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What hypothesis for A is better than B?

Consider the Pearson correlation estimate $r$. I want to do a hypothesis test.

$$\text{Option 1}$$

$H_0: \text{corr}(X,Y) \leq 0$

$H_1: \text{corr}(X,Y) > 0$

$$\text{Option 2}$$

$H_0: \text{corr}(X,Y) = 0$

$H_1: \text{corr}(X,Y) > 0$

I'm wondering if there's any difference between the two in terms of application (and theory)? I want to do $\text{Option 1}$ but the only information I can find is on $\text{Option 2}$.

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Jase
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  • Interestingly, exactly this question was asked just a few hours earlier (in the wider context of hypotheses in general, but the application to $r$ is immediate). The earlier, broader question will remain open--look for answers there. – whuber Dec 13 '12 at 15:34
  • @whuber That question is testing two means and also has an inequality as the null, whereas I have an equality. – Jase Dec 13 '12 at 16:00
  • That question is testing two parameters, Jase, and the correlation is a parameter in your setting. It is explicitly asking whether to use an equality or inequality for the null hypothesis. Thus the only difference I can detect is that you call your parameter "$r$" and the duplicate calls it "$\mu$". – whuber Dec 13 '12 at 16:04

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