We have been learning about Z-test for null hypothesis rejection. It goes as follows:
Suppose $X_1,\ X_2,\ \ldots, X_n$ are samples from $X\sim N(\mu, \sigma^2)$ distribution with sample mean $\overline{X}$. Suppose we take null hypothesis to be $\mu=c$ for some $c<\overline{X}$ and alternative hypothesis to be $\mu>c$.
Now take $Y_1,\ Y_2,\ \ldots, Y_n$ to be samples from $Y\sim N(c,\sigma^2)$ distribution with sample mean $\overline{Y}$.
If $\mathbb{P}(\overline{Y}\geq \overline{X})<\alpha$ for some significance level $\alpha\in (0,1)$, then we reject the null hypothesis.
Now how I try to understand it is, since $\overline{Y}$ is an estimator for $c$, $\mathbb{P}(\overline{Y}\geq \overline{X})$ to be low would imply $\mathbb{P}(c\geq \mu)$ to be low, which actually agrees if the null hypothesis is true. So why are we rejecting null hypothesis in this case?
Is my understanding correct? If not, then what is the correct way to interpret it?
EDIT: One interpretation which makes this correct is, low $\mathbb{P}(\overline{Y}\geq \overline{X})$ would imply low $\mathbb{P}(c\geq \mu)$, which would imply high $\mathbb{P}(c<\mu)$, thus high probability that $\mu$ is far from the $c$ we chose, hence rejecting the null hypothesis.
Is this reasoning correct?