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Let's say you have data from two random variables $X$ and $Y$, and the two random variables are highly correlated (ex: $r=.97$). The p-value, which is testing the null hypothesis that there is no correlation, is low, which indicates that there is a correlation.

Can you test a one-sided hypothesis $H_0: r=0, H_a: r>0$ if previously you had hypothesized a positive (or negative, if that's the case) correlation? If so, would you multiply the p-value calculated above by $0.5$ to achieve the p-value for this directional, one-sided test?

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    $\begingroup$ Just a small comment: in one-sided case, null-hypothesis is $H_0: r\le 0$ and not $H_0:r=0$. Regarding your questions, the answers seem to be Yes and Yes. $\endgroup$ – amoeba says Reinstate Monica Apr 24 '15 at 22:21
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    $\begingroup$ No, the null hypothesis is that correlation is negative or zero, and the alternative hypothesis is that correlation is positive. You want to claim the evidence that the correlation is positive, and for that you need to reject the null that it is negative. $\endgroup$ – amoeba says Reinstate Monica Apr 24 '15 at 22:36
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    $\begingroup$ @amoeba: There's nothing problematic in testing $H_0: \rho= 0$ vs $H_\mathrm{a}: \rho> 0$. For observed $r<0$ the generalized likelihood ratio comes to one, because the value of $\rho$ that maximizes the likelihood is zero under both null & alternative. $\endgroup$ – Scortchi - Reinstate Monica Jul 27 '16 at 12:28
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    $\begingroup$ @amoeba: Oh yes! And there's little practical difference between that & testing $H_0: \rho \leq 0$ vs $H_\mathrm{a}: \rho> 0$. It's just that someone was puzzled by your comment: stats.stackexchange.com/q/225874/17230. $\endgroup$ – Scortchi - Reinstate Monica Jul 27 '16 at 12:35
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    $\begingroup$ @Scortchi I see, thanks for the link. Agree on little practical difference: the test statistic is the same. $\endgroup$ – amoeba says Reinstate Monica Jul 27 '16 at 12:37

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