Question
Explain the method of least square for fitting a regression line.
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Answer
The best fitted regression line of $Y$ on $X$ can be obtained by this method. Suppose n ordered pairs of sample observations on two correlated variables $X$ and $Y$ are $(x_1, y_1), (x_2, y_2), …, (x_n, y_n).$ To understand the least square method we draw the scatter diagram for this data.
If the equation of best fitted line showing the linear regression between $Y$ and $X$ is $ŷ = a + bx$ then the values of constants a and b can be obtained by least square method as follows:
Suppose $ŷ_1, ŷ_2, ………, ŷ_n$ are the estimated values for the values $y_1, y_2, ………. y_n$ of $Y$ obtained by the equation of line $Y$ corresponding to the values $x_1, x_2, ………., x_n$ of variable $X.$
Now, for a given value $x_1$ of $x,$ the corresponding to the value y of $y_1$ the estimated value $ŷ$ of $y$ will be $ŷ = a + bx_1.$
The values of constants a and b of the fitted line $ŷ = a + bx$ are obtained by least square method in such a way so that the sum of squares of errors $\sum e_i^2$ becomes minimum, where $(e = y – ŷ)$. Thus, $\sum e^2 = \sum (y – ŷ)^2 = \sum (y – a – bx)^2$ becomes minimum.
The line $ŷ = a + bx$ obtained passes through close to most of the points on scatter diagram. While obtaining the regression line, the sum of squares of errors is minimised, therefore it is called method of least square.
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