Consider the following statements
I. Both the regression coefficients have same sign
II. If one of the regression coefficient is greater than unity, the other must be less than unity.
Which of the above statements is/are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To determine the correctness of the given statements regarding regression coefficients, we will analyze each statement step by step.

### Step 1: Analyze Statement I
**Statement I:** Both the regression coefficients have the same sign.

- The regression coefficients are denoted as \( b_{yx} \) (the slope of the regression line of \( y \) on \( x \)) and \( b_{xy} \) (the slope of the regression line of \( x \) on \( y \)).
- The relationship between the regression coefficients and the correlation coefficient \( r \) is given by the formula:
\[ r = \sqrt{b_{yx} \cdot b_{xy}} \]
- Since \( r \) (the coefficient of correlation) can only take values between -1 and 1, the product \( b_{yx} \cdot b_{xy} \) must be non-negative. This implies that both \( b_{yx} \) and \( b_{xy} \) must either be both positive or both negative.
- Therefore, **Statement I is correct**.

### Step 2: Analyze Statement II
**Statement II:** If one of the regression coefficients is greater than unity, the other must be less than unity.

- The maximum value of the coefficient of correlation \( r \) is 1. The regression coefficients are related to the correlation coefficient as follows:
\[ b_{yx} = r \cdot \frac{s_y}{s_x} \quad \text{and} \quad b_{xy} = r \cdot \frac{s_x}{s_y} \]
where \( s_y \) and \( s_x \) are the standard deviations of \( y \) and \( x \), respectively.
- If one of the regression coefficients (say \( b_{yx} \)) is greater than 1, then:
\[ b_{yx} > 1 \implies r \cdot \frac{s_y}{s_x} > 1 \]
This would imply that \( r \) must be less than \( \frac{s_x}{s_y} \) which can only happen if \( b_{xy} \) is less than 1.
- Thus, if one regression coefficient is greater than unity, the other must indeed be less than unity. Therefore, **Statement II is also correct**.

### Conclusion
Both statements are correct. Hence, the answer is that both statements I and II are correct.

### Final Answer
Both statements I and II are correct.

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