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According to Wikipedia:

In inferential statistics, the term "null hypothesis" usually refers to a general statement or default position that there is no relationship between two measured phenomena, or no association among groups. ... The statement that is hoped or expected to be true instead of the null hypothesis is the alternative hypothesis

Source: https://en.wikipedia.org/wiki/Null_hypothesis

In my situation, the default position is that there is a correlation between two variables, and the statement that is hoped to be true is that there is no correlation (if some other variable is controlled for).

Questions:

1) Can I state the null hypothesis as "correlation is different from zero", and the alternate hypothesis as "correlation is equal to zero"?

2) Assuming I can do that, then consider situation when I obtain a correlation coefficient of say 0.05 with p-value 0.8. Would it be right to say the null hypothesis is rejected because the correlation coefficient is not different from zero with p-value of 0.8?

tomas
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    Would it not be more informative to present a confidence interval for $\rho$? – mdewey Nov 30 '16 at 12:56
  • The idea was to apply the standard framework of hypothesis testing, where you have the null hypothesis and the alternate hypothesis. Or perhaps I am not understanding your comment? – tomas Nov 30 '16 at 13:42
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    I was suggesting a procedure which might answer your (unstated) scientific question better. – mdewey Nov 30 '16 at 13:54
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    "Correlation different from zero" is useless as a hypothesis: it does not permit one to obtain any observable information about correlation that wouldn't already be known in the absence of that assumption. Please see http://stats.stackexchange.com/questions/31 and search our site for [TOST](http://stats.stackexchange.com/search?q=TOST). – whuber Nov 30 '16 at 17:19

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