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I know that the larger population, of size m, follows a normal distribution. I have some sample that is size n, and follows its own particular distribution, which is not normal. Given my sample sizes, what are the chances that I would get the distribution that I did in my sample?

Additionally, if I know that the larger population does not follow a normal distribution, but I still know what its distribution is, is there a way to do this same calculation?

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  • $\begingroup$ The question is difficult to answer, since it is somewhat imprecise. A Kolmogorov-Smirnov test (see Wikipedia for a start) may be useful to you. More old-school traditional, particularly for the normal, are Chi-squared tests. $\endgroup$ – André Nicolas Aug 28 '14 at 16:05
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For your first question: if a sample is from a continuous distribution, then the probability that you would get your exact sample is zero. However, the p-value from a normality test will give you the probability of getting a sample at least as non-normal as yours.

For the second question: you cannot pick out the "true" distribution using a sample, but there are test analogous to the normality tests for other distributions (e.g., weibull, gamma, etc)

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