Question

If the regression coefficient of x on y and y on x are \(- \frac14\) and \(- \frac19\) respectively, then what is the correlation coefficient between x and y?
1. \(- \frac14\)
2. \(- \frac16\)
3. \(\frac16\)
4. \(\frac14\)

Answer 1

Correct Answer - Option 2 : \(- \frac16\)

Concept:

Correlation coefficient between x and y is given by:

r = \(\rm \sqrt{b_{yx} \times b_{xy}}\)

Here, b_yx and b_xy are regression coefficients.

Or the slopes of the equation y on x and x on y are denoted as byx and bxy

Note: If b_xy and b_yx are negative then the coefficient of correlation would be negative and if If b_xy and b_yx are positive then the coefficient of correlation would be positive.

Calculation:

Given:

b_yx = \(\rm \frac{-1}{9}\)

bxy = \(\rm \frac{-1}{4}\)

As we know that, correlation coefficient between x and y is given by:

r = \(\rm \sqrt{b_{yx} \times b_{xy}}\)

= \(\rm \sqrt{(\frac{-1}{9}) \times (\frac{-1}{4}})\)

= \(\rm \sqrt{\frac{1}{36}}\)

= \(\rm \pm \frac{1}{6}\)

Since b_xy and b_yx are negative so the coefficient of correlation is also negative.

∴ Coefficient of correlation =\(- \frac16\)