Question

What value does \(r\) assume if all the data points fall on the same straight line in these cases? a. The line has positive slope. b. The line has negative slope.

Short answer

Answer: In the case of a positive slope, the value of 'r' is 1, which represents a perfect positive linear relationship. In the case of a negative slope, the value of 'r' is -1, which represents a perfect negative linear relationship.

Step-by-step solution

Case a: Positive Slope

For a straight line with a positive slope, all the data points exhibit a direct relationship. meaning, as one variable increases, the other variable also increases. This relationship results in a correlation coefficient of \(r = 1\). The value \(r=1\) implies a perfect positive linear relationship.

Case b: Negative Slope

For a straight line with a negative slope, all the data points exhibit an inverse relationship, meaning as one variable increases, the other variable decreases. This relationship results in a correlation coefficient of \(r = -1\). The value \(r=-1\) implies a perfect negative linear relationship.

Key concepts

Understanding a Positive Slope

When we talk about a positive slope in the context of data points on a graph, imagine a line that tilts upwards as you move from left to right. This upward tilt reflects a positive slope. A positive slope means that as one variable increases, so does the other. This relationship is often called a "direct relationship." For example:

• If you're looking at a graph of hours studied vs. grades, a positive slope would mean that the more hours you study, the higher your grades.
• In the context of weight and health, consuming more nutritious food might lead to healthier weight levels.

In terms of correlation, a line with a perfect positive slope has a correlation coefficient of 1. This means every increase in one variable perfectly corresponds to an increase in the other. It indicates a perfect positive linear relationship, and visually, all the points would lie exactly on the line. In statistics, seeing an r = 1 is essentially like seeing data in perfect harmony.

Understanding a Negative Slope

A negative slope gives the graph a line that tilts downwards from left to right. In this scenario, as one variable increases, the other decreases. This represents an inverse relationship. For instance:

• Imagine plotting the number of sunny days against electricity consumption for lights. The more sunny days there are, the less artificial light is needed, hence a negative slope.
• If you're graphing exercise time versus body weight, it might show that more exercise time leads to decreased body weight.

With a perfect negative slope, the correlation coefficient is \(-1\). In this case, each increase in one variable results in a perfectly corresponding decrease in the other. Such a setting indicates a perfect negative linear relationship where all the data points align precisely on the line. It's like a perfectly synchronized seesaw – one side goes up while the other goes down, without any deviation.

The Significance of the Correlation Coefficient

The correlation coefficient, often represented by the symbol \(r\), tells us how strongly two variables are related. Ranging between -1 and 1, this value expresses both the direction and the strength of a linear relationship:

• If \(r = 1\), all data points align perfectly along a line with a positive slope, indicating a perfect positive relationship.
• If \(r = -1\), they align along a line with a negative slope, showing a perfect negative relationship.
• An \(r\) close to 0 suggests little to no linear relationship.

Understanding \(r\) is crucial in statistics to predict one variable based on another.

Why is the Correlation Coefficient Useful?

Knowing the correlation coefficient helps determine the reliability of predictions. For example, if you're looking to see how time spent on certain activities impacts performance, a high \(|r|\) value would suggest strong predictability. However, remember, correlation doesn't imply causation. It simply indicates that two variables tend to move together or opposite each other in a linear fashion.