Question

A manager in the human resources department randomly selected five employees files and recorded the following data on X = number of weeks of paid vacation (annually) and Y = number of sick days claimed by the employee in the previous year. An analysis of the linear relationship between X and Y is desired.

a) Find the value of b0 and b1
b) Find the regression equation
c) Interpret the meaning of the Y intercept b0
d) Predict the number of sick days when X = 7

X	1	2	3	4	5
Y	3	1	1	0	0

CALCULATE SST

CALCULATE SSR

CALCULATE SSE

Answer 1

Regression equation is as follows -

Y = b₀ + b₁X

Where,

b_{0 =} - b₀

Observation table -  

X	Y	(X-X_bar)	(Y-Y_bar)	(X-X_bar)(Y-Y_bar)	(X-X_bar)^2	(Y-Y_bar)^2
1	1	-2	0	0	4	0
2	3	-1	2	-2	1	4
3	1	0	0	0	0	0
4	0	1	-1	-1	1	1
5	0	2	-1	-2	4	1
15	5	-	-	-5	10	6

a) -

Values of b₀ & b₁ -

,  ,  

,  

  

b_{0 =} - b₀ = 1 - (-0.5)(3) = 1 + 1.5 = 2.5

b) -

Regression equation is -

Y = 2.5 - 0.5X

c) -

Prediction from Y-intercept -

If the number of weeks of paid vacation is 0, then number of sick days by the employee is nearly(approximately) equal to 3.

Prediction from slope -

The number of sick days by the employee decrease by 0.5 on average per week paid vacation.

d) -

When X =7 -

Y = 2.5 - 0.5X = 2.5 - (0.5)(7) = 2.5 - 3.5 = -1

The number of sick days is -1 which is impossible.

Observation table-

X	Y	y_cap	(Y - )2	(Y - )2	( - )^2
1	1	2	1	0	1
2	3	1.5	2.25	4	0.25
3	1	1	0	0	0
4	0	0.5	0.25	1	0.25
5	0	0	0	1	1
Total	-	-	3.5	6	2.5

Total sum of square -

Sum of square due to regression-

Sum of square due to error -