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Hope this is not too basic:

I understand we used paired testing in situations where, e.g., the same subject is tracked before- and after- an experiment/treatment, e.g., before- and after patient receives a medication.

But there are cases that are not described in this format, so I would like to know if dependence of events tested are enough to use paired tests. Specifically, I am thinking of these 2 experiments:

1)We are testing the parking times for cars C1, C2 of different makes ; we want to see if mean parking times are equal.

We have 10 people park car C1 and we measure parking times for each, we compute the mean $ \mu_1 $ of all parking times. We then have the same 10 people park car C2 in the same spot as C1 , measure parking times, compute the mean $ \mu_2 $. Since parking jobs are done each time by the same group, do we then use paired t-Test to test whether $\mu_1= \mu_2$ ( at a given choice of confidence) , since/because the two times are correlated?

2)We want to test whether right- and left- limbs are of equal length. Do we use paired testing If the limbs are measured in the same person, because the measurememnts are likely correlated ? And if some of the cases we only measured only one limb in one person and left limb in another or we only measured one limb per person we would not used pair testing? Thanks.

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In general, you would use a paired $t$-test when there is variation among observations which is shared (and matchable) between the two samples.

So, in your example #1, yes: use a paired $t$-test since individual drivers have different abilities and pairing each driver with themselves should better estimate if there is a difference in parking car C1 versus C2.

You could also do a paired test if you had drivers of varying experience represented equally in both samples. Then you would compare drivers of C1 and C2 who were new drivers, drivers with more experience, and so on (depending on your grouping of experience. That is less than the clean ideal of comparing each driver to themself, but since we expect experience to affect driving ability (and thus parking time) a paired $t$-test is better than a pooled test.

Note that if you could not pair the observations 1:1 for car C1 and C2, you could instead do a stratified $t$-test. That gets a bit more complicated, however, since you need to correct for different numbers and variation in each group-car combo. This writeup on the stratified $t$-test shows how the bookkeeping gets a bit involved.

In your second example, you would again do well to use a paired $t$-test if you measured both limbs on each person. If you measured some left limbs and some right limbs, you would use a pooled $t$-test unless there were some factor you expected to relate to limb difference. (I'm having a hard time imagining a setup where a paired $t$-test would work for measuring some left limbs and some right limbs.)

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