What does the null hypothesis $H_0:\beta_2=0$ mean?

Question

For the model $$Y=\beta_1+\beta_2X_2+\beta_3X_3+\epsilon$$ what does the null hypothesis $H_0:\beta_2=0$ mean?

I think that then $X_2$ would not affect the expected value of $Y$ since the coefficient os then $0$.

But is that correct?

Answer 1

Yes that is correct. If $\beta_3$ is 0, the variable has no predictive value for the response. No matter what value $X_2$ has, it does not alter the predicted value of the response.