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I want to perform a two sample test, to see whether theirs means are equals. One of the sample is rather big (#60000 values) and represents the scores of a whole population. This population is partitioned in small classes of less that 1000 people. The other sample of the test is a subsamble of the former. It corresponds to people of the same class. So, each time, there is less than 1000 values in it. I have no idea about the variances, so I'd think of using Welch's t-test.

But I'm concerned regarding the independence hypothesis: can it be valid, considering that the second sample is negligible when compared to the overall population?

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  • $\begingroup$ There are two ideas. First, the test is going be a one-sample test since you compare smaller groups' mean to a population (supposed) mean. Second, why would you want to do this given that hypothesis check on a sample statistic is about where the sample is from the specified distribution? $\endgroup$ Commented Oct 27, 2017 at 16:45
  • $\begingroup$ from the specified distribution. Sorry, from specified population. Which you already know as being true. $\endgroup$ Commented Oct 27, 2017 at 17:02
  • $\begingroup$ Could you please clarify the means of what you want to compare? $\endgroup$ Commented Oct 27, 2017 at 17:40
  • $\begingroup$ I do this because I want to reject null hypothesis H_0 is {means are equal} For example : To see whether a class of 500 people that have the same job or the same education (i.e. a subsample of the whole population), do have a mean score different from the average. $\endgroup$
    – user182542
    Commented Oct 27, 2017 at 17:55
  • $\begingroup$ So you want to compare to total population average, do you? Ah, given that subsamples are not randomly drawn, I see now that you ondeed can see significant differences. $\endgroup$ Commented Oct 27, 2017 at 18:15

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But I'm concerned regarding the independence hypothesis: can it be valid, considering that the second sample is negligible when compared to the overall population?

Because you refer to the total population, you no longer need to input a second sample in the test procedure, so you don't need to estimate the second sample's mean and use the degrees of freedom of that sample. The population mean is enough.

The answer is "use a one-sample t-test."

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  • $\begingroup$ In this rephrased question, it says that the subsample and the total population do not have the same variance stats.stackexchange.com/questions/310314/… Is the t-test still valid? $\endgroup$ Commented Oct 28, 2017 at 19:09
  • $\begingroup$ @Benoit Patra, I want to stress it that if your second value of mean comes from the population, by definition this mean does not have its variance at all, so calculating a different variance on the population is not needed at all, as you don't want to use it for standard error calculation (pooled standard error, etc.). You just compare sumbsample mean, using subsample standard error, to a scalar (population mean). $\endgroup$ Commented Oct 29, 2017 at 10:28
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    $\begingroup$ Re "the population mean is enough:" I don't think so, because its calculation includes the data from the subpopulation to which it is compared, thereby ruining the implicit assumption of independence in the one-sample test. Isn't this issue the whole point of the question? $\endgroup$
    – whuber
    Commented Aug 18, 2023 at 13:02

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