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

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Definitions:

Type I Error

The incorrect rejection of a true null hypothesis, also known as a "false positive" in statistical testing.

Measurement Error

Refers to the difference between the true value of a measurement and the recorded value due to various factors such as instrument error or human error.

T-tests

Statistical tests used to compare the means of two groups and determine if they are significantly different from each other.

Statistically Different

Being statistically different means that observed differences between groups or variables in a study are unlikely to have occurred by chance, according to statistical analysis. This distinction supports the idea that the observed effects reflect true differences in the populations being studied.