Question

Calculate the least square regression line where annual savings is the dependent variable and annual income is the independent variable.?

An economist wants to estimate the relationship between a family’s is annual income and the amount that the family saves. The family data for nine families are obtained
Annual income (thousands of Rs.) 12 13 14 15 16 17 18 19 20
Annual Saving (thousands of Rs.) 0.0 0.1 0.2 0.2 0.5 0.5 0.6 0.7 0.8
Calculate the least square regression line where annual savings is the dependent variable and annual income is the independent variable.

Answer 1

`Y = -1.226666 + 0.1016666*X`

Explanation:

`bar X = (12 + 13 + 14 + ... + 20)/9 = 9*(12+20)/(2*9) = 16`
`bar Y = (0 + 0.1 + 0.2 + 0.2 + 0.5 + 0.5 + 0.6 + 0.7 + 0.8)/9 = 0.4`

`hat beta_2 = (sum_{i=1}^{i=9} x_i*y_i) / (sum_{i=1}^{i=9} x_i^2)`

`"with "x_i = X_i - bar X " , and "y_i = Y_i - bar Y`

`=> hat beta_2 =`
`(4*0.4+3*0.3+2*0.2+0.2+0.1+2*0.2+3*0.3+4*0.4)/((4^2+3^2+2^2+1^2)*2)`
`= (1.6+0.9+0.4+0.2+0.1+0.4+0.9+1.6)/60`
`= 6.1/60`
`= 0.10166666`

`=> hat beta_1 = bar Y - hat beta_2 * bar X`
`= 0.4 - (6.1/60)*16`
`= -1.226666`

`"So the regression line is"`

`Y = -1.226666 + 0.1016666*X`