$$H_0:\mu=0 \quad\quad H_1:\mu>0 $$ we assume the distribution of the sample is Gaussian.
If $p$-value is very big, sample mean very small, we still accept null hypothesis.
Isn't it counterintuitive? Why are we still using it?

E.g. $x_1,x_2,\ldots,x_n$, sample mean is $-5$, variance is known as 1.
$$H_0:\mu=0 \quad\quad H_1:\mu>0 $$ It's easy to see that $p$-value is very big, but we still accept mu = 0. But actually it is very intuitive that $\mu$ is less than 0 or at least not equal to zero?

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    $\begingroup$ Your question is not clear - could you elaborate a little bit more? Providing an example illustrating your question would also make it more clear. $\endgroup$ – Tim Oct 20 '15 at 11:12
  • $\begingroup$ p value is calculated based on a statistic(an observable random variable or a function of sample), the statistic may not only depend on $\mu$,it may also depend on standard error. $\endgroup$ – Deep North Oct 20 '15 at 11:31
  • $\begingroup$ This is for your edited question. If the p value is very small, why you still accept $\mu=0$? $\endgroup$ – Deep North Oct 20 '15 at 11:32
  • $\begingroup$ @DeepNorth No, sigma is known to me. This test is only for mu. $\endgroup$ – Li haonan Oct 20 '15 at 11:32
  • $\begingroup$ @DeepNorth Sorry, typo. $\endgroup$ – Li haonan Oct 20 '15 at 11:33

You have written the null hypothesis incorrectly. For a one sided test its'

$H_0 \mu \le 0 \\ H_1 \mu > 0$

So, in your example, where $\bar{X} = -5$ it is clearly much less than 0 and the null cannot be rejected. That's the price of a one tailed test.

Also, you don't accept the null you only fail to reject it.

  • $\begingroup$ Sorry, I check wiki before I raise question here. So it says 'or mu=0' on the one-sided test? $\endgroup$ – Li haonan Oct 20 '15 at 11:34
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    $\begingroup$ I don't know what you checked but if it said that, it's incorrect. $\endgroup$ – Peter Flom Oct 20 '15 at 11:36
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    $\begingroup$ +1 for "you don't accept the null you only fail to reject it." People always forget this. $\endgroup$ – bdeonovic Oct 20 '15 at 11:40
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    $\begingroup$ +1 I just want to mention this (the hypothesis is not correct. $\endgroup$ – Deep North Oct 20 '15 at 11:45

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