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

question 18

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

---

Definitions:

U.S. Department of Labor

A department of the federal government of the United States responsible for occupational safety, wage standards, unemployment insurance benefits, re-employment services, and some economic statistics.

Essential Skills

Fundamental abilities needed for work, learning, and life, such as literacy, numeracy, and problem-solving in technology-rich environments.

Interdisciplinary Influences

The impact and contribution of concepts, methods, and theories from multiple academic disciplines on a specific field of study or project.

Human Behavior

The range of actions and mannerisms exhibited by humans in conjunction with their environment, influenced by culture, attitudes, emotions, values, ethics, and genetics.