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I haven't been able to find a definitive answer to my question in previous topics, so I'll post it here. I am sure this is a trivial question for many, but I do not have ny formal training in statistics/mathematics, and I would like to have a deeper understanding when I apply statistical methods to my research data.

So, from what I understand, the sampling distribution of the t-statistic is the t-distribution defined by the appropriate degrees of freedom. I also understand that the sampling distribution of the difference between two means follows a normal distribution if the sample sizes are big enough, and that we use the t-distribution to approximate the population distribution when the population variance is unknown or when the sample size is too small to reasonably assume that the Central Limit Theorem will kick in. However, I do not understand why we need to concern ourselves with the assumption of normality if what we are concerned with is the sampling distribution of the test statistic, and we know the test statistic to follow a t-distribution with the appropriate degrees of freedom. I think I'm missing a crucial link somewhere, and I would be glad if someone could offer a non-technical explanation of what is it that I am misunderstanding.

Thank you for the feedback!

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    $\begingroup$ "We use the t-distribution to approximate the population distribution when the population variance is unknown or when the sample size is too small" appears to reflect a significant misunderstanding. It is rare for a population to have a t distribution or even for anyone to assume it is. The focus instead is on the sampling distribution of a test statistic. In light of this, could you clarify what you mean by "assumption of normality"? Exactly what do you suppose is assumed to have a Normal distribution? $\endgroup$ – whuber Nov 1 '18 at 12:30
  • $\begingroup$ Yeah, sorry, I might not have been clear enough. I am aware that what actually needs to be normal is the sampling distribution of the mean, which actually approximates a normal distribution as the sample size increases. What I don't understand is why we need to assume that the sampling distribution of the mean is normal, if the test statistic we use to calculate the p-value (t-statistic) follows a t-distribution with the appropriate degrees of freedom. So basically I don't see where normality enters the story. I am sure this is completely trivial, but I can't see the connection yet, $\endgroup$ – Bálint L. Tóth Nov 1 '18 at 18:13
  • $\begingroup$ When the underlying population is non-Normal, the ratio of the sample mean to the sample standard deviation will likely not follow a Student t distribution. Therefore, it is incorrect to compute significance levels or p-values by referring to a Student distribution. That's the story as told to beginners, anyway. In practice, it is well known that certain forms of departures from Normality (such as a bit of kurtosis) are not problematic whereas others (such as high skewness) can be an issue. $\endgroup$ – whuber Nov 1 '18 at 19:14
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    $\begingroup$ Thanks! That is exactly the kind of answer I needed, it makes intuitive sense, and in time, I will dig into the question deeper. $\endgroup$ – Bálint L. Tóth Nov 1 '18 at 20:01
  • $\begingroup$ I am reminded of a specific question about applying the t-test to a highly non-normal (skewed) distribution. The thread is at t-test on highly skewed data. $\endgroup$ – whuber Nov 1 '18 at 21:24

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