Question

If one regression coefficient is less than unity, then the other will be:

• A. less than unity
• B. equal to unity
• C. greater than unity
• D. All of these

Hint:

If one regression coefficient is small (\(<1\)), the other compensates to maintain product \(r^2\).

Answer 1

The Correct Option is C

The question is about understanding the relationship between the two regression coefficients in statistics. Regression coefficients are used to measure the relationship between two variables in linear regression.

Let's denote the two regression coefficients as \(b_{YX}\) and \(b_{XY}\). The product of the regression coefficients is given by the formula:

\(b_{YX} \times b_{XY} = r^2\) where \(r\) is the correlation coefficient.

The key properties to remember are:

• The correlation coefficient \(r\) lies between -1 and 1, i.e., \(-1 \leq r \leq 1\).
• Thus, \(0 \leq r^2 \leq 1\).

Now, let's analyze the options:

1. If one regression coefficient is less than unity, say \(b_{YX} < 1\), then the product \(b_{YX} \times b_{XY} = r^2\) must still equal \(r^2\), which is less than or equal to 1 but non-negative.
2. This implies \(b_{XY}\) must be greater than 1 to maintain the equation. Why? Because:
   • Suppose \(b_{YX} = 0.5\) and \(r^2 = 0.5\). Then, \(b_{XY} = \frac{0.5}{0.5} = 1\).
   • But if \(r^2\) was less than 0.5, say 0.3, then \(b_{XY} = \frac{0.3}{0.5} = 0.6\) which contradicts as both cannot be < 1.

Therefore, when one regression coefficient is less than unity, the other must be greater than unity to satisfy the relation \(b_{YX} \times b_{XY} = r^2\), where \(r^2 \leq 1\).

Thus, the correct answer is: greater than unity.