Probability for Poisson question

Question

Can anyone help me with this question? [1]: https://i.stack.imgur.com/1LJJH.png

this is my solution, is this in the right track? [2]: https://i.stack.imgur.com/wodoa.jpg

Answer 1

I simply would use the Neyman Pearson's lemma

$$\frac{L(\mathbf{x}|\lambda_0)}{L(\mathbf{x}|\lambda_1)}\leq k$$

that is

$$\frac{e^{-14}2^{\Sigma_i X_i}}{e^{-7}}\leq k$$

or equivalently

$$\Sigma_i X_i\leq k^*$$

leading to the following Critical region

$$P(\Sigma_i X_i\leq k^*|\lambda=2)=0.01$$

Where $\Sigma_i X_i\sim Po(14)$

Concluding, you will reject $H_0$ iff

$$\sum_{i=1}^7 X_i\leq5$$