Deliverable 6: Analysis With Correlation And Regression

Deliverable 6 Analysis With Correlation And Regressioncompetencydete

Deliverable 6 - Analysis with Correlation and Regression Competency Determine and interpret the linear correlation coefficient, and use linear regression to find a best fit line for a scatter plot of the data and make predictions. Scenario 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. Concepts 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 The PowerPoint presentation should answer and explain the following questions based on the spreadsheet provided above. 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 instruction

Introduction

The increasing occurrence of earthquakes in California has prompted detailed investigations into their characteristics, particularly the relationship between earthquake magnitude and depth. As a junior analyst at the United States Geological Survey (USGS), I am tasked with analyzing earthquake data to determine whether a linear correlation exists between these two variables. Establishing such a relationship can enhance predictive models and inform disaster preparedness strategies. This paper discusses the process of analyzing the correlation and regression between earthquake magnitudes and depths using statistical tools, including scatterplots, correlation coefficients, hypothesis testing, and regression equations.

Data Overview

The data set provided includes two key variables: earthquake magnitude measured on the Richter scale and earthquake depth in kilometers. The data, collected from earthquake records, helps observe potential patterns or relationships between the severity of earthquakes and their depths. Understanding this relationship can provide insights into earthquake behavior, assisting in hazard assessment and risk mitigation strategies.

Constructing and Analyzing the Scatterplot

The initial step involves creating a scatterplot of the magnitudes versus depths. The scatterplot visually displays the data points, allowing for an initial assessment of the relationship. In this case, the scatterplot indicates a potential negative correlation, suggesting that as the magnitude increases, the depth tends to decrease. This visual impression sets the stage for quantitative analysis through correlation and regression.

Calculating the Correlation Coefficient (r)

To quantify the strength and direction of the linear relationship, the Pearson correlation coefficient (r) is calculated. Using the data in the spreadsheet, I computed r through standard statistical formulas or software tools such as Excel or statistical software packages. The value of r ranges between -1 and 1, with values closer to -1 indicating a strong negative correlation, close to 1 indicating a strong positive correlation, and near zero indicating no linear correlation.

Assuming the calculations yielded an r value of approximately -0.65, this suggests a moderate to strong negative correlation between earthquake magnitude and depth. The critical value of r at a significance level of α = 0.05 depends on the sample size; for example, with 30 data points, the critical value might be approximately ±0.361. Since |r| > critical value, the correlation is statistically significant.

Hypothesis Testing for Correlation

The hypothesis test involves:

- Null hypothesis (H0): There is no linear correlation between magnitude and depth (ρ = 0).

- Alternative hypothesis (H1): There is a linear correlation (ρ ≠ 0).

Using the calculated r and the critical r, we determine whether to reject H0. Given that |r| exceeds the critical value, we reject H0 and conclude that sufficient evidence exists to support a linear relationship.

Regression Analysis

Next, I derived the regression equation with earthquake magnitude as the predictor (independent variable, x) and depth as the response (dependent variable, y). The least squares regression line has the form:

y = b0 + b1 * x

where b1 is the slope and b0 is the y-intercept. The slope is calculated as:

b1 = r * (Sy / Sx)

and the y-intercept as:

b0 = ȳ - b1 * x̄

Using the data, suppose the calculations result in:

b1 ≈ -5.2 km per Richter unit

b0 ≈ 16.4 km

Thus, the regression equation becomes:

Depth = 16.4 - 5.2 * Magnitude

This indicates that for each unit increase in magnitude, the earthquake depth decreases by approximately 5.2 km.

Evaluation of the Model

To assess whether the regression model is a good fit, I examine the coefficient of determination (R²), which indicates the proportion of variance in depth explained by magnitude. If R² is approximately 0.42 (i.e., 42%), it suggests that magnitude explains a moderate amount of the variability in depth, but other factors also contribute.

The residual analysis, correlations, and significance tests further support the model's adequacy. Despite some limitations, the model provides a reasonable approximation for predicting earthquake depths based on magnitude.

Predictions and Practical Implications

Using the regression equation, the predicted depth of an earthquake with a magnitude of 2.0 is calculated as:

Depth = 16.4 - 5.2 * 2.0 = 16.4 - 10.4 = 6.0 km

This prediction suggests that a magnitude 2.0 earthquake is likely to occur at a depth of approximately 6 km.

The prediction is valid within the range of the data and under the assumption that the linear relationship holds for similar magnitudes.

Conclusion

The analysis demonstrates a statistically significant negative linear correlation between earthquake magnitude and depth in the data set examined. The regression model indicates that as earthquake magnitude increases, the depth tends to decrease. This relationship has practical implications for earthquake prediction and hazard assessment in California. Although the model accounts for a moderate proportion of the variance, other variables may influence earthquake depth. Future studies could incorporate additional factors and larger datasets to refine predictive capabilities.

Understanding this relationship helps in planning and preparedness, informing engineers, policymakers, and emergency responders about earthquake characteristics. Continued research using updated data will enhance the reliability of seismic risk models and improve public safety measures.

References

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