What is the coefficient of determination for two variables that have perfect positive linear correlation or perfect negative linear correlation? Interpret your answer.

Verified step by step guidance

1. Recall that the coefficient of determination, denoted as $R²$, is the square of the correlation coefficient $r$. It measures the proportion of variance in one variable that is predictable from the other variable.
2. Understand that a perfect positive linear correlation means the correlation coefficient $r=\; +1$, and a perfect negative linear correlation means $r=\; -1$.
3. Calculate the coefficient of determination by squaring the correlation coefficient: $R²=\; (r)²$. For both $r\; =\; +1$ and $r\; =\; -1$, this will be $1²\; =\; 1$.
4. Interpret the coefficient of determination value of 1 as indicating that 100% of the variance in one variable is explained by the variance in the other variable, meaning a perfect linear relationship.
5. Conclude that whether the correlation is perfectly positive or perfectly negative, the coefficient of determination is 1, showing a perfect linear association in terms of explained variance.

Key Concepts

Coefficient of Determination (R²)

The coefficient of determination, denoted as R², measures the proportion of variance in the dependent variable that is predictable from the independent variable. It ranges from 0 to 1, where 1 indicates perfect prediction and 0 indicates no predictive power.

Correlation Coefficient (r)

The correlation coefficient, r, quantifies the strength and direction of a linear relationship between two variables. It ranges from -1 (perfect negative linear correlation) to +1 (perfect positive linear correlation), with 0 indicating no linear correlation.

Relationship Between R² and r

The coefficient of determination R² is the square of the correlation coefficient r. For perfect positive (r = +1) or perfect negative (r = -1) linear correlation, R² equals 1, meaning the model explains 100% of the variance in the data regardless of the correlation's direction.