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Craig Osborne
08/31/2021 · Primary School
Expert-Verified Answer
Correct Answer is A. SSE can never be larger than SST
To solve this question, we first need to understand the terms involved:
1. SST (Total Sum of Squares): This represents the total variation in the data.
2. SSE (Sum of Squared Errors): This represents the variation that is not explained by the model.
Given these definitions, we can examine each option:
a. SSE can never be larger than SST: This statement is true. SST represents the total variation, and SSE is a portion of this variation (specifically, the unexplained variation). Since SSE is a part of SST, it can never be larger than SST.
b. SSE can never be smaller than SST: This statement is false. As explained, SST is the total variation and includes both the explained and unexplained variation. Therefore, SSE can be smaller than SST.
c. SSE can never be equal to 1: This statement is false. There is no inherent reason why SSE cannot be equal to 1. It depends on the data and the model. If the unexplained variation (error) in the data sums to 1, then SSE can indeed be 1.
d. SSE can never be equal to zero: This statement is false. SSE can be zero if the model perfectly fits the data, meaning there is no unexplained variation (error). In such cases, all the variation is explained by the model, and SSE is zero.
Additional Knowledge:
1. Explained Sum of Squares (SSR): This represents the variation explained by the model. The relationship between these sums is given by:
SST = SSR + SSE
SST is the total variation, and it is split into the variation explained by the model (SSR) and the unexplained variation (SSE).
2. Coefficient of Determination (R²): This is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. It is calculated as:
\[R² = \frac { SSR} { SST} = 1 - \frac { SSE} { SST} \]
R² ranges from 0 to 1. An R² of 1 indicates that the regression predictions perfectly fit the data, meaning SSE is zero. An R² of 0 indicates that the model does not explain any of the variability of the response data around its mean.
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