Analysis of Assertion (A) and Reason (R)

This question involves understanding the relationship between simple regression coefficients and the correlation coefficient.

Evaluating Reason (R)

Reason (R) states that the correlation coefficient ($r$) between two variables $x$ and $y$ is equal to the geometric mean between the two regression coefficients ($b_{yx}$ and $b_{xy}$). The mathematical relationship is given by:

$r = \pm \sqrt{b_{yx} \times b_{xy}}$

The magnitude of the correlation coefficient ($|r|$) is indeed the geometric mean of the magnitudes of the regression coefficients ($|b_{yx}|$ and $|b_{xy}|$). The sign of $r$ matches the signs of both $b_{yx}$ and $b_{xy}$. Therefore, Reason (R) is considered true.

Evaluating Assertion (A)

Assertion (A) claims that if the value of one regression coefficient ($b_{yx}$) is less than 1.00, the value of the second regression coefficient ($b_{xy}$) must also be less than 1.00.

From the relationship in Reason (R), we know that $|r^2| = |b_{yx} \times b_{xy}|$. Since $|r| \le 1$, it follows that $r^2 \le 1$. Thus, the product of the magnitudes of the regression coefficients cannot exceed 1:

$|b_{yx} \times b_{xy}| \le 1$

Let's consider a scenario:

• Suppose $b_{yx} = 0.5$. This value is less than 1.00.
• For the condition $|b_{yx} \times b_{xy}| \le 1$ to hold, we need $|0.5 \times b_{xy}| \le 1$, which implies $|b_{xy}| \le \frac{1}{0.5} = 2$.
• It is possible to have $b_{xy} = 1.5$. In this case, $b_{yx} = 0.5$ (less than 1) and $b_{xy} = 1.5$ (greater than 1). The product is $0.5 \times 1.5 = 0.75$, which is less than or equal to 1. The correlation coefficient would be $r = \sqrt{0.75} \approx 0.866$, which is valid.

Since we found a valid case where $b_{yx} < 1$ but $b_{xy} > 1$, Assertion (A) is false.

Conclusion

Assertion (A) is false, and Reason (R) is true. Reason (R) correctly states a fundamental property relating regression coefficients and the correlation coefficient.

Therefore, the correct option is that (A) is false but (R) is true.