Which one of the following statements is false?

Question

BlackTom AI · Accepted Answer

Pearson correlation r measures the strength and direction of a linear relationship between two quantitative variables, so the form of the relationship (linear vs. nonlinear) is crucial to whether r is an appropriate summary.

Relevant concepts:
- r ranges from –1 to 1. The magnitude |r| indicates strength of a linear association; the sign indicates direction (positive or negative).
- r measures linear association; nonlinearity can make r small even when a strong nonlinear relationship exists.
- The least-squares slope formula b1 = r*(sy/sx) shows the sign of r determines the sign of the regression slope.
- Correlation describes association, not perfect pairwise behavior or distributional skewness.

Step-by-step reasoning addressing each option:
A: True. A positive r indicates a general tendency for y to increase as x increases, but individual points can deviate; only r = 1 implies every increase in x corresponds to an increase in y in a perfectly linear way.

B: False. The statement claims the form of the relationship is not as important as strength and direction when deciding to compute r. This is incorrect because r specifically quantifies linear relationships. If the relationship is nonlinear, r can be misleading (e.g., a strong U-shaped relationship can yield r ≈ 0). Thus the form (linear vs nonlinear) is essential before computing Pearson r.

C: True. Compare magnitudes: |–0.52| = 0.52 versus |0.20| = 0.20, so –0.52 represents a stronger linear correlation than 0.20.

D: True. From b1 = r*(sy/sx), sy and sx are positive standard deviations, so b1 and r share the same sign; a negative r gives a negative slope.

E: True. A negative r indicates the direction of association, not the skew of marginal distributions. Skewness of a variable’s distribution is unrelated to the sign of the correlation.

Conclusion: The false statement is "When deciding to compute r, the form of the relationship is not as important as strength and direction." This matches the provided answer.

Which one of the following statements is false?单项选择题

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