Show First That the Regression $R ^ { 2 }$ Is the Square of the Sample Correlation Coefficient

question 6

Essay

Show first that the regression $R ^ { 2 }$ is the square of the sample correlation coefficient. Next, show that the slope of a simple regression of Y on X is only identical to the inverse of the regression slope of X on Y if the regression $R ^ { 2 }$ equals one.

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Definitions:

Rapid Alternating Movements

A neurological test that assesses the ability to perform fast, repeating movements to detect coordination problems.

Astereognosis

The inability to recognize objects by touch alone, often due to damage to the brain's sensory cortex.

Upper Motor Neuron

nerve cells that originate in the brain's cerebral cortex and project to the spinal cord, involved in the initiation of voluntary movements.

Hyperreflexia

An exaggerated response of the muscles following a stimulus, indicative of disturbances in the central nervous system.