Question

The probable error of correlation coefficient \((r)\) is used for :

• A. Measuring the magnitude of error in \(r\)
• B. Testing the significance of \(r\)
• C. Both (A) and (B) are correct
• D. Neither (A) nor (B) is correct

Hint:

Probable error of \(r\) fixes the range in which the true population correlation lies, and it also supplies the \(r < PE\) / \(r > 6PE\) rule for judging significance.

Answer 1

The Correct Option is C

Step 1: Recall the definition. For a sample correlation coefficient \(r\) based on \(n\) paired observations, the probable error is \[PE(r) = 0.6745\times\frac{1-r^2}{\sqrt{n}}.\]

Step 2: First use of \(PE(r)\), measuring the magnitude of error. It gives the limits within which the population correlation coefficient \(\rho\) may be expected to lie, with about 50% probability: \[\rho = r \pm PE(r).\] This is exactly what option (A) describes.

Step 3: Second use of \(PE(r)\), testing significance. A standard rule of thumb states: if \(r < PE(r)\), there is no evidence of correlation in the population; if \(r > 6\times PE(r)\), the correlation is taken as significant, provided \(n\) is reasonably large. This is exactly what option (B) describes.

Step 4: Combine both uses. Since \(PE(r)\) is used both to fix the probable limits of \(\rho\) (magnitude of error) and to judge whether \(r\) is significant, both statements (A) and (B) are correct.

Step 5: Conclusion. \[\boxed{\text{Both (A) and (B) are correct}}\]