Question

Given the following data
X 1 3 4 6 8 9 11 14
y 1 2 4 4 5 7 8 9
Determine the least square line using x as independent variable.
Find the estimated value for given values of x and show that
(a) ∑▒Y=∑▒Y ̂ (b) ∑▒〖e_i=o〗

Answer 1

Solution :

The independent variable is X, and the dependent variable is Y. In order to compute the regression coefficients, the following table needs to be used:

$\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}=\frac{56}{8}=7$

$\overline{Y}=\frac{1}{n}\sum_{i=1}^{n}Y_{i}=\frac{40}{8}=5$

$SS_{XX}=\sum_{i=1}^{n}X_{i}^2 -\frac{1}{n}(\sum_{i=1}^{n}X_{i})^2= 524- \frac{56^2}{8}=132$

$SS_{YY}=\sum_{i=1}^{n}Y_{i}^2 -\frac{1}{n}(\sum_{i=1}^{n}Y_{i})^2= 256- \frac{40^2}{8}=56$

$SS_{XY}=\sum_{i=1}^{n}X_{i}Y_{i} -\frac{1}{n}\sum_{i=1}^{n}X_{i}\sum_{i=1}^{n}Y_{i}= 364- \frac{56*40}{8}=84$

Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n are obtained as follows:

$m=\frac{SS_{XY}}{SS_{XX}}=\frac{84}{132}=0.6364$

$n=\overline{Y}-\overline{X}*m = 5-7*0.6364=0.5455$

Therefore, we find that the regression equation is:

Y=0.5455+0.6364X

b)$\frac{1}{n}\sum_{i=1}^{n}\widehat{e_{i}}=0$ because the intercept of the model absorbs the mean of the residuals.

a)By definition $Y_{i}=\widehat{Y_{i}}+\widehat{\epsilon _{i}}$

Since

$\sum_{i=1}^{n}\widehat{e_{i}}=0$

$\frac{1}{n}\sum_{i=1}^{n}Y_{i}= \frac{1}{n}\sum_{i=1}^{n}\widehat{Y}_{i}+\frac{1}{n}\sum_{i=1}^{n}\widehat{e_{i}}$

$\frac{1}{n}\sum_{i=1}^{n}Y_{i}= \frac{1}{n}\sum_{i=1}^{n}\widehat{Y}_{i}+0$

$=$ $\frac{1}{n}\sum_{i=1}^{n}\widehat{Y}_{i}$

Hence the proof.