Question

1. determine the line of best fit for this data. enter in y = mx + b form. round to the nearest tenth

Explanation:

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

Let \(x\) values be \(x_1=-1,x_2 = 0,x_3=1,x_4=2,x_5 = 4\) and \(y\) values be \(y_1=-1,y_2 = 1,y_3=1,y_4=2,y_5 = 5\).
The mean of \(x\), \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{-1 + 0+1+2 + 4}{5}=\frac{6}{5}=1.2\)
The mean of \(y\), \(\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{-1+1 + 1+2+5}{5}=\frac{8}{5}=1.6\)

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

The formula for \(m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\)
\((x_1-\bar{x})(y_1-\bar{y})=(-1 - 1.2)(-1-1.6)=(-2.2)(-2.6) = 5.72\)
\((x_2-\bar{x})(y_2-\bar{y})=(0 - 1.2)(1 - 1.6)=(-1.2)(-0.6)=0.72\)
\((x_3-\bar{x})(y_3-\bar{y})=(1 - 1.2)(1 - 1.6)=(-0.2)(-0.6)=0.12\)
\((x_4-\bar{x})(y_4-\bar{y})=(2 - 1.2)(2 - 1.6)=(0.8)(0.4)=0.32\)
\((x_5-\bar{x})(y_5-\bar{y})=(4 - 1.2)(5 - 1.6)=(2.8)(3.4)=9.52\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})=5.72+0.72 + 0.12+0.32+9.52=16.4\)

\((x_1-\bar{x})^2=(-1 - 1.2)^2=(-2.2)^2 = 4.84\)
\((x_2-\bar{x})^2=(0 - 1.2)^2=1.44\)
\((x_3-\bar{x})^2=(1 - 1.2)^2=0.04\)
\((x_4-\bar{x})^2=(2 - 1.2)^2=0.64\)
\((x_5-\bar{x})^2=(4 - 1.2)^2=7.84\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})^2=4.84+1.44+0.04+0.64+7.84 = 14.8\)

\(m=\frac{16.4}{14.8}\approx1.1\)

Step3: Calculate \(b\)

Using the formula \(b=\bar{y}-m\bar{x}\), substitute \(m = 1.1\) and \(\bar{x}=1.2,\bar{y}=1.6\)
\(b=1.6-(1.1\times1.2)=1.6 - 1.32 = 0.3\)

Answer:

\(y = 1.1x+0.3\)