solution:
\sum ei =0
that is
\sum (yi-yiˆ)=0
means,
\sumyi = \sum yiˆ
yi = \alpha +\beta xi+ei
yiˆ=\alphaˆ+\betaˆxi
so,
\sumei=0
hence proved
Answer:
Properties of Fitted Regression Line
Step-by-step explanation:
We know that,
[tex]\sum{e_{i} } = 0[/tex]
In turn we understand that
[tex] e_{i}= Y_{i}-Y'_{i} [/tex]
The third property of Fitted Regression Line tells us that: The sum of the observed values [tex]Y_{i}[/tex] equals the sum of the fitted values [tex]Y'_{i}[/tex], so:
[tex]\sum{Y_{i}} = \sum{Y'_{i}}[/tex] (1)
We further understand that the values given for [tex]Y_{i}[/tex], is equivalent to:
[tex]Y_{i}= \beta_{0} + \beta_{1}X_{i}+\epsilon_{i}[/tex] (2)
On the other hand for the definition of the value for the regression function of [tex]Y'_{i}[/tex] is,
[tex]Y'_{i}= \beta_{0}+\beta_{1}X_{i}[/tex] (3)
By replacing (3) and (2) in (1), we get that
[tex]\sum{ (\beta_{0} + \beta_{1}X_{i}+\epsilon_{i})} = \sum{(\beta_{0}+\beta_{1}X_{i})}[/tex]
Since the sum is distributive
[tex]\sum \beta_{0} + \sum \beta_{1}X_{i}+\sum \epsilon_{i} = \sum\beta_{0}+ \sum \beta_{1}X_{i}[/tex]
Equal values on opposite sides of an equation are canceled, we get that
[tex]\sum \epsilon_{i} = 0[/tex]
