The first statement is true: "The strength of a correlation is independent of whether the correlation of a scatterplot is positive or negative. A scatterplot with positive correlation can have either weak association or strong association."
To elaborate:
- The strength of the correlation (weak, moderate, or strong) refers to how closely the data points cluster around a line of best fit, regardless of whether the correlation is positive (upward slope) or negative (downward slope).
- A scatterplot with a positive correlation can display weak correlation if the points are widely spread out, or strong correlation if the points cluster tightly around the line.
- The other statements are incorrect for various reasons:
- The slope of a line indicates the direction of the relationship (positive or negative), but does not affect the nature of the correlation itself; thus, a negative slope indicates a negative correlation.
- Nonlinear correlations do not necessarily imply strong correlation; a scatterplot can exhibit a nonlinear association without showing a strong correlation.
- Nonlinear relationships can occur in both positive and negative correlations, and a scatterplot can exhibit nonlinear correlations regardless of the slope's direction.
Overall, the first statement captures an essential aspect of correlation accurately.
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