Problem 14 - Algebra 2 and Trigonometry

Question

In \(7-14,\) for each of the given correlation coefficients, describe the linear correlation as strong positive, moderate positive, none, moderate negative, or strong negative. \(r=-0.95\)

Step-by-Step Solution

Answer

The correlation is strong negative.

Step 1: Identify the Range for Correlations

Correlation coefficients range from -1 to 1. Positive values indicate a positive correlation, and negative values indicate a negative correlation. The closer the coefficient is to either -1 or 1, the stronger the correlation.

Step 2: Assess the Strength of the Correlation

Refer to standard thresholds to determine the strength of the correlation. Typically, \(|r| > 0.8\) indicates a strong correlation, \(0.6 < |r| \leq 0.8\) indicates a moderate correlation, and \(|r| \leq 0.5\) indicates a weak correlation.

Step 3: Determine the Type and Strength

Given that \(r = -0.95\), which falls into the range for a strong correlation and has a negative sign, we can classify the correlation as strong negative.

Key Concepts

Linear Correlation

When talking about linear correlation, think about how two variables relate to each other in a straight-line manner. Linear correlation is a term used in statistics to describe the strength and direction of a relationship between two variables.
For example, if you're looking at the height and weight of a group of people, you're trying to see if taller people tend to weigh more, or less.​

To measure linear correlations, we use something called the correlation coefficient, usually represented as \(r\). This coefficient tells us if the relationship is positive or negative:

• A positive \(r\) means as one variable increases, the other one tends to increase too.
• A negative \(r\) indicates that as one variable increases, the other tends to decrease.

A perfect positive linear correlation has an \(r\) value of 1, while a perfect negative linear correlation has an \(r\) of -1.

Strong Negative Correlation

A strong negative correlation occurs when an increase in one variable strongly corresponds to a decrease in another. It means the two variables have a very close inverse relationship.​

Imagine you're observing how temperature and the sales of hot cocoa fluctuate. You might find that as the temperature drops, hot cocoa sales rise significantly.

In statistical terms, a strong negative correlation typically corresponds to correlation coefficient values close to -1. In our exercise, the given correlation coefficient \(r = -0.95\) means there's almost a perfect inverse relationship.

• This strong negative value indicates high reliability in predicting one variable based on the other.
• There is a clear and predictable pattern between the two.

Remember, a strong negative correlation doesn’t mean one variable causes the other to decrease, just that they consistently move in opposite directions.

Correlation Strength

Correlation strength refers to how closely two variables move in relation to one another. It's all about how tightly the data fits a straight-line pattern.

There are some general guidelines to help you determine the strength of a correlation:
- Strong correlation: \(|r| > 0.8\)
- Moderate correlation: \(0.6 < |r| \leq 0.8\)
- Weak correlation: \(|r| \leq 0.5\)

These rules apply whether the correlation is positive or negative.
In our given solution, since \(|r| = 0.95\) (which is greater than 0.8), it easily qualifies as a strong correlation. The direction of the correlation (negative or positive) doesn’t affect the strength - only the magnitude of \(r\) does. So, whether two variables move together or apart robustly defines if they're strongly correlated.

Correlation Coefficient Interpretation

Understanding how to interpret the correlation coefficient is crucial for insights in data analysis. ​

The correlation coefficient, represented by \(r\), sheds light on not only the strength but also the direction of the relationship between two variables.

Here are the things to consider:

• Value Range: \(r\) ranges from -1 to 1. ​
• Direction: Positive \(r\) indicates a positive relationship; negative \(r\) suggests a negative relationship.
  ​​
• Strength: The closer \(r\) is to 1 or -1, the stronger the correlation.​​

For example, in our step-by-step solution, the \(r = -0.95\) indicates a strong negative correlation, reflecting a very reliable, predictable relationship where one variable tends to decrease as the other increases. Understanding these aspects allows you to make informed decisions based on data relationships.