Question

If the correlation coefficient is negative, then the regression coefficient is:

• A. $0$
• B. $1$
• C. Negative
• D. Positive

Answer 1

The correct option is C

Negative

Explanation for the correct option:

We know that the regression line of $y$ on $x$ is $y-\overline{y}$ & for $x$ on $y$ is $x-\overline{x}$.

Also $r=\frac{\sum \left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}{\sqrt{\sum {\left(x_i-\overline{x}\right)}^2\sum {\left(y_i-\overline{y}\right)}^2}}$

$r=$ correlation coefficient
$x_i=$values of the x-variable in a sample
$\overline{x}=$ mean of the values of the $x$-variable
$y_i=$ values of the $y$-variable in a sample
$\overline{y}=$ mean of the values of the $y$-variable

By the above correlation coefficient, we can say that if the correlation coefficient is negative only if either $y-\overline{y}$ or $x-\overline{x}$ is negative.

A negative correlation accurs when the correlation coefficient is less than $0$, Hence, if both the regression are negative, the correlation coefficient will also be negative

Therefore, if the correlation coefficient is negative, then the regression coefficient is negative.

Hence, option C is correct.