Problem 6 - An Introduction to Psychological Statistics

Question

What is the direction and magnitude of the following correlation coefficients a. -0.81 b. 0.40 c. 0.15 d. -0.08 e. 0.29

Step-by-Step Solution

Answer

-0.81: strong negative, 0.40: moderate positive, 0.15, 0.29: weak positive, -0.08: weak negative.

Step 1: Understanding Correlation Coefficients

Correlation coefficients measure the strength and direction of a linear relationship between two variables. They range from -1 to 1. A positive value indicates a positive correlation, while a negative value indicates a negative correlation. A value of 0 indicates no correlation.

Step 2: Identifying the Direction

Look at the sign of each coefficient: - Negative values (such as -0.81 and -0.08) mean a negative correlation, where as one variable increases, the other tends to decrease. - Positive values (such as 0.40, 0.15, and 0.29) mean a positive correlation, where as one variable increases, the other also tends to increase.

Step 3: Assessing the Magnitude

The magnitude describes the strength of the correlation: - Strong correlation: Coefficients closer to -1 or 1, like -0.81. - Moderate correlation: Typically between 0.4 and 0.6 or -0.4 and -0.6, such as 0.40. - Weak correlation: Coefficients closer to 0, such as 0.15, 0.29, and -0.08.

Key Concepts

linear relationship

When we talk about a linear relationship, we're looking at how two variables change together. Imagine two lines on a graph. If one line moves as the other line moves, either up or down, they're showing a linear relationship. This connection can be described using correlation coefficients. These coefficients help determine if the relationship is:

• Positive,
• Negative,
• Or non-existent (zero).

The coefficent ranges between -1 and 1, showing how closely matched these movements are. A linear relationship is all about the pattern these numbers create when plotted on a chart. If they form a straight line, the relationship is perfectly linear.

positive correlation

A positive correlation indicates that as one variable increases, the other also increases. It's like saying the more you study, the higher your grades might be. This kind of relationship is identified when the correlation coefficient is a positive number.

For instance:

• If the coefficient is close to +1, it's a strong positive correlation. You can expect a nearly direct line upwards on a graph.
• If it's closer to 0 but still positive, like 0.15 or 0.29, this indicates a weak positive correlation. The line won’t be as straight or steep.

The key takeaway is that both variables try to move in the same direction.

negative correlation

Negative correlation happens when one variable increases and the other decreases. Imagine a seesaw where one side goes up while the other goes down. This is the essence of a negative correlation.

• The correlation coefficient will be a negative number.
• A coefficient close to -1 shows a strong negative correlation. For example, -0.81 suggests a strong downward relationship.
• Values like -0.08, closer to zero, imply a weak negative correlation, with the change not being as pronounced.

Negative correlations help us identify inversely related variables.

magnitude of correlation

The magnitude of a correlation coefficient tells us how strong the relationship is between two variables. Whether the number is positive or negative, what matters is how close it is to 1 or -1.

• A coefficient like 0.40 is considered a moderate correlation.
• Numbers hovering close to zero, such as 0.15 or -0.08, indicate weak correlations.
• Closer to 1 or -1, such as -0.81, points to strong correlations.

In sum, the magnitude is all about this closeness to the ends of the range, signifying how powerful or mild the connection between the two variables is.