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Currently hypothesis tests for normality are setup so that the null hypothesis is that the data is normally distributed and the alternative hypothesis is that the data is not. This seems to have problems where either you have a small sample size the test is underpowered and will fail to detect a meaningful deviation from normality, or you have a large sample size and the test rejects normality even when the departure is inconsequential.

Doesn't this problem mainly stem from the way normality tests are set up? To me it seems to be turning the hypothesis test logic on its head. Would it not make more sense if the tests were specified the other way around so that the null hypothesis represents the data deviating meaningfully from normality and the alternative hypothesis that the data is sufficiently normal? Then you would need to achieve enough power to be able to reject the data deviating from normality, so underpowered tests would not let you get away with assuming normality, and very high power wouldn't hurt because the test only rejects if the deviation from normality is big enough to be a problem.

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    $\begingroup$ You seem to misunderstand how hypothesis testing works. Most importantly, the null must be specific enough to enable calculation of the sampling distribution of the test statistic when the null is assumed true. "Data deviating meaningfully from normality" is such a broad, vague (and subjective) description that basing any kind of test on this would be impossible. You might find it helpful to study our higher-voted threads concerning these issues. $\endgroup$ – whuber Apr 29 '15 at 16:12
  • $\begingroup$ Yes, I know the null must be specific. Specifying the null would be done in the context you are using the normality test. If you are testing for normality because you wish to use some other test that assumes normality, then you can use that other tests sensitivity to deviations from normality to choose a specific null. Sure, it will still require judgement, but I don't think hypothesis testing is ever a license to leave judgement at the door. $\endgroup$ – Mr Alpha Apr 29 '15 at 17:56
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As indicated in whuber's comment, the null hypothesis is always the one used to compute a test statistic, so it is not surprising that it is a concrete statement ("$\mu = 10$", "distribution is normal", etc.).

More to the point, the normality test follows the same logic as other tests, which are usually done in a context. ("Is the product modification worthwhile", "Did the earth get warmer", ...) When you reject the null hypothesis, you usually want to do something (change the product, campaign for reduction in greenhouse emissions, transform the data in a regression analysis, etc.) The null hypothesis represents the status quo (no action required), and the normality test is no exception to this.

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