Question

question help: d post to forum based on the data shown below, calculate the regression line (each value to at least two decimal places) x: 4 5 6 7 8 9 y: 22.6 20.48 19.36 16.84 17.72 15.8

Explanation:

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

Let \(x = [4,5,6,7,8,9]\), \(y=[22.6,20.48,19.36,16.84,17.72,15.8]\)
\(\bar{x}=\frac{4 + 5+6+7+8+9}{6}=\frac{39}{6}=6.5\)
\(\bar{y}=\frac{22.6+20.48+19.36+16.84+17.72+15.8}{6}=\frac{112.8}{6}=18.8\)

Step2: Calculate the slope \(m\)

\[

$$\begin{align*} m&=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\\ \sum_{i = 1}^{6}(x_i - 6.5)(y_i-18.8)&=(4 - 6.5)(22.6-18.8)+(5 - 6.5)(20.48 - 18.8)+(6 - 6.5)(19.36-18.8)+(7 - 6.5)(16.84 - 18.8)+(8 - 6.5)(17.72-18.8)+(9 - 6.5)(15.8 - 18.8)\\ &=(- 2.5)\times3.8+(-1.5)\times1.68+(-0.5)\times0.56 + 0.5\times(-1.96)+1.5\times(-1.08)+2.5\times(-3)\\ &=-9.5-2.52 - 0.28-0.98 - 1.62-7.5\\ &=-22.4 \end{align*}$$

\]
\[

$$\begin{align*} \sum_{i = 1}^{6}(x_i - 6.5)^2&=(4 - 6.5)^2+(5 - 6.5)^2+(6 - 6.5)^2+(7 - 6.5)^2+(8 - 6.5)^2+(9 - 6.5)^2\\ &=(-2.5)^2+(-1.5)^2+(-0.5)^2+0.5^2+1.5^2+2.5^2\\ &=6.25 + 2.25+0.25+0.25+2.25+6.25\\ &=17.5 \end{align*}$$

\]
\(m=\frac{-22.4}{17.5}=-1.28\)

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

\(b=\bar{y}-m\bar{x}=18.8-(-1.28)\times6.5=18.8 + 8.32=27.12\)

Answer:

\(y=-1.28x + 27.12\)