# Finding values for some test and power for uniform distribution

I have the random sample $$X_1, X_2, \dots, X_n$$ drawn from the uniform distribution on $$[\varphi, \varphi + 1]$$. To test the null hypothesis $$H_0 : \varphi = 0$$ against the alternative hypothesis $$H_1 : \varphi > 0$$, we have the test

$$\text{Reject} \ H_0 \ \ \ \text{if} \ \ \ X_{(n)} \ge 1 \ \text{or} \ X_{(1)} \ge g,$$

where $$g$$ is a constant, $$X_{(1)} = \min\{X_1, X_2, \dots, X_n\}, X_{(n)} = \max\{X_1, X_2, \dots, X_n\}$$.

How do I find $$n$$ and $$g$$ so that the $$0.05$$ level will have power at least $$0.8$$ if $$\varphi > 0.1$$?

I think we have to begin by finding the CDF of $$X_{(1)}$$ and $$X_{(n)}$$, but I've only just started learning this stuff, so I really don't know what I'm supposed to do. I would appreciate it if any answers would please explain the reasoning behind the steps, so that I may better understand.

For $$x \in [\phi, \phi+1]$$:
• $$\mathbb P(X_{(n)} \le x) = \left(x-\phi\right)^n$$ as you want all the values to be in $$[\phi,x]$$
• $$\mathbb P(X_{(1)} \le x) = 1 -\left(\phi +1- x\right)^n$$ as you want the complement of the event that all the values are in $$(x,\phi+1]$$