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Is the magnitude of an earthquake related to the depth below the surface at which the quake occurs? Let \(x\) be the magnitude of an earthquake (on the Richter scale), and let \(y\) be the depth (in kilometers) of the quake below the surface at the epicenter. The following is based on information taken from the National Earthquake Information Service of the U.S. Geological Survey. Additional data may be found by visiting the Brase/Brase statistics site at college.hmco.com/pic/braseUs9e and finding the link to earthquakes. $$ \begin{array}{c|crrrrrr} \hline x & 2.9 & 4.2 & 3.3 & 4.5 & 2.6 & 3.2 & 3.4 \\ \hline y & 5.0 & 10.0 & 11.2 & 10.0 & 7.9 & 3.9 & 5.5 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that \(\Sigma x=24.1, \Sigma x^{2}=85.75, \Sigma y=53.5, \Sigma y^{2}=\) \(458.31\), and \(\Sigma x y=190.18\). Compute \(r\). As \(x\) increases, does the value of \(y\) imply that \(y\) should tend to increase or decrease? Explain.

Step by step solution

01

Plot the Scatter Diagram

First, create a scatter plot with the given data points. Place the magnitude of the earthquake on the x-axis and the depth on the y-axis. Plot each corresponding pair \((x, y)\) on the graph. This visual representation will help determine the correlation pattern between the two variables.

02

Draw the Best Fit Line

Observe the scatter plot and draw a line that you believe best fits the trend of the data points. This line should capture the overall direction and pattern of the data. Consider if most of the points lie close to a straight line, which suggests a linear relationship.

03

Assess Correlation Strength and Direction

Evaluate the scatter plot to describe the correlation. If points are clustered closely around the line, the correlation is strong; if not, it is weak. Check if the line has a positive slope (upward from left to right), indicating a positive correlation, or a negative slope, indicating a negative correlation.

04

Compute Correlation Coefficient r

Using the provided sums, calculate the correlation coefficient \( r \). Use the formula: \[r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}\]where \(n\) is the number of data points. Plug in \(n=7\), \(\Sigma x=24.1\), \(\Sigma y=53.5\), \(\Sigma xy=190.18\), \(\Sigma x^2=85.75\), and \(\Sigma y^2=458.31\) to calculate \(r\).

05

Interpret the Correlation Coefficient

Once you've calculated \(r\), interpret its value. If \(r\) is close to 1 or -1, the correlation is strong; if it's close to 0, the correlation is weak. A positive \(r\) value indicates that as the earthquake magnitude increases, the depth tends to increase. A negative \(r\) suggests the opposite.

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Key Concepts

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

Correlation Coefficient

The correlation coefficient, often represented by the symbol \( r \), is a statistical measure that calculates the strength and direction of a linear relationship between two variables on a scatter plot. It ranges from -1 to 1. If \( r \) is close to 1, this indicates a strong positive relationship, meaning when one variable increases, the other tends to increase as well. Conversely, if \( r \) is close to -1, it signifies a strong negative relationship, where an increase in one variable tends to lead to a decrease in the other. When \( r \) is closer to 0, it suggests that there is little to no linear correlation between the variables.

In the context of earthquake data analysis, calculating the correlation coefficient helps in understanding the relationship between the magnitude of earthquakes and their depth. This analysis can provide insights into whether stronger earthquakes are generally deeper or shallower compared to weaker ones. By computing \( r \), students can assess whether there's a tendency for the depth to increase or decrease as the magnitude increases.

Scatter Plot

A scatter plot is a graph used to visualize the relationship between two numerical variables. On this graph, each point represents an observation from the dataset with one variable plotted on the x-axis and the other on the y-axis. This allows us to observe potential relationships or patterns.

In this exercise, the scatter plot is used to show the relationship between earthquake magnitude (on the x-axis) and depth (on the y-axis). By examining the distribution and pattern of the data points, you can get an initial sense of whether there's a relationship between these two variables. If the points roughly form a linear shape, either increasing or decreasing, it suggests a linear relationship. The scatter plot can reveal outliers, clusters, or trends that aren't immediately apparent from the raw data alone.

Linear Relationship

A linear relationship suggests that there is a constant rate of change between two variables. If one variable increases or decreases by a specific amount, the other variable changes by a consistent and proportionate amount as well.

In the case of our earthquake data, a linear relationship would mean that as the magnitude of an earthquake changes, the depth changes consistently. After plotting the scatter plot, if you notice a trend where the data points tend to align closely around a straight line, this indicates a linear relationship.

The notion of a linear relationship is crucial in predicting and understanding patterns within data. For instance, if we determine a positive linear relationship between magnitude and depth, it implies that deeper earthquakes tend to be stronger. This can have real-world implications for how we prepare for and respond to such natural events.

Earthquake Data Analysis

Earthquake data analysis involves examining various aspects of seismic activity, which primarily includes the magnitude and depth of earthquakes. By analyzing this data, researchers and statisticians can uncover patterns and relationships that help in understanding seismic behavior.

The exercise presented aims to investigate whether there's a significant correlation between an earthquake's magnitude and its depth. By assessing this relationship using tools such as scatter plots and correlation coefficients, students can develop insights into whether stronger earthquakes generally occur at greater depths or not.

This kind of analysis is essential not just for students learning statistical methods, but it also provides value in seismology and disaster preparedness. Understanding the relationship between magnitude and depth is key to predicting potential impacts of earthquakes and can guide construction practices in earthquake-prone areas. This analysis demonstrates the application of statistical tools in real-world scenarios, emphasizing the importance of data in decision-making and policy planning.