The half-life of 100 electronic components is 260 hours. Suppose that the life of the components is like an exponential of $\lambda$. For $\alpha = 0.05$ I want to do the next test ($\mu$ is the half-life) $H_0 : \mu = 300$ and $H_1 : \mu < 300$.
Then $$ \overline{X} = \frac{X_1 + \cdots + X_{100}}{100} \sim exp(\lambda). $$ The relationship between $\lambda$ and $\mu$ is $\mu = 1/\lambda$ because $E[\overline{X}] = 1/\lambda$. $H_0$ is true if $|\overline{X}| \leq k$. Then I am going to calculate this $k$. $$ \alpha = 0.05 = P(|\overline{X}| > k | \mu = 300) = P(\overline{X}< k_1|\lambda = 1/300) + P(\overline{X}>k_2|\lambda = 1/300). $$ If $$ P(\overline{X}<k_1) = 0.025 $$ then $k_1 = 7.6$ and with the same argument $k_2 = 1106,66$. I think that this acceptance region is so large and I am not sure that it's correct. And there is another question in this exercises because it says that if $\mu = 250$ really then what is the probability that this test detect this fact.