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.

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Step 1: Observe the scatterplot provided. The data points are closely clustered along a downward-sloping line, indicating a relationship between the variables.

Step 2: Understand the concept of correlation. A negative linear correlation means that as one variable increases, the other variable decreases. The strength of the correlation depends on how closely the points follow a straight line.

Step 3: Analyze the pattern of the data points. In this scatterplot, the points are tightly clustered along a downward slope, suggesting a strong negative linear correlation.

Step 4: Compare the observed pattern to definitions of correlation types. A perfect negative linear correlation would have all points exactly on a straight line, while a strong negative linear correlation allows for slight deviations from the line.

Step 5: Conclude that the scatterplot demonstrates a strong negative linear correlation between the variables, as the points are closely aligned but not perfectly on a straight line.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Correlation

Correlation measures the strength and direction of a linear relationship between two variables. It is quantified by 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 one variable may predict another.

Scatterplot

A scatterplot is a graphical representation of two quantitative variables, where each point represents an observation. The position of the points indicates the relationship between the variables. In the context of correlation, scatterplots help visualize whether the relationship is positive, negative, or nonexistent. The pattern of the points can reveal the strength of the correlation, with tighter clusters indicating stronger relationships.

Linear Relationship

A linear relationship between two variables means that as one variable changes, the other variable changes in a consistent manner, either increasing or decreasing. This relationship can be represented by a straight line in a scatterplot. Identifying whether a relationship is linear is crucial for applying linear regression techniques and for making predictions based on the data.

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