Question

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. Two different tests are designed to measure employee productivity and dexterity. Several employees are randomly selected and tested with these results.
Productivity 23 25 28 21 21 25 26 30 34 36

Dexterity 49 53 59 42 47 53 55 63 67 75

Answer 1

Answer:

The equation of the regression line is: $y=5.055 + 1.905 \cdot x$

Step-by-step explanation:

We have the following data:

$\begin{array}{c|cccccccccc}Productivity&23&25&28&21&21&25&26&30&34&36\\Dexterity&49&53&59&42&47&53&55&63&67&75\end{array}$

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 $x^2$ and xy.

Step 2: Sum all x, y, $x^2$ and xy, which gives us Σx, Σy, $\sum x^2$ and Σxy.

Step 3: Calculate Slope b:

$b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2}$

where n is the number of points.

Step 4: Calculate Intercept a:

$a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2}$

Step 5: Assemble the equation of a line

$y=a + b \cdot x$

Following the above steps we get:

Step 1: Find $x^2$ and xy as it was done in the table

Step 2: Find the sum of every column:

$\sum{X} = 269 ~,~ \sum{Y} = 563 ~,~ \sum{X \cdot Y} = 15596 ~,~ \sum{X^2} = 7473$

Step 3: Use the following equations to find a and b:

$a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 563 \cdot 7473 - 269 \cdot 15596}{ 10 \cdot 7473 - 269^2} \approx 5.055$

$b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 10 \cdot 15596 - 269 \cdot 563 }{ 10 \cdot 7473 - \left( 269 \right)^2} \approx 1.905\end{aligned}$

Step 4: Assemble the equation of a line

$y=5.055 + 1.905 \cdot x$

The graph of the regression line is: