When analyzing a data set we frequently graph one metric as a function of another. If the slope of the Correlation line is -2.5 we would say the two metrics are ___________ correlated?

In statistical analysis, the slope of the regression line (or correlation line) quantifies the relationship between two variables. The sign of the slope indicates the direction of this relationship. A negative slope, such as -2.5, signifies a negative or inverse correlation. This means that for every one-unit increase in the independent variable (plotted on the x-axis), the dependent variable (plotted on the y-axis) is expected to decrease by 2.5 units. Essentially, as one metric goes up, the other tends to go down.

A. Positively: A positive correlation would be represented by a positive slope, indicating that both metrics tend to increase or decrease together.
B. Not: "Not correlated" implies no linear relationship between the metrics, which would be represented by a slope of or very close to zero.
D. None: This option is incorrect because the slope value of -2.5 provides a clear and standard interpretation of the relationship between the two metrics.

1. MIT OpenCourseWare: In the course 15.075J Statistical Thinking and Data Analysis, the lecture notes on Simple Linear Regression state that the sign of the slope coefficient determines the direction of the relationship. A negative coefficient (slope) indicates a negative relationship between the predictor and response variables.

Source: MIT OpenCourseWare, 15.075J Statistical Thinking and Data Analysis, Fall 2011. Lecture 10: Simple Linear Regression, Section 10.2 "Interpreting the Coefficients".

2. Pennsylvania State University, Eberly College of Science: The online course materials for STAT 501 - Regression Methods explain that the sign of the slope parameter indicates the direction of the association. A negative slope signifies that as the predictor variable increases, the response variable tends to decrease.

Source: PennState Eberly College of Science, STAT 501 Course Notes, Lesson 2: Simple Linear Regression, Section 2.1 "The Model".

3. Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (7th ed.). Wiley. This foundational textbook, often used in university engineering and quality management programs, explains that in the simple linear regression model Y = β₀ + β₁x + ε, the slope β₁ represents the expected change in Y for a unit change in x. "If the slope is negative, the line moves from upper left to lower right, and the relationship is such that increasing x results in decreasing y."

Source: Chapter 11, "Simple Linear Regression and Correlation," Section 11-2 "Simple Linear Regression Model".