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I've read some articles about t-test and figured out that there are a couple of assumptions for t-test to be valid, and one of them is 'random sampling' assumption.

What does this assumption actually mean? What is the exact criteria that distinguishes a 'randomly sampled' variable from those who are not. I want to understand the precise meaning. Please help.

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  • $\begingroup$ Possible duplicate of Can non-random samples be analyzed using standard statistical tests? $\endgroup$ Commented Oct 28, 2016 at 11:45
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    $\begingroup$ I'm not convinced this thread is a duplicate, since the answers in the other thread seem to take it for granted that the reader knows what a "random sample" is. Clearly the two are related, but that thread has more of a focus on the effect of violations. $\endgroup$
    – Silverfish
    Commented Oct 28, 2016 at 13:06
  • $\begingroup$ I'm not entirely sure, I found the first answer in the other thread to be a clear explanation of what constituted a random sample under the assumptions defined for a t-test. If the question were more generic, e.g. "What is a random sample?", I would entirely agree with you. Perhaps the author could make this more clear? $\endgroup$ Commented Oct 28, 2016 at 17:40

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Let's first write out the assumptions of a T-test.

  • The distribution of the mean of the sample being tested is normal.
  • The variance of this distribution is unknown.
  • The sample is random.

In the common situation where you sample a bunch of people and want to make inference on the mean of a characteristic of the population such as height, weight, income etc... a random sample implies that you are using a non-deterministic method to select a sample from the population. In simple cases, addressed in most text books, you are looking at taking what is known as a "simple random sample"; a sample where each element of the population has equal probability of inclusion in a sample.

In a more general case, this sample need not be a simple random sample. The only requirement is that there is a non-deterministic method of generating the sample. However, it is important to note that the mean you are using as your test statistic is the population mean under this sampling method. In statistical theory this can be seen by taking random samples from non-uniform distributions, and in statistical practice this can be seen as non-uniform sampling from a finite population to adjust for peculiarities in the elements being sampled.

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