Question

Suppose that, for a sample of pairs of observations from two variables, the linear correlation coefficient r, is negative. Does this result necessarily imply that the variables are negatively linearly correlated ? Explain

Short answer

Correlation coefficient rnegative doesn't necessarily imply that variables are negatively correlated.

Step-by-step solution

Basic 

Linear correlation means that variables change at constant rate, the relationship represented by a straight line graph.

p(rho)is statistical measure of variables' population correlation. r is a sample based 'statistic' estimate of p(rho)

p(rho) $<\hspace{0.17em}0$implies that variables are negatively linearly correlated; they move in opposite direction - one variable increase and other variable decrease, vice versa

Explanation 

rnegative (r $<0$) for sample needs to be tested using hypothesis testing, to ensure that the population correlation parameter p(rho) is negative

$t=\sqrt{r/[(1-r^2)\hspace{0.17em}/(n-2)]}$

If t value leads to rejection of null hypothesis p(rho) $=0$, and non rejection of alternate hypothesisp(rho) $<\hspace{0.17em}0$. Only then, we can confirm that the variables are neccesarily negatively correlated, otherwise we can't.