Question

Which of the following correlation coefficients would indicate the strongest relationship between two variables? A. .58 B. 19 C. -.97 D. -.05

Short answer

Answer: C. -0.97

Step-by-step solution

Understand correlation coefficients

A correlation coefficient measures the strength and direction of the relationship between two variables. The value of the correlation coefficient ranges from -1 to 1. A value close to 1 indicates a strong positive relationship, a value close to -1 shows a strong negative relationship, while a value near 0 indicates a weak or no relationship between the variables.

Identify the strongest correlation coefficient

Since the strongest relationship is either a positive or a negative relationship, we need to find the correlation coefficient with the highest absolute value among the given options. Let's analyze the coefficients: A. .58 -> Moderate positive relationship B. 19 -> Invalid coefficient (the value exceeds the -1 to 1 range) C. -0.97 -> Extremely strong negative relationship D. -0.05 -> Very weak negative relationship Only option B (19) is not a valid correlation coefficient, because it exceeds the valid range (-1 to 1).

Compare absolute values

Now, let's compare the absolute values of the remaining coefficients: A. |.58| = 0.58 C. |-.97| = 0.97 D. |-.05| = 0.05 Among all the remaining coefficients, 0.97 (option C) has the highest absolute value, which indicates the strongest relationship between the two variables.

Answer the question

Based on the analysis, we can conclude that option C, with a correlation coefficient of -0.97, represents the strongest relationship between two variables among all the given options.

Key concepts

Relationship Between Variables

When we talk about the 'relationship between variables', what we are referring to is how one variable changes in response to another. Imagine you are tracking the amount of time students study and their resulting test scores. If you notice that more study time often corresponds with higher test scores, you're observing a relationship between the two variables.

In statistics, this relationship is quantified using correlation coefficients. These coefficients, which always range between -1 and 1, tell us not only whether there is a relationship but also how strong that relationship is. When we see a correlation coefficient that's close to 1 or -1, it indicates a significant connection between the variables at play.

Strength of Correlation

The magnitude of the correlation coefficient indicates the 'strength of correlation'. A correlation coefficient of -1 or 1 signifies an absolute linear relationship, meaning that the two variables move perfectly in tandem - they increase or decrease exactly in proportion to each other. For instance, a coefficient of 1 means that for every positive increase in one variable, there's a positive increase in the second variable at a consistent rate.

A coefficient closer to 0 means a weaker correlation; the changes in one variable are less consistently matched in the second variable. In our example involving study time and test scores, a coefficient of 0.58 suggests a moderate relationship: as study times increase, test scores tend to increase as well, though not as consistently as they might with a higher coefficient.

Positive and Negative Correlation

Correlation coefficients can be 'positive' or 'negative', indicating the direction of the relationship. A 'positive correlation' means that as one variable increases, the other variable also increases. Conversely, a 'negative correlation' means that as one variable increases, the other decreases.

Take, for example, two scenarios: In one, a coefficient of 0.58 shows that more study time is moderately associated with better test scores (positive correlation). In another situation, a coefficient of -0.97 means that as the number of hours of television watched increases, perhaps students' test scores significantly decrease (negative correlation). Thus, a strong negative correlation like -0.97 in the original exercise suggests a powerful inverse relationship between the two variables under consideration.