Want to compare the average lifetime of LED bulbs under two different temperature settings, namely, $S$ degree Celsius and $T$ degree Celsius. Suppose that the distribution of the bulb’s lifetime is known to follow an exponential distribution with rate parameter $\lambda > 0$. The density of this distribution is $f(x) = \lambda e^{−\lambda𝑥}$, $x \geq 0$. The mean and standard deviation of the exponential distribution is $1/\lambda$. This parameterization is in units corresponding to the reciprocal of time.
The hypothesis $H_a$ is that the expected lifetime of LED bulbs is $3$ years under temperate $S$ and $1$
year under temperature, $T$. Randomly generate a sample of $18$ data points to form the
observations under two experimental designs: a completely randomized design and a randomized paired design, to compare the average lifetimes between the two groups $S$
and $T$.
$\textbf{My Question:}$
I just don't know how to do a randomized paired design on the given info above.
Right now I can randomly generate $9$ observations for $Exp(\lambda=1/3)$
and randomly generate 9 for $Exp(\lambda=1)$ by using the $rexp()$ function in R.
Which means right now I have $9$ bulbs and $9$ observations for each the two treatments (i.e. $S$ and $T$) in total of $18$ observations.
$\textbf{But I can't think of a way to paired it up for each bulbs.}$
$\textbf{(i.e. I don't know how to do the blocking or randomized within a pair}$
$\textbf{which I don't even know what the name of the pair can it be)}$
I found a example of randomized paired design on "https://scidesign.github.io/designbook/completely-randomized-designs-comparing-two-treatments.html#the-randomization-test-for-a-randomized-paired-design" which is the Boy's Shoe Experiment. In this design it pair under the left and right foot of a boy.
$\textbf{But I still cannot think of a way for pair in the experiment of LED bulbs.}$
