Larger values of R2 imply that the observations are more closely grouped about the

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BlackTom AI · Accepted Answer

When evaluating what larger R^2 values imply in a simple linear regression context, we want to consider what R^2 represents and how it relates to the data fit.
Option 1: 'average value of the independent variables' — This is not what R^2 measures. R^2 does not describe where the x-values cluster around; it concerns how well the regression model accounts for variability in the dependent variable. So this choice misidentifies the role of the independent variable averages.
Option 2: 'average value of the dependent variables' — Again, R^2 is not about the central tendency (mean) of the y-values. It concerns the proportion of variance explained by the model, not the mean of y. This option confuses variance explained with a simple mean.
Option 3: 'none of the other statements are correct' — This would be a tempting catch-all, but one of the statements does align with a standard interpretation of R^2, so this cannot be true if we accept conventional definitions.
Option 4: 'least squares line' — R^2 measures how well the data points fit the regression line produced by least squares. Higher R^2 means observations are closer to the fitted line, i.e., the model explains more of the variability around the line of best fit. This description matches the core idea of R^2 as the proportion of variance explained by the regression line, indicating tighter clustering around that line.
In summary, the essence of a higher R^2 is that the data align more closely with the regression relationship, which is captured by the least squares line, rather than being tied to averages of x or y alone.

Larger values of R2 imply that the observations are more closely grouped about the

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