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 mean of \(x\) values

Let \(x\) values be \(x_1 = 4,x_2=3,x_3 = 3,x_4=1,x_5 = 0,x_6=- 3\).
\(\bar{x}=\frac{4 + 3+3 + 1+0+( - 3)}{6}=\frac{8}{6}\approx1.3\)

Step2: Calculate the mean of \(y\) values

Let \(y\) values be \(y_1 = 7,y_2=-1,y_3 = 0,y_4=1,y_5 = 2,y_6=4\).
\(\bar{y}=\frac{7+( - 1)+0 + 1+2 + 4}{6}=\frac{13}{6}\approx2.2\)

Step3: Calculate the numerator for slope \(m\)

\(\sum_{i = 1}^{6}(x_i-\bar{x})(y_i-\bar{y})=(4 - 1.3)(7 - 2.2)+(3 - 1.3)(-1 - 2.2)+(3 - 1.3)(0 - 2.2)+(1 - 1.3)(1 - 2.2)+(0 - 1.3)(2 - 2.2)+(-3 - 1.3)(4 - 2.2)\)
\(=2.7\times4.8+1.7\times(-3.2)+1.7\times(-2.2)+(-0.3)\times(-1.2)+(-1.3)\times(-0.2)+(-4.3)\times1.8\)
\(=12.96-5.44 - 3.74+0.36 + 0.26-7.74\)
\(=-3.34\)

Step4: Calculate the denominator for slope \(m\)

\(\sum_{i = 1}^{6}(x_i-\bar{x})^2=(4 - 1.3)^2+(3 - 1.3)^2+(3 - 1.3)^2+(1 - 1.3)^2+(0 - 1.3)^2+(-3 - 1.3)^2\)
\(=2.7^2+1.7^2+1.7^2+(-0.3)^2+(-1.3)^2+(-4.3)^2\)
\(=7.29+2.89+2.89 + 0.09+1.69+18.49\)
\(=33.34\)

Step5: Calculate slope \(m\)

\(m=\frac{\sum_{i = 1}^{6}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{6}(x_i-\bar{x})^2}=\frac{-3.34}{33.34}\approx - 0.1\)

Step6: Calculate \(b\)

Using the formula \(b=\bar{y}-m\bar{x}\), substitute \(m\approx - 0.1\) and \(\bar{x}\approx1.3,\bar{y}\approx2.2\)
\(b = 2.2-(-0.1)\times1.3=2.2 + 0.13=2.3\)

Answer:

\(y=-0.1x + 2.3\)