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I have rainfall time series of 85 points from 1901-2013. I want to check whether the mean rainfall during 1901-1970 is significantly different than that of the mean rainfall during 1971-2013.

Here, the data would not 'necessarily' follow the normal distribution. So, one of the main assumption for applying t-test may not be fulfilled.

Now as the length of the dataset for two samples would be 70 and 43, can I use the paired t-test or Welch's t-test assuming that the length is sufficient to circumvent the assumption of normality.

What should be the minimum length of data to overcome the normality condition in t-test, is 43 good enough?

In case, it's not then what test should be applied?

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    $\begingroup$ I doubt there's a good reason to think you'll have near-independence across time, so I wouldn't suggest either. $\endgroup$ – Glen_b Mar 29 '16 at 5:37
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I would use the Mann-Whitney U test, which is quite robust and reasonably powerful. Otherwise you can first check for normality (eg with the Shapiro-Wilks test) and then use the t test if the hypothesis of normality is not rejected.

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