Write it the way you'd explain it to a classmate

Consider the following hypothesis testing for population mean $\mu$ when the population variance is $\textbf{unknown}$; \begin{align*} H_{o}:\mu=\mu_{0}\\ H_{1}:\mu > \mu_{0}. \end{align*}

1. Derive the expression of type-II error $\beta_{\alpha}(\mu_{t}),$ when the given lavel of significance is $\alpha\in (0, 1)$ and the true mean is $\mu_{t}.$
2. Show that $\beta_{\alpha}(\mu_{t})$ is a decreasing function of $\alpha.$
3. Provide an intuitive explanation for the relationship established in part 2. In particular, discuss how increasing the level of significance $\alpha$ affects the likelihood of Type II error. Your explanation should clearly describe the trade-off between Type I and Type II errors without using formal derivations.