# Solution verification for a hypothesis testing question

I am posting this question as a solution checking.

Let $$X_1,...,X_{30}$$ be a random sample from the exponential distribution with unknown mean $$\mu\in \{1,1/\delta\}$$ (where $$\delta>1)$$. Consider the best test of $$H_0:\mu=1$$ v.s. $$H_1:\mu=1/\delta$$ with power given by $$\beta=\int_0^{17}\dfrac {\delta^{30}}{\Gamma(30)}y^{29}e^{-\delta y} dy$$. What is the critical region of the test ?

My work: By Neyman-Pearson lemma, the best test is given by $$L(1;\mathbb x)/L(1/\delta;\mathbb x)\le k$$ i.e. $$e^{-\sum x_i}/e^{-\delta \sum x_i} \le c$$ i.e. $$e^{(\delta-1)\sum x_i}\le c$$ i.e. $$\sum x_i \le c_1$$ for some constant $$c_1$$. Now given $$P_{H_1}(\sum_{i=1}^{30} X_i \le c_1)=\int_0^{17}\dfrac {\delta^{30}}{\Gamma(30)}y^{29}e^{-\delta y} dy$$. Now under $$H_1$$, $$X_i \sim exp (\delta)$$ , so $$\sum_{i=1}^{30} X_i \sim Gamma(30,1/\delta)$$, hence $$c_1=17$$.

Thus the critical region is $$\sum_{i=1}^{30} X_i\le 17$$.

Am I correct ?