} & Sigma mathrm{X}_{1} times X_{2} \
hline
end{array}

Coefficient of Multiple Determination

begin{equation*}
mathrm{S}_{mathrm{e}}=sigma_{1.23}=mathrm{S}_{1.23}=sqrt{frac{Sigma Y^{2}-a Sigma Y-b_{1} Sigma Y X_{1}-b_{2} Sigma Y X_{2}}{n-3}}
end{equation*}
If (S_{e} neq 0), estimation equation cannot be perfect estimator of dependent variable (Y)
If (S_{e}=0) estimation equation cannot perfect estimator of dependent variable (Y)

Coefficient of Multiple Determination

begin{equation*}
R_{Y .12}^{2}=frac{a Sigma Y+b_{1} Sigma X_{1} X_{2}+b_{2} Sigma X_{1} X_{2}-n bar{Y}^{2}}{Sigma X_{1}^{2}-n bar{X}_{1}{ }^{2}}=mathrm{n} %(text { say })
end{equation*}
Means (n %) of total variation dependent variable (Y) can be explained by independent variable (X_{1}) and (X_{2}), and remaining ((100-n) %) effected of other factors (unexplained variation)

Statistics Free Resource

Theory of Probability Theoretical Probability Distribution Sampling & estimation Testing of Hypothesis Correlation & Regression