Does the response depend on X?

In a normal linear model, the response has a distribution whose mean, µy, depends linearly on the explanatory variable,

Y  ~  normaly , σ)

If the slope parameter, β1, is zero, then the response has a normal distribution that does not depend on X.

Y   ~   normal0 ,  σ)

This can be tested formally with a hypothesis test for whether β1 is zero.