Question

identify the least squares regression line.
consider the following scatter plot:
which of the following provides the most reasonable approximation to the least squares regression line?
$\hat{y}=50 + 10x$
$\hat{y}=50 + x$
$\hat{y}=10 + 50x$
$\hat{y}=1 + 50x$
$\hat{y}=10 + x$

Explanation:

Step1: Recall the form of regression line

The least - squares regression line has the form \(\hat{y}=b_0 + b_1x\), where \(b_0\) is the y - intercept and \(b_1\) is the slope.

Step2: Estimate the y - intercept

Looking at the scatter - plot, when \(x = 0\), the regression line is close to \(y=50\), so \(b_0\approx50\).

Step3: Estimate the slope

When \(x = 1\), if we assume a rough estimate. Let's check the change in \(y\) for a unit change in \(x\).
If \(x = 0,y\approx50\) and if \(x = 1,y\approx60\) (a rough visual estimate). The slope \(b_1=\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}\). Here, \(x_1 = 0,y_1 = 50,x_2=1,y_2 = 60\), so \(b_1=\frac{60 - 50}{1-0}=10\)

Answer:

\(\hat{y}=50 + 10x\)