Population Regression Coefficient

The population least-squares slope minimizes $E[(Y-a-bX{)}^2]$ over $b$.

The minimizing slope is $\text{Cov}(X,Y)/\text{Var}(X)$.

1. 1. Recall the slope formula

   The population regression slope of $Y$ on $X$ is given by $\beta =\text{Cov}(X,Y)/\text{Var}(X)$.

2. 2. Substitute the given values

   Plugging in the covariance and variance gives the slope.

   $$\beta =\frac{6}{4}=\frac{3}{2}$$

Answer: 3/2