In a Simple Regression with an Intercept and a Single $\left( T S S = \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 } \right)$

Essay

In a simple regression with an intercept and a single explanatory variable, the variation in Y $\left( T S S = \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 } \right)$ can be decomposed into the explained sums of squares $\left( E S S = \sum _ { i = 1 } ^ { n } \left( \hat { Y } _ { i } - \bar { Y } \right) ^ { 2 } \right)$ and the sum of squared residuals $\left( \operatorname { SSR } = \sum _ { i = 1 } ^ { n } \hat { u } _ { i } ^ { 2 } = \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \hat { Y } \right) ^ { 2 } \right)$ (see, for example, equation (4.35)in the textbook).
Consider any regression line, positively or negatively sloped in {X,Y} space. Draw a horizontal line where, hypothetically, you consider the sample mean of Y $( = \bar { Y } )$ to be. Next add a single actual observation of Y.
In this graph, indicate where you find the following distances: the
(i)residual
(ii)actual minus the mean of Y
(iii)fitted value minus the mean of Y