This question asks about the relationship between two regression coefficients, specifically what the value of the regression coefficient of X on Y (denoted as $b_{yx}$) is, given that the regression coefficient of Y on X (denoted as $b_{xy}$) is greater than 1.

Understanding Regression Coefficients

In statistics, regression coefficients quantify the relationship between variables. When we talk about the relationship between two variables, say Y and X, we can consider two regression coefficients:

• $b_{xy}$: This is the regression coefficient of Y on X. It tells us how much Y changes for a one-unit change in X.
• $b_{yx}$: This is the regression coefficient of X on Y. It tells us how much X changes for a one-unit change in Y.

The Relationship Between $b_{xy}$ and $b_{yx}$

There's a fundamental relationship connecting these two coefficients and the correlation coefficient ($r$) between the two variables. The correlation coefficient ($r$) measures the strength and direction of the linear relationship between two variables and ranges from -1 to +1 (i.e., $-1 \le r \le +1$).

The relationship is given by the formula:

$$ b_{xy} \times b_{yx} = r^2 $$

Since the square of the correlation coefficient ($r^2$) must always be between 0 and 1 (inclusive), we have:

$$ 0 \le r^2 \le 1 $$

Therefore, the product of the two regression coefficients must also satisfy:

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

Note: This holds true as long as both $b_{xy}$ and $b_{yx}$ have the same sign (which they typically do, mirroring the sign of $r$). If $r=0$, both coefficients are 0.

Determining $b_{yx}$ when $b_{xy} > 1$

The question states that $b_{xy} > 1$. We need to find the possible value of $b_{yx}$ using the relationship $b_{xy} \times b_{yx} \le 1$.

Let's rearrange the inequality to solve for $b_{yx}$:

$$ b_{yx} \le \frac{1}{b_{xy}} $$

Since we are given that $b_{xy} > 1$, when we take the reciprocal of $b_{xy}$, the value becomes less than 1.

For example:

• If $b_{xy} = 1.5$, then $b_{yx} \le \frac{1}{1.5}$, which means $b_{yx} \le 0.667$. In this case, $b_{yx} < 1$.
• If $b_{xy} = 2$, then $b_{yx} \le \frac{1}{2}$, which means $b_{yx} \le 0.5$. Again, $b_{yx} < 1$.

In general, if $b_{xy}$ is a number greater than 1, then $\frac{1}{b_{xy}}$ will be a number less than 1. Because $b_{yx}$ must be less than or equal to $\frac{1}{b_{xy}}$, it follows that $b_{yx}$ must be less than 1.

Conclusion

Given the condition $b_{xy} > 1$, and knowing that the product $b_{xy} \times b_{yx} \le 1$, the regression coefficient $b_{yx}$ must be less than 1 ($b_{yx} < 1$).