Question

We have consumption (Y) of ten people and their savings(X) data is given below. Pleaseestimate the regression model ( ) for the variables by using normal equations and showing all calculation steps clearly in the Ms World file. (25p). Also interpret the coefficients (10p)

X 10 30 15 20 12 10 18 22 25 26

Y 24 72 36 48 28.8 24 43.2 52.8 60 62.4

Answer 1

Here the given data is

	X	Y
	10	24
	30	72
	15	36
	20	48
	12	28.8
	10	24
	18	43.2
	22	52.8
	25	60
	26	62.4
Sum	188	451.2
Average	18.8	45.12

Obtaining the equation

From the given data obtaining the means

= 188 / 10

= 18.8

= 451.2 /10

= 45.12

Calculating the , and  

	X	Y
	10	24	-8.8	-21.12	185.856	77.44	446.0544
	30	72	11.2	26.88	301.056	125.44	722.5344
	15	36	-3.8	-9.12	34.656	14.44	83.1744
	20	48	1.2	2.88	3.456	1.44	8.2944
	12	28.8	-6.8	-16.32	110.976	46.24	266.3424
	10	24	-8.8	-21.12	185.856	77.44	446.0544
	18	43.2	-0.8	-1.92	1.536	0.64	3.6864
	22	52.8	3.2	7.68	24.576	10.24	58.9824
	25	60	6.2	14.88	92.256	38.44	221.4144
	26	62.4	7.2	17.28	124.416	51.84	298.5984
Sum	188	451.2		0	1064.64	443.6	2555.136

Therefore the = 443.6   , = 2555.136 and   = 1064.64

Obtaining the coefficients

= 2.4

Now obtaining

= 45.12 - 2.4 * 18.8

= 0

The regression equation is

Substituting the values

Y = 0 + 2.4 X

Consumption =  0+ 2.4 *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 , 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 2.4 unit increase in consumption (Y).(and vice -versa)