What would you conclude about a set of quantitative bivariate data with a linear correlation coefficient of -1? How would the scatterplot for such data appear?

Verified step by step guidance

1. Understand that the linear correlation coefficient, denoted as \(r\), measures the strength and direction of a linear relationship between two quantitative variables. It ranges from \(-1\) to \$1$.
2. Recognize that a correlation coefficient of \(r = -1\) indicates a perfect negative linear relationship between the two variables. This means that as one variable increases, the other decreases in a perfectly linear manner.
3. Conclude that the data points lie exactly on a straight line with a negative slope. There is no deviation from this line, so the relationship is deterministic rather than just strongly correlated.
4. Visualize the scatterplot: all points will be aligned on a straight line that slopes downward from left to right, showing a perfect inverse relationship.
5. Summarize that such a scatterplot reflects a perfect negative linear association, where knowing the value of one variable allows you to predict the other variable exactly using a linear equation with a negative slope.

Key Concepts

Linear Correlation Coefficient

The linear correlation coefficient, denoted as r, measures the strength and direction of a linear relationship between two quantitative variables. It ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 means no linear relationship, and 1 indicates a perfect positive linear relationship.

Interpretation of r = -1

A correlation coefficient of -1 means that the two variables have a perfect negative linear relationship. This implies that as one variable increases, the other decreases in a perfectly consistent manner, with all data points lying exactly on a straight line with a negative slope.

Scatterplot Representation

A scatterplot for data with r = -1 will show all points aligned precisely along a straight line sloping downward from left to right. This visual pattern confirms the perfect negative linear relationship, with no deviations or scatter around the line.