Prove That the Regression $R ^ { 2 }$ Is Identical to the Square of the Correlation Coefficient Between

question 28

Essay

Prove that the regression $R ^ { 2 }$ is identical to the square of the correlation coefficient between two variables Y and X . Regression functions are written in a form that suggests causation running from X to Y . Given your proof, does a high regression $R ^ { 2 }$ present supportive evidence of a causal relationship? Can you think of some regression examples where the direction of causality is not clear? Is without a doubt?

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Definitions:

Newman Projection

A method of representing the spatial arrangement of atoms in a molecule, visualizing the conformation of a chemical bond from front to back.

C2-C3 Bond

The chemical bond between the second and third carbon atoms in an organic molecule, critical in determining the molecule's shape and reactivity.

2,3-Dimethylbutane

A branched alkane with the formula C6H14, notable for its use as a standard in octane rating tests and its presence in various fuels.

Ring Strain

The destabilization that occurs in a cyclic molecule due to the geometric or electronic constraints of the ring system.