Question

raquel throws darts at a coordinate grid centered at the origin. her goal is to create a line of darts. her darts actually hit the coordinate grid at (-5,0), (1,-3), (4,5), (-8,-6), (0,2), and (9,6). which equation best approximates the line of best fit of the darts?

o y = 0.6x + 0.6
o y = 0.1x + 0.8
o y = 0.8x + 0.1
o y = 0.5x + 0.6

Explanation:

Step1: Recall the form of a linear regression line

The line of best - fit is of the form $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We can use a statistical method (least - squares regression) or a quick estimate by looking at the general trend of the points. A rough way is to consider the rise - over - run for the general trend of the points and the approximate y - intercept.

Step2: Analyze the y - intercept

The y - intercept is the value of $y$ when $x = 0$. We have the point $(0,2)$ among our data points. This gives us an idea that the y - intercept $b$ is around 2. But we can also use a more formal approach. For a set of points $(x_i,y_i)$ with $i = 1,\cdots,n$, the formulas for the slope $m$ and y - intercept $b$ in least - squares regression are:
\[m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\]
\[b=\frac{\sum_{i = 1}^{n}y_i - m\sum_{i = 1}^{n}x_i}{n}\]
Let's calculate the sums:
For the points $(-5,0),(1, - 3),(4,5),(-8,-6),(0,2),(9,6)$:
\(n = 6\)
\(\sum_{i=1}^{6}x_i=-5 + 1+4-8 + 0+9=1\)
\(\sum_{i=1}^{6}y_i=0-3 + 5-6 + 2+6=4\)
\(\sum_{i=1}^{6}x_i^{2}=(-5)^{2}+1^{2}+4^{2}+(-8)^{2}+0^{2}+9^{2}=25 + 1+16+64+0+81=187\)
\(\sum_{i=1}^{6}x_iy_i=(-5)\times0+1\times(-3)+4\times5+(-8)\times(-6)+0\times2+9\times6=0-3 + 20+48+0+54=119\)
\[m=\frac{6\times119-1\times4}{6\times187 - 1^{2}}=\frac{714 - 4}{1122-1}=\frac{710}{1121}\approx0.63\]
\[b=\frac{4-0.63\times1}{6}=\frac{4 - 0.63}{6}=\frac{3.37}{6}\approx0.56\approx0.6\]

Answer:

A. \(y = 0.6x+0.6\)