Question

We have consumption (Y) of ten people and their savings(X) data is given below. estimate the regression model (?̂? = ? − ??_? ) for the variables by using normal equations and showing all the calculation steps clearly in the.. Ms World file.. Also interpret the coefficients

X Y

13 39

17 51

19 57

22 66

21.6 64.8

22.4 67.2

18 54

20.8 62.4

22 66

24.2 72.6

Answer 1

Here the given data is

	X	Y
	13	39
	17	51
	19	57
	22	66
	21.6	64.8
	22.4	67.2
	18	54
	20.8	62.4
	22	66
	24.2	72.6
Sum	200	600
Average	20	60

Obtaining the equation

From the given data obtaining the means

= 200 / 10

= 20

= 600 / 10

= 60

Calculating the , and  

	X	Y
	13	39	-7	-21	147	49	441
	17	51	-3	-9	27	9	81
	19	57	-1	-3	3	1	9
	22	66	2	6	12	4	36
	21.6	64.8	1.6	4.8	7.68	2.56	23.04
	22.4	67.2	2.4	7.2	17.28	5.76	51.84
	18	54	-2	-6	12	4	36
	20.8	62.4	0.8	2.4	1.92	0.64	5.76
	22	66	2	6	12	4	36
	24.2	72.6	4.2	12.6	52.92	17.64	158.76
Sum	200	600		0	292.8	97.6	878.4

Therefore the = 97.6   , = 878.4 and   = 292.8

Obtaining the coefficients

= 3

Now obtaining 'a'

= 60 - 3* 20

= 0

The regression equation is

Substituting the values

Y = 0 + 3 X

Consumption =  0+ 3 *savings

Interpretation

Since the     is zero ,the regression line passes through origin .i.e the expected value of consumption (Y) is zero if all savings(X) are zero.

Since b = 3   , it is positive , hence there is positive correlation between consumption (Y) and savings(X) . i.e for every additional unit in savings(X) there is 3 unit increase in consumption (Y).(and vice -versa)