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Hi, I am studying two population hypothesis test in statistic.

I added two Photos related to the topic which include the descripion of the topics. First one is related to depended sample. Second one is related to independent sample with known population variance.

However, I am stack with a point. When I read the question related to two population hypothesis test, how can I decide whether the population is dependent or independent? Please explain me this explicitly? I am confused so much and I have an exam tommorow.

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It does not make sense to talk about dependence of populations. You need to ask yourself: "if I take item #1 from sample set #1, can I find an item in sample #1 to which it has a real relationship, i.e., can two such items logically be paired?"

Example: say, you own a pizza shop and you decide to change your recipe. You take 10 customers and ask each of them to rate your old-style pizzas and the, ask the same 10 people, to rate the new pizzas. Then you can match these two sample sets into pairs, because they involve the same people: Sarah's score for old-style forms a pair with Sarah's score for the new pizzas.

This is when you will apply a paired-sample t-test (for average scores).

However, if you asked 10 (or other number) DIFFERENT people after changing the recipe, the score of John for old-style pizza cannot be related to any of the new-style scores, because John is not part of the new scores. More strongly, if at least one of the person in the survey has been added/deleted/exchanged, you cannot have 10 matching pairs of scores anymore, hence you cannot apply paired-sample t-test, but you must recourse to an independent-samples t-test.

While independent-samples test can always be applied, it gives slightly larger uncertainties (larger p-value) than matching-pairs test, so you should apply the latter whenever it is feasible.

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