Let $(X_1, \dots , X_n)$ be a random sample of i.i.d. random variables distributed as uniform $U[0,\theta]$
Verify null hypothesis $H_{0} :\theta = \theta_{0}$ versus $H_{1} :\theta \not=\theta_{0}$ through likelihood ratio test, with level of signicance fixed equal to $\alpha$
Verify null hypothesis $H_{0} : \theta = 7$ versus $H_1: \theta = 8$ considering a test with the following rejection region: $R = ({(x_1, x_2, \dots, x_6) : max(X_1, X_2, X_3, X_4, X_5, X_6) = X_{(6)} > 6})$
I have calculated as $\alpha = P(X_{(n)} > 6|\theta = 7) = 1- \left(\frac{6}{7}\right)^{6}$ = $0.6034$
and $\beta = P(X_{(n)} < 6|\theta = 8) = \left(\frac{6}{8}\right)^{6} = 0.178$
From here how can I verify the null hypothesis? Thanks in advance