The best-fitted regression line is found by the method of least squares.

The regression line is chosen so that the sum of the squares of the deviations of the observed $y$ from the estimated $\hat y$ is minimum, i.e. $\Sigma(y-\hat y)^2$ is least. This is the least square method; Karl Pearson's method is for correlation, not for fitting a line.

The regression line passes through $(\bar x,\bar y)$; putting $\bar x=10$ gives $\bar y=15$.

The regression line of $Y$ on $X$ always passes through the point of averages $(\bar x,\bar y)$. Substituting $\bar x=10$ in $\hat y=30-1.5x$:

$\bar y=30-1.5(10)=30-15=15.$

A regression line always passes through the point of means $(\bar{x}, \bar{y})$ — option (a).

In the GSEB Class-12 Statistics Linear Regression chapter, the regression line of $Y$ on $X$ is written as:

$\hat{y} - \bar{y} = b_{yx}(x - \bar{x})$

Substituting $x = \bar{x}$ gives $\hat{y} = \bar{y}$, so the line passes through $(\bar{x}, \bar{y})$. The same is true for the line of $X$ on $Y$; hence both lines intersect at the mean point.

Each regression line is constructed to pass through the point of averages $(\bar{x}, \bar{y})$, which is also where the two regression lines intersect.

Reasoning. The regression line of $Y$ on $X$ is $y-\bar{y} = b_{yx}(x-\bar{x})$ and of $X$ on $Y$ is $x-\bar{x} = b_{xy}(y-\bar{y})$. Substituting $x=\bar{x},\,y=\bar{y}$ satisfies both equations, so both lines pass through $(\bar{x},\bar{y})$.

The best-fitted regression line is obtained by the Method of Least Squares.

Explanation. The method of least squares chooses the line for which the sum of the squares of the deviations (errors) of the observed values from the estimated values is minimum. This criterion yields the best-fitted regression line of $Y$ on $X$ (and, similarly, of $X$ on $Y$).

Error of estimation $e = y - \hat{y}$ (observed minus estimated) — option (a).

For the regression line of $Y$ on $X$, we estimate $Y$ as $\hat{y} = a + bx$. For a given observation the error (residual) of estimation is defined as the actual observed value of $Y$ minus the value estimated by the line:

$e = y - \hat{y}$

The method of least squares chooses $a$ and $b$ so that $\sum e^2 = \sum (y - \hat{y})^2$ is minimum. The other options either interchange the sign or use the $X$-on-$Y$ residual.

$a$ = intercept ($\hat{y}$ at $x = 0$); $b$ = regression coefficient $b_{yx}$ = slope = change in $\hat{y}$ per unit change in $x$.

In the regression line of $Y$ on $X$,

$\hat{y} = a + bx,$

• $a$ is the intercept constant: the value the line predicts for $\hat{y}$ when $x = 0$. It fixes the position of the line on the $Y$-axis.
• $b$ is the regression coefficient of $Y$ on $X$, denoted $b_{yx}$: it is the slope of the line, i.e. the amount by which the estimated $\hat{y}$ changes for a one-unit increase in $x$.

Both are determined by the method of least squares from the data, using

$b = b_{yx} = \frac{\sum(x-\bar{x})(y-\bar{y})}{\sum(x-\bar{x})^2}, \qquad a = \bar{y} - b\bar{x}.$