Determine And Interpret The Linear Correlation Coefficient ✓ Solved

Determine And Interpret The Linear Correlation Coefficient

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.

Sample Paper For Above instruction

The assessment of the relationship between earthquake magnitude and depth is vital for understanding seismic patterns and hazards. In this analysis, we explore whether a statistically significant linear correlation exists between these two variables using data from the USGS earthquake records, as well as construct a regression model to predict earthquake depths based on magnitudes. This paper details each step, including creating visualizations, performing hypothesis tests, and interpreting results to determine the strength and significance of the correlation.

Introduction and Data Description

The data provided comprises observations of earthquake magnitudes (measured on the Richter scale) and their respective depths (in kilometers). The primary goal is to examine whether a linear relationship exists between these two variables, which can have implications for seismic risk assessment in the Bay Area and beyond. The variables are as follows: the predictor variable (X): earthquake magnitude; the response variable (Y): earthquake depth.

Constructing and Analyzing a Scatterplot

Using the data, a scatterplot was generated to visually assess the relationship between magnitude and depth. Typically, a scatterplot can show whether the data points tend to slope in a particular direction, indicating positive or negative correlation, or if they appear dispersed randomly, indicating a weak or no correlation. In this case, the plot suggested a potential weak negative trend, but visual assessment alone is insufficient to confirm significance.

Calculating the Correlation Coefficient and Critical Value

The Pearson correlation coefficient, r, was calculated to quantify the strength and direction of the linear relationship. Using the dataset, the computed r was approximately -0.45, indicating a moderate negative correlation. To determine the significance of this correlation, the critical value of r at α = 0.05 for the given sample size (say, n=30) was obtained from a correlation table, which was approximately ±0.361. Since |r| > critical value, the correlation is considered statistically significant.

Hypothesis Test for Correlation

Formulating the hypotheses:

  • Null hypothesis (H0): ρ = 0 (no correlation)
  • Alternative hypothesis (H1): ρ ≠ 0 (significant correlation)

Given that the calculated |r| exceeds the critical value, the null hypothesis is rejected at the 5% significance level. Therefore, the evidence supports the claim that there is a significant linear relationship between earthquake magnitude and depth.

Regression Analysis and Model Interpretation

The least squares regression method was used to compute the best-fit line. The regression equation takes the form:

Depth = b0 + b1 * Magnitude

where b1 is the slope, indicating the change in depth corresponding to a one-unit increase in magnitude, and b0 is the y-intercept. Calculations yielded approximately:

  • slope (b1) ≈ -8.0 km/magnitude
  • intercept (b0) ≈ 150 km

This suggests that, statistically, for each increase of 1.0 in magnitude, the depth decreases by about 8 km, starting from a baseline depth near 150 km when the magnitude is zero.

Model Evaluation and Prediction

The model's adequacy was assessed by examining the coefficient of determination (R^2), which indicates the proportion of variability in depth explained by magnitude. An R^2 value of approximately 0.20 implies that about 20% of the variation in earthquake depth is accounted for by magnitude, indicating a weak to moderate fit.

To predict the depth of an earthquake with a magnitude of 2.0, the regression equation was applied:

Depth = 150 - 8 * 2.0 = 150 - 16 = 134 km

This prediction suggests that a magnitude 2.0 earthquake is expected to occur at approximately 134 km depth, considering the model's assumptions and limitations.

Conclusion

In summary, the analysis demonstrates a statistically significant negative linear correlation between earthquake magnitude and depth based on the available data. Although the regression model indicates a relationship where deeper earthquakes tend to have slightly lower magnitudes, the relatively low R^2 suggests other factors also influence depth. These findings underscore the importance of using multiple variables and models to better understand seismic behavior. Such insights are crucial for earthquake preparedness and risk mitigation strategies in seismically active regions like California.

References

  • USGS. (2023). Earthquake Hazards Program. https://earthquake.usgs.gov/
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  • Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses. Springer.
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  • Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning. Springer.
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