Question

Roller coasters. Roller coasters get all their speed by dropping down a steep initial incline, so it makes sense that the height of that drop might be related to the speed of the coaster. Here's a scatterplot of top Speed and largest Drop for 75 roller coasters around the world. a) Does the scatterplot indicate that it is appropriate to calculate the correlation? Explain. b) In fact, the correlation of Speed and Drop is \(0.91\). Describe the association.

Short answer

a) Yes, if the scatterplot shows a linear trend with no extreme outliers. b) There is a strong positive linear relationship.

Step-by-step solution

Assessing If Correlation Is Appropriate

To decide if it is appropriate to calculate the correlation, we need to evaluate the scatterplot for linearity, presence of outliers, and strength of the relationship. The data should show a reasonably linear trend without any extreme outliers that could distort the correlation measure.

Describing the Association Based on Correlation Coefficient

The given correlation coefficient of 0.91 suggests a strong positive linear relationship between the speed and the largest drop of roller coasters. A correlation of 0.91 is very close to 1, indicating that as the height of the drop increases, the speed of the coaster also tends to increase.

Key concepts

Scatterplot

A scatterplot is a valuable tool for visualizing the relationship between two numerical variables. In the case of the roller coasters, the scatterplot plots each coaster's top speed against its largest drop. Each point on the scatterplot represents a different roller coaster.

When analyzing a scatterplot, you should look for overall patterns, such as whether the data follow a linear form, and identify any potential outliers.

• Linear Patterns: On the scatterplot, a linear pattern appears as a collection of points that could be approximately covered with a straight line.
• Outliers: These are individual points that deviate significantly from the overall trend of the data.

The scatterplot must show a linear trend without extreme outliers for it to be appropriate to calculate a correlation coefficient.

Linear Relationship

A linear relationship between two variables means that as one variable changes, the other variable tends to change at a constant rate. In the given problem of roller coasters, there's a query about whether the height of the drop is related to the speed in a linear manner.

A linear relationship is indicated when you see that data points in a scatterplot trend upward or downward along a path that resembles a line without unusual curving.

• Positive Linear Relationship: As the largest drop increases, the roller coaster speed tends to increase.
• Negative Linear Relationship: As one value increases, the other decreases (not typical in the context of roller coasters).

A well-defined linear relationship in the scatterplot suggests that computing the correlation is meaningful.

Correlation Coefficient

The correlation coefficient quantifies the strength and direction of a linear relationship between two variables. The value of the correlation coefficient ranges from -1 to 1.

• A coefficient of 1 indicates a perfect positive linear relationship, where both variables move in the same direction.
• A coefficient of -1 indicates a perfect negative linear relationship, where as one variable increases, the other decreases.
• A coefficient of 0 means no linear relationship exists between the variables.

In the roller coaster example, a correlation coefficient of 0.91 indicates a very strong positive linear relationship. This suggests that as the largest drop height increases, the speed of the roller coaster also increases. A high correlation value close to 1 suggests that the changes in one variable are highly predictable based on changes in the other.