Answer:
The equation of the regression line is: [tex]y=5.055 + 1.905 \cdot x[/tex]
Step-by-step explanation:
We have the following data:
[tex]\begin{array}{c|cccccccccc}Productivity&23&25&28&21&21&25&26&30&34&36\\Dexterity&49&53&59&42&47&53&55&63&67&75\end{array}[/tex]
We can use the Least Squares Regression to find the line of best fit for a set of paired data.
To find the line of best fit for n points:
Step 1: For each (x,y) point calculate [tex]x^2[/tex] and xy.
Step 2: Sum all x, y, [tex]x^2[/tex] and xy, which gives us Σx, Σy, [tex]\sum x^2[/tex] and Σxy.
Step 3: Calculate Slope b:
[tex]b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2}[/tex]
where n is the number of points.
Step 4: Calculate Intercept a:
[tex]a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2}[/tex]
Step 5: Assemble the equation of a line
[tex]y=a + b \cdot x[/tex]
Following the above steps we get:
Step 1: Find [tex]x^2[/tex] and xy as it was done in the table
Step 2: Find the sum of every column:
[tex]\sum{X} = 269 ~,~ \sum{Y} = 563 ~,~ \sum{X \cdot Y} = 15596 ~,~ \sum{X^2} = 7473[/tex]
Step 3: Use the following equations to find a and b:
[tex]a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 563 \cdot 7473 - 269 \cdot 15596}{ 10 \cdot 7473 - 269^2} \approx 5.055[/tex]
[tex]b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 10 \cdot 15596 - 269 \cdot 563 }{ 10 \cdot 7473 - \left( 269 \right)^2} \approx 1.905\end{aligned}[/tex]
Step 4: Assemble the equation of a line
[tex]y=5.055 + 1.905 \cdot x[/tex]
The graph of the regression line is:
