Assume that the flaws along with a magnetic tape of length $t$ have $\operatorname{Poisson}(\lambda \, t)$ distribution. Suppose that 2 flaws were found in a piece of tape of lenght $500$. Using UMP test at level $\alpha \leq 0.01$ to test the following hypothesis: $H_0:\lambda \geq 0.02$ against $H_1:\lambda < 0.02 $?
To construct the UMP test, we have to construct the corresponding N-P test. Hence, the LR is given by \begin{align} \frac{L(X_1,\ldots,X_n\mid\lambda_1 t)}{L(X_1,\ldots,X_n\mid\lambda_0 t)} = \frac{\prod_{i=1}^n \frac{e^{-\lambda_1 t} (\lambda_1 t)^{x_i}}{x_i!}}{\prod_{i=1}^n \frac{e^{-\lambda_0 t} (\lambda_0 t)^{x_i}}{x_i!}} &= \exp\{n t (\lambda_0 - \lambda_1)\}\left(\frac{\lambda_1}{\lambda_0}\right)^{\sum_i^nx_i}\\ &= \exp\{nt(\lambda_0 - \lambda_1)\}\left(\frac{\lambda_1}{\lambda_0}\right)^{\sum_i^nx_i} > c\,. \end{align}
Therefore this statistic dependent on $\sum_i^nx_i$ and since $\lambda_{0} > \lambda_{1}$ it is a decreasing function of $\sum_i^nx_i$. Hence, $\mathcal{P}(\sum_i^nx_i \leq c \, \mid \lambda_{0} = 0.02) \leq 0.01 $.
This is a self-study question. Is that it in terms of UMP test? does it look good to you? How would I calculate $p$-value for this test? Thank you in advance.