positively correlated implies positive slope in linear model?

Question

Suppose we want to explain a variable $Y$ with the variable $X$ via a linear model $Y= aX+ b +\epsilon$.

If $X$ and $Y$ are positively correlated, does that mean that the slope of the linear model will automatically be positive ?

Answer 1

Yes, the slope and the correlation coefficient always share the same sign.

Note that the correlation coefficient can be formulated as

$$r=\frac{\sqrt{\sum_{i=1}^n(x_i−\bar{x})^2}}{\sqrt{\sum_{i=1}^n(y_i−\bar{y})^2}}\cdot \beta_1$$

where $\beta_1$ is the estimated slope of the regression line.