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I've been trying to get my head around z and t statistics. and I almost have a matra in my head that "when the sample are small, use the t test, when the samples are big, use either the t or the z test".

Now As I understand it, the z test requires a large number of samples, because it assumes you have a normal distribution, and you need a certain number of samples before your samples will start to look like a normal distribution.

But Why does the t test allow us to deal with smaller samples ? what does it have (or what assumptions doesn't it have) which allow us to deal with smaller samples ?

Is it that in the z test the standard error of the mean distribution is determined from the KNOWN population variance, whereas the t test the standard error of the mean distribution is determined from the ESTIMATE of the population variance, and in the limit of large number of samples the ESTIMATE of the population variance will approach the TRUE population variance ?

If this is indeed the case, does the central limit theorem show us that in the limit of a large number of samples the estimate of the population variance will approach the true population variance ?

Thanks

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# Why can t statistic deal with small numbers ?

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