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Chapter 4: Problem 5

Explain how to interpret the absolute value of a correlation coefficient.

Short Answer

Expert verified
The absolute value of a correlation coefficient measures the relationship's strength between two variables, with values closer to 1 indicating a stronger relationship.

Step by step solution

01

Understand the Correlation Coefficient

The correlation coefficient, typically denoted as \( r \), measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1.
02

Define Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering the direction. For the correlation coefficient \( r \), the absolute value is represented as \( |r| \).
03

Interpretation of \( |r| \)

The absolute value \( |r| \) reveals the strength of the relationship between the two variables. A value of \( |r| = 1 \) implies a perfect linear relationship, whereas \( |r| = 0 \) indicates no linear relationship. The closer \( |r| \) is to 1, the stronger the linear relationship.
04

Assessing Strength Using \( |r| \)

Interpret \( |r| \) within common ranges to quickly ascertain the relationship's strength: **0** suggests no correlation, **0.1 to 0.3** suggests a weak correlation, **0.3 to 0.5** suggests a moderate correlation, **0.5 to 0.7** suggests a strong correlation, and **0.7 to 1** indicates a very strong correlation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Relationship
A linear relationship between two variables occurs when their values follow a straight line when plotted on a graph. In simpler terms, if you double the value of one variable, the other one should ideally double too if the relationship is perfectly linear. This is characterized by a correlation coefficient, denoted as \( r \).

Understanding this concept is key when working with correlation coefficients because \( r \) captures how these two variables move together in a linear fashion.
  • If \( r \) is positive, both variables increase together, meaning a positive linear relationship.
  • If \( r \) is negative, as one variable increases, the other decreases, indicating a negative linear relationship.
The strength and direction of this linear relationship can help in making predictions and analyzing the relationships within data sets.
Absolute Value
Absolute value in mathematics is simply the non-negative value of a number without regards to its sign. For example, the absolute value of both -5 and 5 is 5.

When it comes to a correlation coefficient, the absolute value is not only about the number itself but its magnitude and what that magnitude tells us. For a correlation coefficient \( r \), its absolute value \( |r| \) shows us how far \( r \) is from 0 in either direction on the number line.
  • If \( |r| \) is close to 1, a strong relationship exists between the two variables, regardless of whether it's positive or negative.
  • If \( |r| \) is close to 0, this suggests a weaker or no linear relationship between the variables.
Thus, the absolute value does not care whether the relationship is increasing (positive \( r \)) or decreasing (negative \( r \)), just its strength.
Statistical Interpretation
Interpreting a correlation coefficient involves understanding what the number really means in terms of the data it describes. The absolute value \( |r| \) gives us a quantifiable sense of this strength, but how do we interpret these numbers practically?

Here's a quick guide to understanding \( |r| \):
  • 0 suggests no linear relationship.
  • 0.1 to 0.3 indicates a weak linear relationship; this could mean the two variables do not significantly affect each other.
  • 0.3 to 0.5 implies a moderate relationship, showing some predictive ability.
  • 0.5 to 0.7 demonstrates a strong relationship, indicating a more dependable predictive relation.
  • 0.7 to 1 represents a very strong relationship; the variables are closely likened linearly.
It is important to note that even if \( |r| \) suggests a strong relationship, correlation does not equate to causation. Statistical tools such as these describe associations rather than cause-and-effect dynamics.

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Most popular questions from this chapter

For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted. If the function \(C\) is graphed, find and interpret the slope of the function.

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither. $$ \begin{array}{l} 4 x-7 y=10 \\ 7 x+4 y=1 \end{array} $$

For the following exercises, consider this scenario: A town's population has been increased at a constant rate. In 2010 the population was \(46,020 .\) By 2012 the population had increased to \(52,070 .\) Assume this trend continues. Predict the population in \(2016.\)

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Graph \(f(x)=0.5 x+10 .\) Pick a set of five ordered pairs using inputs \(x=-2,1,5,6,9\) and use linear regression to verify that the function is a good fit for the data.
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