According to (1.17), lei = 0 when regression model (1.1) is fitted to a set of n cases by the method of least squares. is it als

Question

Phoenix

2 months ago

11

According to (1.17), lei = 0 when regression model (1.1) is fitted to a set of n cases by the method of least squares. is it als

o true that l e:i = o? comment.

Mathematics

2 answers:

Inessa [12.5K] · Answer 1 · 2026-03-21T15:04:29+03:00

Inessa [12.5K]2 months ago

4 0

Solution:

\sum ei =0

which implies

\sum (yi-yiˆ)=0

this means,

\sumyi = \sum yiˆ

yi = \alpha +\beta xi+ei

yiˆ=\alphaˆ+\betaˆxi

therefore,

\sumei=0

consequently demonstrated

Inessa [12.5K] · Answer 2 · 2026-03-21T15:03:46+03:00

Answer:

Characteristics of the Fitted Regression Line

Step-by-step elucidation:

It is established that

$\sum{e_{i} } = 0$

From this, we learn that

$e_{i}= Y_{i}-Y'_{i}$

The third characteristic of the Fitted Regression Line indicates: The total of the observed values $Y_{i}$ equals the total of the fitted values $Y'_{i}$ , hence:

$\sum{Y_{i}} = \sum{Y'_{i}}$ (1)

We further comprehend that the values given for $Y_{i}$ correspond to:

$Y_{i}= \beta_{0} + \beta_{1}X_{i}+\epsilon_{i}$ (2)

Conversely, the definition relating to the value of the regression function of $Y'_{i}$ is

$Y'_{i}= \beta_{0}+\beta_{1}X_{i}$ (3)

By substituting (3) and (2) in (1), the outcome can be derived

$\sum{ (\beta_{0} + \beta_{1}X_{i}+\epsilon_{i})} = \sum{(\beta_{0}+\beta_{1}X_{i})}$

Since the sum exhibits distributive properties

$\sum \beta_{0} + \sum \beta_{1}X_{i}+\sum \epsilon_{i} = \sum\beta_{0}+ \sum \beta_{1}X_{i}$

Equal terms on opposite sides of an equation cancel each other, leading us to conclude that

$\sum \epsilon_{i} = 0$