Math Problem Statement

We are interested in predicting the concentration of a chemical in the soil (y) based on the distance from the site of chemical spill.  Suppose the following regression line was obtained based on data collected at 8 different distances from the spill site.  Find and interpret the value of the correlation coefficient. 

Solution

To find and interpret the value of the correlation coefficient (rr), we can start with the R2R^2 value provided in the summary of fit. The R2R^2 value is the square of the correlation coefficient rr.

From the provided data, we have:

R2=0.941917R^2 = 0.941917

To find the correlation coefficient rr:

r=R2r = \sqrt{R^2} r=0.941917r = \sqrt{0.941917} r0.9705r \approx 0.9705

Since the slope of the regression line is negative, the correlation coefficient will also be negative:

r0.9705r \approx -0.9705

Interpretation: The correlation coefficient of approximately 0.9705-0.9705 indicates a strong negative linear relationship between the concentration of the chemical in the soil and the distance from the site of the chemical spill. As the distance from the spill site increases, the concentration of the chemical decreases significantly.

Would you like to ask for more details or have any other questions? Here are some additional questions you might consider:

  1. What is the significance of the slope in this regression analysis?
  2. How do you interpret the p-values in the parameter estimates table?
  3. What steps would you take if you suspected multicollinearity in a multiple regression model?
  4. How can you use this regression model to make predictions?
  5. What are some potential limitations of this regression analysis?

Tip: When interpreting regression results, always consider both the statistical significance (p-values) and the practical significance (magnitude and direction of coefficients) of the predictors.

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Math Problem Analysis

Mathematical Concepts

Correlation Coefficient
Linear Regression
Statistics

Formulas

Correlation Coefficient (r) = sqrt(R^2)
Coefficient of Determination (R^2)

Theorems

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Suitable Grade Level

Advanced High School