The slope between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $\frac{y_2-y_1}{x_2-x_1}$.
Applying this to points $P$ and $Q$ gives $\frac{y-7}{4-(-2)}=\frac{-3}{2}$.
Simplifying, we have $\frac{y-7}{6}=\frac{-3}{2}$.
Cross-multiplying, we get $2(y-7)=-3(6)$.
Expanding, we have $2y-14=-18$.
Adding 14 to both sides gives $2y=-4$.
Finally, dividing both sides by 2 gives $y=\boxed{-2}$.
The answer is: -2
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Given points $P(-2,7)$ and $Q(4,y)$ in a coordinate plane, for what value of $y$ is the slope of the line through $P$ and $Q$ equal to $\frac{-3}{2}$?

### Answer:
