Your Textbook Defines the Correlation Coefficient as Follows Another Textbook Gives an Alternative Formula

question 25

Essay

Your textbook defines the correlation coefficient as follows: $$r = \frac { \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 } \left( X _ { i } - \bar { X } \right) ^ { 2 } } { \sqrt { \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 } } \sqrt { \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } } }$$ Another textbook gives an alternative formula: $$r = \frac { n \sum _ { i = 1 } ^ { n } Y _ { i } X _ { i } - \left( \sum _ { i = 1 } ^ { n } Y _ { i } \right) \left( \sum _ { i = 1 } ^ { n } X _ { i } \right) } { \sqrt { n \sum _ { i = 1 } ^ { n } Y _ { i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } Y _ { i } \right) ^ { 2 } } \sqrt { n \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } X _ { i } \right) ^ { 2 } } }$$ Prove that the two are the same.
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Definitions:

Thermic Effect

The increase in metabolic rate after eating, due to the energy required for digestion, absorption, and distribution of nutrients.

Food

Nutritious substances that organisms eat or absorb in order to maintain life and growth.

Basal Metabolic Rate

The speed at which the body consumes energy when in a resting state to uphold essential operations like breathing and maintaining body heat.

Vasoconstriction

The narrowing of blood vessels, which increases blood pressure and reduces blood flow to certain areas.