Question

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

Explanation:

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

The formula for the mean of \(x\) values \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), and for \(y\) values \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\).
Here \(n = 6\), \(\sum_{i=1}^{6}x_{i}=-2-1 + 0+1+2+3=3\), so \(\bar{x}=\frac{3}{6}=0.5\).
\(\sum_{i = 1}^{6}y_{i}=4 + 2+1-2-1-2=2\), so \(\bar{y}=\frac{2}{6}\approx0.3\).

Step2: Calculate the numerator and denominator for \(m\)

The formula for \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\((x_{1}-\bar{x})(y_{1}-\bar{y})=(-2 - 0.5)(4 - 0.3)=(-2.5)\times3.7=-9.25\)
\((x_{2}-\bar{x})(y_{2}-\bar{y})=(-1 - 0.5)(2 - 0.3)=(-1.5)\times1.7=-2.55\)
\((x_{3}-\bar{x})(y_{3}-\bar{y})=(0 - 0.5)(1 - 0.3)=(-0.5)\times0.7=-0.35\)
\((x_{4}-\bar{x})(y_{4}-\bar{y})=(1 - 0.5)(-2 - 0.3)=0.5\times(-2.3)=-1.15\)
\((x_{5}-\bar{x})(y_{5}-\bar{y})=(2 - 0.5)(-1 - 0.3)=1.5\times(-1.3)=-1.95\)
\((x_{6}-\bar{x})(y_{6}-\bar{y})=(3 - 0.5)(-2 - 0.3)=2.5\times(-2.3)=-5.75\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})=-9.25-2.55 - 0.35-1.15-1.95-5.75=-21\)

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

So \(m=\frac{-21}{17.5}=-1.2\)

Step3: Calculate \(b\)

Using the formula \(y=mx + b\), substitute \(x = \bar{x}\) and \(y=\bar{y}\)
\(0.3=-1.2\times0.5+b\)
\(0.3=-0.6 + b\)
\(b=0.3 + 0.6=0.9\)

Answer:

\(y=-1.2x + 0.9\)