Question

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

Short answer

Correlation coefficient rpositive doesn't necessarily imply that variables are positively correlated.

Step-by-step solution

Basics 

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 positively linearly correlated, they move in same direction, one variable increase or decrease and other variable the same.

Explanation 

r positive ( $r>\hspace{0.17em}0$) for sample needs to be tested using hypothesis testing, to ensure that the population correlation parameter p(rho) is positive

localid="1653544930587" $t=r/\sqrt{[(1-r^2)\hspace{0.17em}/(n-2)\hspace{0.17em}]}$

If t value leads to rejection of null hypothesis p(rho) $=0$, and non rejection of alternate hypothesisp(rho)$>0$. Only then, we can confirm that the variables are neccesarily positively correlated, otherwise we can't.