The slope and correlation will always have ____________.

Question

BlackTom AI · Accepted Answer

The question asserts a relationship between the slope of the regression line and the correlation. In Pearson correlation, the slope b1 of the least-squares regression line of Y on X is given by b1 = r * (sy / sx), where r is the correlation between X and Y, sy is the standard deviation of Y, and sx is the standard deviation of X. Since standard deviations sx and sy are nonnegative (and typically treated as positive in standard cases), the sign of the slope b1 matches the sign of the correlation r.

Option consideration:
- If r is positive, b1 is positive, indicating a positive slope and a positive association.
- If r is negative, b1 is negative, indicating a negative slope and a negative association.
- If r is zero, then b1 = 0 (assuming sx and sy are nonzero), meaning the slope is zero and there is no linear association.

However, caveats exist. The sign match holds when both X and Y have nonzero variability (sx > 0 and sy > 0). In degenerate cases where one variable has zero variance, correlation is undefined and the slope is not interpretable. When r = 0, the slope is zero, which technically does not have a positive or negative sign, so saying the slope and correlation will “always” have the same sign is not strictly true in that boundary case. In typical, well-behaved data with a meaningful linear relationship, the slope and correlation share the same sign.

In summary, under normal conditions with nonzero variability, the slope and the correlation tend to have the same sign because b1 = r * (sy / sx) and sx, sy > 0, making the sign of b1 equal to the sign of r. The edge case where r = 0 yields a zero slope, for which a “sign” is not defined, illustrating that the statement is conditionally true rather than absolute in all possible datasets.

STAT 1450.02 SP2025 (28132) Checkpoint 5.4

The slope and correlation will always have ____________.

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