Question

The least-squares regression line is:

Answer 1

Answer:

a. the line that makes the sum of the squares of the vertical distances of the data points from the line (the sum of squared residuals) as small as possible.

Step-by-step explanation:

If we have N points $(x_1, y_1),. ...(x_N, y_N)$ and we want to adjust a model $y = ax+b$

We can define the error associated to this like that:

$E(a,b) = \sum_{n=1}^N [y_n -(ax_n +b)]^2$

So as we can see here we are adding the square distances between the real and the adjusted values in order to minimize the error for this reason the correct answer is:

a. the line that makes the sum of the squares of the vertical distances of the data points from the line (the sum of squared residuals) as small as possible.

For this case we need to calculate the slope with the following formula:

$a=\frac{S_{xy}}{S_{xx}}$

Where:

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$

And we can find the intercept using this:

$b=\bar y -m \bar x$