Light Bulb hypothesis testing 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!

You are testing $H_0: \mu \ge 1000$ against
$H_a: \mu < 1000.$ With $\mu =\mu_0 = 1000,$ one has
$$\frac{\bar X}{1000} \sim \mathsf{Gamma}(\mathrm{shape}=4,\mathrm{rate}=4).$$
So the test statistic is $Y=\frac{\bar X}{\mu_0} = 900/1000 = 0.9$ and we would reject $H_0$ at the 5% level
if $Y < c = 0.3416,$ where the critical value $c = 0.3416$ can be found using R as shown below.
qgamma(.05, 4, 4)
[1] 0.3415796

The P-value of the test is 0.4848 > 0.05
pgamma(.9, 4, 4)
[1] 0.4847839

The figure below shows the density function of
$\mathsf{Gamma}(4,4).$ The critical value
is shown by a dotted red line and the P-value
is the area under the density curve to the left
of the vertical black line.

R code for figure:
hdr="Density of GAMMA(4,4)"
curve(dgamma(x,4,4), 0, 3, lwd=2, ylab="Density", 
      xlab="y", col="blue", main=hdr)
 abline(v = 0.3416, lwd=3, lty="dotted", col="red")
 abline(v = 0.9, lwd=2)
 abline(h=0, col="green2")
 abline(v=0, col="green2")

Note: This test with only $n = 4$ observations has very poor power.
To give
a rough idea of the lack of precision
from a sample of size $n=4:$ a 90% CI
for $\mu$ based on $\bar X=900$ from
a sample of four is $(464, 2635)$ hours.
900/qgamma(c(.95,.05), 4,4)
[1]  464.2971 2634.8178

