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? $y = 0.1x + 0.8$ $y = 0.8x + 0.1$ $y = 0.6x + 0.6$ $y = 0.5x + 0.6$

Explanation:

Step1: Calcular la media de \(x\) y \(y\)

Dados los puntos \((-5,0)\), \((1,-3)\), \((4,5)\), \((-8,-6)\), \((0,2)\), \((9,6)\)
Media de \(x\): \(\bar{x}=\frac{-5 + 1+4-8 + 0+9}{6}=\frac{1}{6}\approx0.17\)
Media de \(y\): \(\bar{y}=\frac{0-3 + 5-6+2 + 6}{6}=\frac{4}{6}\approx0.67\)

Step2: Calcular la pendiente \(m\)

\(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})\):
\((-5 - 0.17)(0 - 0.67)+(1 - 0.17)(-3 - 0.67)+(4 - 0.17)(5 - 0.67)+(-8 - 0.17)(-6 - 0.67)+(0 - 0.17)(2 - 0.67)+(9 - 0.17)(6 - 0.67)\)
\(=(-5.17)(-0.67)+(0.83)(-3.67)+(3.83)(4.33)+(-8.17)(-6.67)+(-0.17)(1.33)+(8.83)(5.33)\)
\(=3.4639-3.0461 + 16.5839+54.5939-0.2261+47.0639\)
\(\approx118.4\)

\(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}\):
\((-5 - 0.17)^{2}+(1 - 0.17)^{2}+(4 - 0.17)^{2}+(-8 - 0.17)^{2}+(0 - 0.17)^{2}+(9 - 0.17)^{2}\)
\(=(-5.17)^{2}+(0.83)^{2}+(3.83)^{2}+(-8.17)^{2}+(-0.17)^{2}+(8.83)^{2}\)
\(=26.7289+0.6889+14.6689+66.7489+0.0289+77.9689\)
\(\approx186.8\)

\(m=\frac{118.4}{186.8}\approx0.63\approx0.6\)

Step3: Calcular la intersección \(b\)

Usando la ecuación \(y = mx + b\) y \(\bar{x}\approx0.17\), \(\bar{y}\approx0.67\), \(m\approx0.6\)
\(0.67=0.6\times0.17 + b\)
\(b=0.67 - 0.102=0.568\approx0.6\)

Answer:

\(y = 0.6x+0.6\)