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:

Carl Rogers

An influential American psychologist who founded the humanistic approach to psychology, emphasizing empathy, personal growth, and the inherent goodness of individuals.

Total Acceptance

Unconditional positive regard and acceptance of an individual, regardless of their actions or beliefs.

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Scholars and psychologists who focus on individual potential and stress the importance of growth and self-actualization.

Inconsistency

The lack of stability or uniformity in actions, decisions, or conditions.