Answer :
Answer:
The function is [tex]y~=~12 ~-~ 0.5 \cdot x[/tex].
The slope is [tex]m=-0.5[/tex].
The y-intercept is [tex]b=12[/tex].
Step-by-step explanation:
Our aim is to calculate the values m (slope) and b (y-intercept) in the equation of a line :
[tex]y=mx+b[/tex]
We have the following data:
[tex]\begin{array}{c|cccc}x&2&4&6&12\\y&11&10&9&6\end{array}[/tex]
To find the line of best fit for the points given, you must:
Step 1: Find [tex]X\cdot Y[/tex]and [tex]X\cdot X[/tex] as it was done in the table below.
Step 2: Find the sum of every column:
[tex]\sum{X} = 24 ~,~ \sum{Y} = 36 ~,~ \sum{X \cdot Y} = 188 ~,~ \sum{X^2} = 200[/tex]
Step 3: Use the following equations to find b and m:
[tex]\begin{aligned} b &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 36 \cdot 200 - 24 \cdot 188}{ 4 \cdot 200 - 24^2} \approx 12 \\ \\m &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 4 \cdot 188 - 24 \cdot 36 }{ 4 \cdot 200 - \left( 24 \right)^2} \approx -0.5\end{aligned}[/tex]
Step 4: Assemble the equation of a line
[tex]\begin{aligned} y~&=~b ~+~ m \cdot x \\y~&=~12 ~-~ 0.5 \cdot x\end{aligned}[/tex]
The graph of the regression line is:
