If $b_{yx}=0.8$ and $b_{xy}=0.2$ for a set of bivariate data, find the coefficient of correlation $r$, and verify that the arithmetic mean of the two regression coefficients is not less than $r$.

Question

Accepted Answer

**Step 1 — Correlation from regression coefficients.**
$$r = \pm\sqrt{b_{yx}\cdot b_{xy}} = \pm\sqrt{0.8 	imes 0.2} = \pm\sqrt{0.16} = \pm 0.4$$
Both $b_{yx}$ and $b_{xy}$ are positive, so $r$ takes the positive sign: $r = 0.4$.

**Step 2 — Verify AM $\ge$ GM.**
Arithmetic mean of the two coefficients: $\dfrac{0.8+0.2}{2} = \dfrac{1.0}{2} = 0.5$.
Geometric mean (which equals $r$): $0.4$.
Since $0.5 \ge 0.4$, the property $\dfrac{b_{yx}+b_{xy}}{2}\ge r$ holds, as it must for any real pair of regression coefficients. …

Q.If $b_{yx}=0.8$ and $b_{xy}=0.2$ for a set of bivariate data, find the coefficient of correlation $r$ , and verify that the arithmetic mean of the two regression coefficients is not less than $r$ .

Concept understanding — Relationship between Regression Coefficients and Correlation

Q.If byx=0.8b_{yx}=0.8byx​=0.8 and bxy=0.2b_{xy}=0.2bxy​=0.2 for a set of bivariate data, find the coefficient of correlation rrr, and verify that the arithmetic mean of the two regression coefficients is not less than rrr.

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Q.If $b_{yx}=0.8$ and $b_{xy}=0.2$ for a set of bivariate data, find the coefficient of correlation $r$ , and verify that the arithmetic mean of the two regression coefficients is not less than $r$ .

Unlock everything free for 14 days