Question

Regression coefficient ranges from

• A. -1 to +1
• B. 0 to 1
• C. -1 to +2
• D. - \(\infty\) to + \(\infty\)

Hint:

Statistical Bounds:
Correlation Coefficient ($r$) $\rightarrow -1 \le r \le +1$.
Coefficient of Determination ($r^2$) $\rightarrow 0 \le r^2 \le 1$.
Regression Coefficient ($b$) $\rightarrow -\infty \le b \le +\infty$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Mathematical domain of linear regression slopes: regression coefficients ($b_{yx} = ext{Cov}(X,Y)/\sigma_x^2$) represent dimensional rates of change spanning all real numbers ($-\infty \le b \le +\infty$).
Key Formula or Approach:
\[ -\infty < b_{yx} < +\infty \quad | \quad \text{Compare with Correlation: } -1 \le r \le +1 \]

Step 2: Detailed Explanation:
Comparing statistical metrics:
1. Correlation Coefficient (r): A dimensionless, standardized measure of linear association strictly bounded between $-1$ and $+1$ ($-1 \le r \le +1$).
2. Regression Coefficient (b Slope): An absolute measure of the rate of change of $Y$ per unit change in $X$ ($b = r \cdot \frac{\sigma_y}{\sigma_x}$). Because it carries physical units and depends on the scales of measurement of $X$ and $Y$, its value is mathematically unbounded and ranges from $-\infty$ to $+\infty$.

Step 3: Final Answer:
Hence, the regression coefficient ranges from - \(\infty\) to + \(\infty\), matching option (D).