Question

In order to determine a relationship between an employee’s age and absenteeism following
random data was selected and presented as following.

Age in years(x) 42 27 36 25 22 39
No. of days absent / years 5 10 8 12 13 7

Required

a) Compute the line of regression
b) Calculate coefficient of correlation (r)

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} i & x & y & xy & x^2 & y^2 \\ \hline 1 & 42 & 5 & 210 & 1764 & 25 \\ \hdashline 2 & 27 & 10 & 270 & 729 & 100 \\ \hdashline 3 & 36 & 8 & 288 & 1296 & 64 \\ \hdashline 4 & 25 & 12 & 300 & 625 & 144 \\ \hdashline 5 & 22 & 13 & 286 & 484 & 169 \\ \hdashline 6 & 39 & 7 & 273 & 1521 & 49 \\ \hdashline & \sum x_i=191 & \sum y_i=55 & \sum x_iy_i=1627 & \sum x_i^2= 6419& \sum y_i^2=551 \end{array}$$

a) Compute the line of regression 

Calculating the mean $(\bar{x}, \bar{y})$

$$\overline{x}={\sum x_i \over n}={191 \over 6}, \overline{y}={\sum x_i \over n}={55 \over 6}$$

The equation of a simple linear regression line (the line of best fit) is y = mx + b,

$$m=slope={n\sum x_iy_i-\sum x_i \sum y_i \over n\sum x_i^2-(\sum x_i)^2}$$

$$m={6\cdot 1627-191\cdot 55 \over 6\cdot 6419-(191)^2}\approx -0.365470$$

$$b=\overline{y}-m\overline{x}$$

$$b\approx {55 \over 6}-(-0.365470)\cdot {191 \over 6}\approx 20.800787$$

The line of regression 

$$y= -0.365470x+20.800787$$

b) Calculate coefficient of correlation (r)

$$r={n\sum x_iy_i-\sum x_i \sum y_i \over \sqrt{n\sum x_i^2-(\sum x_i)^2}\cdot \sqrt{n\sum y_i^2-(\sum y_i)^2}}$$

$$r={6\cdot 1627-191\cdot 55 \over \sqrt{6\cdot 6419-(191)^2} \sqrt{6\cdot 551-(55)^2}}\approx -0.983030$$

$$r=-0.983030$$