Graphical Analysis In Exercises 11–14, determine whether there is a perfect positive linear correlation, a strong positive linear correlation, a perfect negative linear correlation, a strong negative linear correlation, or no linear correlation between the variables.

Verified step by step guidance

1. Observe the scatterplot provided. The data points appear to be randomly distributed without forming any discernible pattern or trend.
2. Recall that a linear correlation refers to a relationship where data points tend to follow a straight line, either positively (upward slope) or negatively (downward slope).
3. In this scatterplot, there is no visible clustering of points along a line or a clear direction of increase or decrease. This indicates the absence of a linear relationship.
4. Based on the random distribution of points, conclude that there is no linear correlation between the variables.
5. To confirm, you could calculate the correlation coefficient (r). If r is close to 0, it supports the conclusion of no linear correlation.

Key Concepts

Correlation

Correlation measures the strength and direction of a linear relationship between two variables. It is quantified using the correlation coefficient, which ranges from -1 to 1. A value of 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation. Understanding correlation is essential for interpreting scatterplots and determining how closely related two variables are.

Scatterplot

A scatterplot is a graphical representation of two quantitative variables, where each point represents an observation. The position of each point indicates the values of the two variables, allowing for visual assessment of relationships. In the context of correlation, scatterplots help identify patterns, such as linear relationships, clusters, or randomness, which are crucial for determining the type of correlation present.

Linear Correlation Types

Linear correlation can be classified into several types: perfect positive, strong positive, perfect negative, strong negative, and no correlation. Perfect positive correlation means that as one variable increases, the other does so in a perfectly linear manner. Strong correlations indicate a close relationship but with some variability. No correlation suggests that changes in one variable do not predict changes in the other, as seen in the provided scatterplot.