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I have a series of values and I need to test if the returns are > 0. The t-test assumes normality, so I was wondering if there are other alternatives. Note that the standard suggested sample size is >30, what if you have less than that? What other statistical procedures can one apply to determine significance in small sample sizes which can ether be parametric or non-parametric.

gung - Reinstate Monica
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user1234440
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    There is a thread here that's relevant for you about [how to decide between a t-test or some non-parametric alternative for a small sample size](http://stats.stackexchange.com/q/121852/22228) ... But do note Peter Flom's comment about time series, as if your data are autocorrelated you need a different strategy. – Silverfish Jan 10 '15 at 17:31
  • *Note: while the linked answer is a nice complementary resource, they are not actually asking the same question / duplicates.* – gung - Reinstate Monica Jan 10 '15 at 17:58
  • good point, any ideas on the other methods that take in to account the dependence of the data? – user1234440 Jan 10 '15 at 19:42
  • I initially thought this question was essentially asking how to choose between a t-test and an alternative - and if so, which - when sample size is small but I now see there's some more subtle aspects going on, which limit the usefulness of the linked thread. Is the hypothesis being tested because we want to see if *expected total returns* are positive? In this case, alternative tests using only rank or sign aren't going to be helpful, and you need to use one based on the mean. – Silverfish Jan 10 '15 at 22:07

2 Answers2

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First, there is no lower limit to sample size for the t-test; in fact, it's mostly used when N < 30. When N > 30 the t-distribution looks just like the Normal.

Second, if, by "series of values" you mean a time series, then the t-test is not appropriate - you need some method that accounts for the dependence of the data.

Peter Flom
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Nonparametric analyses typically use ranks. If you just want to know if >50% are greater than zero, then you could use a sign test, or the Wilcoxon signed rank test (which is an augmented sign test). Such tests don't tell you about the mean, though, they tell you about the median. If you want to know about the mean and your data are not sufficiently normal to trust the $t$-test, you may be able to use a bootstrapping approach.

(Note that Peter's point regarding time series still applies here. These suggestions are inappropriate if you have serial correlation.)

gung - Reinstate Monica
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    (+1) As the question specifically concerns small samples, it might be worth adding that the nonparametrics would have lower power than t (power being a particular concern with small samples) if the data were drawn from a normal population, and perhaps stating the smallest sample size at which they can attain significance (eg at the 5% level). – Silverfish Jan 10 '15 at 18:25