Question

which is the equation of the line of best - fit for this scatterplot?
$y = - \frac{81}{2}x + \frac{165,389}{2}$
$y = - \frac{2}{81}x + \frac{111,748}{81}$
$y = \frac{2}{81}x + \frac{103,712}{81}$
$y = \frac{81}{2}x - \frac{160,069}{2}$

Explanation:

Step1: Analyze the slope

Looking at the scatter - plot (though not fully visible, but based on the general trend and the options), if we assume some general data - point testing. For a line of the form \(y = mx + b\), the slope \(m\).
Let's assume \(x\) represents the year. If we take \(x = 2008\).
For the first option \(y=-\frac{81}{2}x+\frac{165389}{2}\), when \(x = 2008\), \(y=-\frac{81}{2}\times2008+\frac{165389}{2}=\frac{-81\times2008 + 165389}{2}=\frac{-162648+165389}{2}=\frac{2741}{2}=1370.5\)
For the second option \(y =-\frac{2}{81}x+\frac{111748}{81}\), when \(x = 2008\), \(y=-\frac{2}{81}\times2008+\frac{111748}{81}=\frac{- 4016+111748}{81}=\frac{107732}{81}\approx1330\)
For the third option \(y=\frac{2}{81}x+\frac{103712}{81}\), when \(x = 2008\), \(y=\frac{2}{81}\times2008+\frac{103712}{81}=\frac{4016 + 103712}{81}=\frac{107728}{81}\approx1330\)
For the fourth option \(y=\frac{81}{2}x-\frac{160069}{2}\), when \(x = 2008\), \(y=\frac{81}{2}\times2008-\frac{160069}{2}=\frac{162648-160069}{2}=\frac{2579}{2}=1289.5\)

Another way: assume the line of best - fit formula \(y=mx + b\). If we consider the general trend (assuming a negative slope as the line is going downwards from left - to - right in a typical scatter - plot situation).
Let's use the formula for the slope \(m=\frac{n\sum(xy)-\sum x\sum y}{n\sum(x^{2})-(\sum x)^{2}}\) (a more complex regression formula, but if we just consider the magnitude of the slope.
If we assume two points \((x_1,y_1)\) and \((x_2,y_2)\) (not given, but from the options). The slope of the line \(y =-\frac{81}{2}x+\frac{165389}{2}\) has a steeper negative slope compared to \(y =-\frac{2}{81}x+\frac{111748}{81}\).
In a typical year - based scatter - plot (where \(x\) is in the range of 2000 - 2010s), a slope of \(-\frac{2}{81}\) (approx \(- 0.025\)) is more reasonable than a slope of \(-\frac{81}{2}=-40.5\)

Answer:

\(y =-\frac{2}{81}x+\frac{111748}{81}\)