The Correlation Coefficient of y 5 - 3x and Its Significance

Understanding the correlation coefficient between two variables, especially in the context of a linear relationship, is crucial for analyzing how one variable changes with another. In this article, we will focus on the equation y 5 - 3x and the steps to determine the correlation coefficient.

Understanding the Linear Relationship

The equation y 5 - 3x is a linear equation, which can be expressed in the standard form:

y mx c

Here, m is the slope and c is the y-intercept. For the given equation:

m -3 (slope) c 5 (y-intercept)

Calculating the Correlation Coefficient

The correlation coefficient, denoted as r, measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1:

r 1 indicates a perfect positive linear relationship. r -1 indicates a perfect negative linear relationship. r 0 indicates no linear relationship.

For the equation y 5 - 3x, the slope m is negative (-3). Therefore, the correlation coefficient r is:

r -1

This indicates a perfect negative linear correlation between x and y.

Mathematical Framework for Calculating Correlation Coefficient

The correlation coefficient can also be calculated using the formula:

r frac{text{Cov}(x, y)}{sqrt{text{Var}(x)cdottext{Var}(y)}}

Where:

Cov(x, y) is the covariance between x and y. Var(x) and Var(y) are the variances of x and y, respectively.

For a linear relationship y mx c, we have:

Cov(x, y) mcdottext{Var}(x)

Var(y) m^2cdottext{Var}(x)

Substituting these into the correlation coefficient formula, we get:

r frac{mcdottext{Var}(x)}{sqrt{(m^2cdottext{Var}(x))cdottext{Var}(x)}} frac{mcdottext{Var}(x)}{sqrt{m^2cdot(text{Var}(x)^2)}} frac{mcdottext{Var}(x)}{|m|cdottext{Var}(x)} frac{m}{|m|}

Since the slope m is -3, which is negative:

r -1

Significance and Interpretation

The correlation coefficient of -1 indicates a perfect negative linear relationship between the variables x and y. This means that as x increases, y decreases proportionally, and vice versa. The relationship is fully deterministic in this case, as the slope is fixed at -3.

Understanding the correlation coefficient is essential in various fields such as economics, finance, and data science. It helps in making predictions and understanding the dynamics between variables.