位置1的问题 Consider two variables, X and Y. The sign of the correlation coefficient describing the linear relationsip between X and Y has the same sign as the corresponding simple linear regression coefficient.TrueFalseDon't Know清除选择

Question

BlackTom AI · Accepted Answer

This question asks about the relationship between the sign of the Pearson correlation coefficient and the sign of the simple linear regression slope between two variables X and Y.
Option 1: True. In simple linear regression, the slope coefficient is proportional to the covariance of X and Y, and its sign matches the sign of the correlation coefficient r between X and Y. Intuitively, if X and Y tend to increase together (positive r), the regression line slopes upward (positive beta1); if one tends to increase while the other decreases (negative r), the slope is negative. This alignment of signs holds regardless of scale, since both r and beta1 reflect the direction of the linear relationship.
Option 2: False. A common misconception would be that the slope and the correlation could have opposite signs; however, the mathematical relationship shows they share the same sign. While their magnitudes differ because r is a standardized measure and beta1 depends on the units and variance of X, the direction (sign) remains in agreement due to the underlying covariance structure.
Option 3: Don’t Know. While it’s tempting to doubt a straightforward property, statistical theory provides a clear answer: the sign of r and the sign of the regression slope are the same. The “Don’t Know” option ignores this established relationship and is not a valid choice once one recalls the definitions of r and beta1.
In summary, the intuitive and formal reasoning both lead to the conclusion that the signs coincide when assessing the linear relationship between X and Y. The false and don’t know options fail to capture this fundamental connection, making them incorrect.

位置1的问题 Consider two variables, X and Y. The sign of the correlation coefficient describing the linear relationsip between X and Y has the same sign as the corresponding simple linear regression coefficient.TrueFalseDon't Know清除选择

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