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

Question 18

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In a simple regression with an intercept and a single explanatory variable, the variation in Y (TSS=∑i=1n(Yi−Yˉ)2)\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 (ESS=∑i=1n(Y^i−Yˉ)2)\left( E S S = \sum _ { i = 1 } ^ { n } \left( \hat { Y } _ { i } - \bar { Y } \right) ^ { 2 } \right) and the sum of squared residuals (SSR⁡=∑i=1nu^i2=∑i=1n(Yi−Y^)2)\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 (=Yˉ)( = \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

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