Consider a population of independent light bulbs with an exponential lifetime distribution with mean $\mu$. It is claimed that their expected lifetime is 1000 hours. A definition of a 100(1−$\alpha$)% confidence interval obtained from an observation t0 is the set of all $\mu_0$ which are not rejected in a test of a null hypothesis $\mu$= $\mu_0$ against an alternative hypothesis $\mu$$\neq$$\mu_0$.
One particular light bulb fails after 622 hours. Solve the equations of the two significance probabilities Pr(T ≥ 622 |$\mu_0$) = 0.05 (for a test of $\mu$= $\mu_0$ versus $\mu$> $\mu_0$) and Pr(T ≤ 622 |$\mu_0$) = 0.05 (for a test of $\mu$= $\mu_0$ versus $\mu$< $\mu_0$) for $\mu$. Determine the range of values of $\mu$ such that both of the probabilities Pr(T ≥ 622 | $\mu$) and Pr(T ≤ 622 |$\mu$) are at least 0.05. (This range gives an equi-tailed 90% confidence interval for $\mu$.)
I don't seem to understand what they mean by 'solve the equations'. Do I have to find a specific value for T or compute Pr(T ≥ 622 |$\mu_0$), Pr(T ≤ 622 |$\mu_0$) and compare with 0.05? I believe I will get the second part after I understand this bit.