Identifying the Correct Statistical Statement

This section analyzes several statements related to correlation and regression coefficients to determine the correct one.

Analyzing Statistical Statements

Option 1: Karl Pearson's Coefficient of Correlation Appropriateness

• Statement: When both variables X and Y are measured on an interval or ratio scale, Karl Pearson's coefficient of correlation is most appropriate.
• Analysis: Karl Pearson's coefficient of correlation (r) quantifies the linear association between two continuous variables. It assumes that the data can be measured on an interval or ratio scale, allowing for meaningful calculations of means, variances, and covariances. Therefore, this statement is correct.

Option 2: Probable Error Formula

• Statement: The probable error of the coefficient of correlation is $\frac{\sigma_P}{\sqrt{n}}$.
• Analysis: The standard formula for the probable error (PE) of the coefficient of correlation is $PE = 0.6745 \times \frac{1-r^2}{\sqrt{n}}$, where r is the correlation coefficient and n is the sample size. The formula provided in the option is incorrect.

Option 3: Signs of Regression Coefficients and Correlation

• Statement: If the sign of two regression coefficients (Y on X) and (X on Y) is negative, the sign of the correlation coefficient is positive.
• Analysis: The relationship between the correlation coefficient (r) and the regression coefficients ($b_{xy}$ and $b_{yx}$) is given by $r = \text{sign}(b_{xy}) \sqrt{|b_{xy} \times b_{yx}|}$. If both regression coefficients ($b_{xy}$ and $b_{yx}$) are negative, their product ($b_{xy} \times b_{yx}$) is positive. However, the sign of the correlation coefficient r matches the common sign of the regression coefficients. Thus, if both are negative, r must also be negative, not positive. This statement is incorrect.

Option 4: Magnitudes of Regression Coefficients

• Statement: If one of the regression coefficients is greater than one, the other regression coefficient must also be greater than one.
• Analysis: From the relationship $|r| = \sqrt{|b_{xy} \times b_{yx}|}$ and knowing that $|r| \le 1$, it follows that $|b_{xy} \times b_{yx}| \le 1$. This implies that it is impossible for both regression coefficients to have an absolute value greater than 1 simultaneously. If $|b_{xy}| > 1$, then $|b_{yx}|$ must be less than 1 (specifically, $|b_{yx}| \le \frac{1}{|b_{xy}|}$) for the condition to hold. This statement is incorrect.

Conclusion

Based on the analysis, only the first statement regarding the appropriateness of Karl Pearson's coefficient of correlation for interval or ratio scales is correct.