Deliverable 6 - Analysis with Correlation and Regression ✓ Solved

According to the U.S. Geological Survey (USGS), the probability of a magnitude 6.7 or greater earthquake in the Greater Bay Area is 63%, about 2 out of 3, in the next 30 years. In April 2008, scientists and engineers released a new earthquake forecast for the State of California called the Uniform California Earthquake Rupture Forecast (UCERF). As a junior analyst at the USGS, you are tasked to determine whether there is sufficient evidence to support the claim of a linear correlation between the magnitudes and depths from the earthquakes.

Your deliverables will be a PowerPoint presentation you will create summarizing your findings and an Excel document to show your work.

Concept being Studied:

  • Correlation and regression

  • Creating scatterplots

  • Constructing and interpreting a Hypothesis Test for Correlation using r as the test statistic

You are given a spreadsheet that contains the following information:

  • Magnitude measured on the Richter scale

  • Depth in km

Using the spreadsheet, you will answer the problems below in a PowerPoint presentation.

What to Submit:

  • Slide 1: Title slide

  • Slide 2: Introduce your scenario and data set including the variables provided.

  • Slide 3: Construct a scatterplot of the two variables provided in the spreadsheet. Include a description of what you see in the scatterplot.

  • Slide 4: Find the value of the linear correlation coefficient r and the critical value of r using α = 0.05. Include an explanation on how you found those values.

  • Slide 5: Determine whether there is sufficient evidence to support the claim of a linear correlation between the magnitudes and the depths from the earthquakes. Explain.

  • Slide 6: Find the regression equation. Let the predictor (x) variable be the magnitude. Identify the slope and the y-intercept within your regression equation.

  • Slide 7: Is the equation a good model? Explain. What would be the best predicted depth of an earthquake with a magnitude of 2.0? Include the correct units.

  • Slide 8: Conclude by recapping your ideas by summarizing the information presented in context of the scenario.

Along with your PowerPoint presentation, you should include your Excel document which shows all calculations.

Paper For Above Instructions

Earthquakes pose serious threats to communities, primarily through their potential destructiveness. To develop insights into their characteristics, the U.S. Geological Survey (USGS) has made predictions regarding earthquake probabilities in various regions, including the Greater Bay Area of California. This analysis will focus on investigating the correlation between earthquake magnitudes and depths using the given data. Utilizing statistical methods such as correlation coefficients and regression analysis, this study aims to comprehend the patterns that emerge from earthquake data and predict future earthquake depths based on a specific magnitude.

Understanding Correlation and Regression

First, it is essential to understand what correlation and regression signify in this context. Correlation measures the strength and direction of a linear relationship between two variables. In this case, we aim to analyze the relationship between earthquake magnitudes (independent variable) and their depths (dependent variable). The linear correlation coefficient (r) quantifies the degree of this relationship, where values close to 1 or -1 indicate a strong positive or negative correlation, respectively, while values around 0 imply little to no correlation.

On the other hand, regression analysis involves fitting a line (regression line) to the data points observed, enabling us to predict values for one variable based on another. The equation derived from this analysis typically takes the form of y = mx + b, where m represents the slope of the line, and b signifies the y-intercept.

Gathering Data

The data used for this analysis consists of various earthquakes measured by their magnitudes on the Richter scale and their corresponding depths, provided in kilometers. The scatterplot, which visually depicts this data, serves as the first step in our analysis. Constructing the scatterplot allows us to observe any discernible patterns or correlations in the magnitude-depth relationship.

Upon plotting the data points, one could recognize the trends: if the data suggests that as magnitude increases, depth tends to increase or decrease correlatively. This foundational visual analysis serves to provide insight into the subsequent calculations of the correlation coefficient and regression equation.

Calculating the Linear Correlation Coefficient

To calculate the linear correlation coefficient r, one can employ statistical software or formulas. The formula for r is given by:

r = Σ[(xi - x̄)(yi - ȳ)] / (√Σ[(xi - x̄)²] * Σ[(yi - ȳ)²])

where xi and yi represent the individual data points for magnitude and depth, and x̄ and ȳ are the means of the respective datasets.

Once calculated, it is crucial to compare this obtained value of r to the critical value at α = 0.05 for a significance level. This comparison discern whether the correlation is statistically significant – informing our assessment regarding the existence of a linear correlation between the two variables.

Determining Sufficient Evidence for Correlation

Upon finding the correlation coefficient, the next step is identifying whether there exists sufficient statistical evidence to support the claim of a linear correlation. This determination can be made by evaluating the statistical significance, wherein if the absolute value of the calculated r exceeds the critical value, we can accept the alternative hypothesis that a correlation exists. Conversely, if the absolute value is less than the critical value, we fail to reject the null hypothesis, suggesting no significant correlation between magnitude and depth.

Creating the Regression Equation

Next, we seek to derive the regression equation from the dataset. This process will yield a linear equation defining the best-fit line that describes the relationship. The slope (m) indicates how much depth is expected to change for each unit increase in magnitude, while the y-intercept (b) reveals the extrapolated depth when the magnitude is zero. This regression equation is paramount, as it allows us to make predictions regarding depths based on specific magnitudes observed.

Evaluating the Model's Effectiveness

The effectiveness of the regression model can be assessed based on multiple metrics, including the coefficient of determination (R²), which indicates how well the independent variable (magnitude) explains the variation in the dependent variable (depth). A higher R² value suggests that the model is a good fit for the data, thus supporting its predictive capability.

To exemplify the practical application of the regression equation, we can predict the depth of an earthquake with a magnitude of 2.0 utilizing the regression model derived. This predicted value enables valuable insights into the expected impact of low-magnitude earthquakes, guiding safety protocols and preparedness measures.

Conclusion

In conclusion, employing statistical tools to analyze the correlation and regression in earthquake data equips us with essential insights into potential behaviors of earthquake magnitudes and depths. Through constructing scatterplots, calculating correlation coefficients, developing regression equations, and evaluating the model's integrity, this analysis promotes a deeper understanding of seismic phenomena, essential for effective disaster management strategies.

References

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