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Chapter 3: Problem 26

Strong association but no correlation The gas mileage of an automobile first increases and then decreases as the speed increases. Suppose that this relationship is very regular, as shown by the following data on speed (miles per hour) and mileage (miles per gallon). Make a scatterplot of mileage versus speed. $$ \begin{array}{llll}{\text { Speed: }} & {20} & {30} & {40} & {50} & {60} \\\ {\text { Mileage: }} & {24} & {28} & {30} & {28} & {24}\end{array} $$ The correlation between speed and mileage is \(r=0\) . Explain why the correlation is 0 even though there is a strong relationship between speed and mileage.

Short Answer

Expert verified
The correlation is 0 because it measures linear relationships, while the data shows a strong non-linear relationship.

Step by step solution

01

Understand the table of values

We have two variables: speed (in miles per hour) and mileage (in miles per gallon). The values given are: \( \{(20, 24), (30, 28), (40, 30), (50, 28), (60, 24)\} \). We will create a scatterplot in the subsequent steps to visualize these values.
02

Plot the scatterplot

Create a scatterplot using a graph with the x-axis representing speed and the y-axis representing mileage. Plot the points (20, 24), (30, 28), (40, 30), (50, 28), and (60, 24). Once plotted, you'll see that the graph has a parabolic shape where mileage first increases with speed and then decreases.
03

Analyze the trend

The plotted points form an inverted U-shaped pattern. This indicates that as speed increases, mileage increases to a maximum point and then decreases. This kind of pattern signifies a non-linear relationship.
04

Understand the meaning of correlation

Correlation measures the strength and direction of a linear relationship between two variables. A correlation of \( r = 0 \) means there is no linear relationship between the variables. Non-linear relationships like the parabola don't get captured by the correlation coefficient.
05

Explain the zero correlation

Even though the scatterplot shows a strong non-linear relationship (as speed increases, mileage increases to a peak then decreases), it doesn't exhibit a linear trend. This is why the correlation coefficient \( r = 0 \), indicating no linear association despite the evident pattern.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Non-Linear Relationship
In examining data, recognizing how variables relate to each other is crucial. A "non-linear relationship" means that as one variable changes, the other does not change in a proportionate, straight-line fashion. This is important when interpreting data because not all relationships are linear, or straight-lined.
When the relationship is non-linear, as seen in our data on speed and mileage, you may notice that changes in one variable do not consistently cause proportional changes in the other.
  • The pattern might be curved or irregular.
  • The speed and mileage pattern forms a curve, specifically a parabola, demonstrating that they do not increase or decrease at a constant rate.
Understanding non-linear relationships can help in identifying trends that linear models might miss.
Correlation Coefficient
The correlation coefficient is a measure often used in statistical analysis to determine the level of association between two variables. It is represented by the letter "r".
The value of "r" can range from -1 to 1:
  • An "r" of 1 indicates a perfect positive linear relationship.
  • An "r" of -1 signifies a perfect negative linear relationship.
  • An "r" of 0 indicates no linear relationship.
Why is "r=0" for the speed and mileage despite a clear pattern? It's because correlation specifically measures linear relationships. Therefore, any non-linear pattern, like our parabola, results in an "r" equal to zero. This highlights the limitation of the correlation coefficient鈥攊t accurately measures only straight-line relationships.
Parabolic Data Pattern
A parabolic data pattern is a specific type of non-linear relationship that forms a U-shape or inverted U-shape on a graph. Our speed and mileage data creates such a pattern where:
  • Mileage initially increases with speed.
  • Reaches a peak.
  • Decreases as speed continues to rise.
This kind of pattern is symmetrical which typically results in no slope鈥攁 factor necessary for a correlation.
This symmetry is why our data has a zero correlation coefficient despite the clear relationship between speed and mileage. Recognizing parabolic patterns is essential in data interpretation because they suggest maximum or minimum points which can be critical in optimization scenarios and decision-making.
Statistical Analysis
Statistical analysis is the set of techniques used to interpret, analyze and make inferences from data. In this context, when analyzing our data on speed and mileage, several important considerations arise.
First, understanding the type of relationship鈥攍inear versus non-linear鈥攇uides which statistical methods to apply.
For linear relationships, regression and correlation coefficients are powerful tools. However, when it comes to non-linear relationships:
  • More complex models like quadratic models or non-linear regression might be necessary.
  • Scatterplots can effectively reveal non-linear relationships.
  • More intricate statistical tools, beyond basic correlation, can show the full scope of the data relationship.
Statistical analysis is broad and can include everything from simple visualizations like scatterplots to advanced modeling techniques, allowing for a comprehensive interpretation of data.

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Most popular questions from this chapter

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