Question (Neyman-Pearson lemma [5 marks]). Suppose \( X_{1}, \cdots, X_{n} \) is a random sample from a population follows the normal distribution \( N(\mu, 16) \). Apply the Neyman-Pearson lemma to find the most powerful test, with a sample size of 10 and with the significance level \( \alpha=0.05 \), for testing
\[
\begin{array}{ll}
H_{0}: & \quad \mu=30 \\
H_{1}: & \quad \mu=40 .
\end{array}
\]
