One claims that the life time distribution of its Everyday light bulbs is exponential with mean 1000 hours. If you test a random sample of 4 light bulbs and find that the average life time is 900 hours, do you have significant evidence against the this claim? Set up an appropriate hypothesis testing problem. Use the nominal level of 0.05 for your test. (Hint: You may view the exponential distribution as a gamma distribution. With data collected from a sample of 4 light bulbs, what is the power of your test if the actual mean life time is only 900 hours?
My attempt:
$\mu = 1000, \sigma = 1000, \bar X_4 = 900$
$H_0: \mu \geq 1000$
$H_a: \mu \lt 1000$
Since $\bar X_4 = 900, \sum_{i=1}^{4} X_i = 3600$
I then conduct the integration $$\int_{3600}^{\infty} \frac{(\frac{1}{1000})^4}{\gamma(4-1)}x^{4-1}e^{-0.001x} dx \approx 1.546$$
compare this result with $\alpha = 0.05$, which is 2.353 after checking the t-test table. So I should not reject $H_0$
Is everything alright? I have no idea about the second part. Any hint or suggestion would be appreciated!