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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.

$\textbf{But I still cannot think of a way for pair in the experiment of LED bulbs.}$

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    $\begingroup$ I don't believe your experiment lends itself to a paired design. You do not have any way to associate any bulb from condition S to any particular bulb in condition T. The best you can do is compare the distribution of condition S to the distribution of condition T $\endgroup$
    – Dave2e
    Feb 14, 2021 at 19:28
  • $\begingroup$ @Dave2e So, do you mean that I can't do a Randomized Paired Design on this experiment ? If I can't, then can I do a Completely Randomized Design which without any blocking/pairing for this experiment ? $\endgroup$
    – xxxxxx
    Feb 14, 2021 at 22:05
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    $\begingroup$ As far as pairing and blocking, you have not provided enough information to answer that question. This will all depend on the limitations of the experimental setup. Are you running all 18 bulbs at the same time or only 1 at a time? Can you run samples at both temperature at the same time? Is there a limit to the number of runs per day? Are the bulbs from the same production lot or different production lots? $\endgroup$
    – Dave2e
    Feb 15, 2021 at 2:32
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    $\begingroup$ This seems like a homework/self-study question rather than a question where there is an actual experiment needed to be answered. If so, then you'd have two different data generation processes, depending on whether you're simulating an unpaired or paired design. The paired design would have two sources of variability: between-pair and within-pair, and you'd simulate both. Also, I believe you're supposed to tag it "self-study" if it's a homework question. $\endgroup$ Feb 19, 2021 at 19:12

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