Question

1. determine the line of best fit. enter it in y = mx + b format. round to the nearest tenth.

type a response

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

The mean of \(x\) values: \(\bar{x}=\frac{-3 - 2-1 + 0+1 + 2}{6}=\frac{-3}{6}=-0.5\)
The mean of \(y\) values: \(\bar{y}=\frac{1-1 + 0-2-4-5}{6}=\frac{-11}{6}\approx - 1.8\)

Step2: Calculate the slope \(m\)

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\((x_{1}-\bar{x})(y_{1}-\bar{y})=(-3 + 0.5)(1+1.8)=(-2.5)\times2.8=-7\)
\((x_{2}-\bar{x})(y_{2}-\bar{y})=(-2 + 0.5)(-1 + 1.8)=(-1.5)\times0.8=-1.2\)
\((x_{3}-\bar{x})(y_{3}-\bar{y})=(-1 + 0.5)(0 + 1.8)=(-0.5)\times1.8=-0.9\)
\((x_{4}-\bar{x})(y_{4}-\bar{y})=(0 + 0.5)(-2 + 1.8)=0.5\times(-0.2)=-0.1\)
\((x_{5}-\bar{x})(y_{5}-\bar{y})=(1 + 0.5)(-4 + 1.8)=1.5\times(-2.2)=-3.3\)
\((x_{6}-\bar{x})(y_{6}-\bar{y})=(2 + 0.5)(-5 + 1.8)=2.5\times(-3.2)=-8\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})=-7-1.2-0.9-0.1-3.3-8=-20.5\)

\((x_{1}-\bar{x})^{2}=(-3 + 0.5)^{2}=(-2.5)^{2}=6.25\)
\((x_{2}-\bar{x})^{2}=(-2 + 0.5)^{2}=(-1.5)^{2}=2.25\)
\((x_{3}-\bar{x})^{2}=(-1 + 0.5)^{2}=(-0.5)^{2}=0.25\)
\((x_{4}-\bar{x})^{2}=(0 + 0.5)^{2}=0.25\)
\((x_{5}-\bar{x})^{2}=(1 + 0.5)^{2}=2.25\)
\((x_{6}-\bar{x})^{2}=(2 + 0.5)^{2}=6.25\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=6.25+2.25 + 0.25+0.25+2.25+6.25 = 17.5\)

\(m=\frac{-20.5}{17.5}\approx - 1.2\)

Step3: Calculate the y - intercept \(b\)

Using the formula \(b=\bar{y}-m\bar{x}\), substitute \(m=-1.2\) and \(\bar{x}=-0.5\), \(\bar{y}\approx - 1.8\)
\(b=-1.8-(-1.2)\times(-0.5)=-1.8 - 0.6=-2.4\)

Answer:

\(y=-1.2x-2.4\)