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I have a model: $$Y=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3X_3+\beta_4x_4$$ and I want to test these two hypotheses: $$H_0: \beta_1>-3 \qquad H_a: \beta_1<-3$$
So if I conduct a one-sided test at the $\alpha=1\%$ significance level, the value of the $t$ statistic is over $4$. For reference, the critical value of the test statistic is $-2.326$.

I'm struggling to find the intuition for rejecting or accepting the $H_0$. The $p$ value seems to suggest reject, which I don't think the data shows.

$x ̅=-2.186$

$SE=0.2$

$(-2.186--3)/0.2=4.07$

So the critical value is $-2.326$ and the test statistic is $4.07$.

I think I reject below this, and so I would fail to reject the null hypothesis. But if the $p value$ is $0.000024$, shouldn't that mean the null hypothesis is rejected?

enter image description here

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    $\begingroup$ Could you elaborate a little on how you find a p-value of "almost 1" for a t statistic of "over 4"? Exactly which statistic is that, anyway? The one reported by software or one computed from your null hypothesis (which will be very different)? $\endgroup$
    – whuber ♦
    Commented Nov 29, 2021 at 21:57
  • $\begingroup$ I would double check whether the critical value for your t-statistic is indeed 2.326, but not, say -2.326 (and you reject $H_0$ for any t-statistic lower than the critical value). $\endgroup$
    – B.Liu
    Commented Nov 29, 2021 at 22:03
  • $\begingroup$ Sorry, I'm using R and I have got it the wrong way around. it should be 0.000024 $\endgroup$
    – user8000
    Commented Nov 29, 2021 at 22:07
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    $\begingroup$ My guess is that you are getting tripped up by confusing the default t-stat and p-value that come from the regression (corresponding to a two-sided null that $\beta_1=0$ against $\beta_1 \ne 0$ with the ones from your one-sided test that $\beta_1 \ge -3$ against $\beta_1 < -3$. Editing your question with the commands and output would be helpful if you want a good answer. $\endgroup$
    – dimitriy
    Commented Nov 29, 2021 at 22:47
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    $\begingroup$ That's a truly strange drawing, because it appears to place $-2.326$ at the same distance from $0$ as $4.07.$ And what does "x" represent? The main point here is that the t-statistic you need to use is $(\hat\beta_1 - (-3)) / \operatorname{se}(\hat\beta_1),$ which will not be the t-statistic reported by the software. $\endgroup$
    – whuber ♦
    Commented Nov 30, 2021 at 18:20

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You are miscalculating the p-value. If your $H_a$ is $\beta < -3$ the p-value is the area from $-\infty$ to 4.07, or 1 - 0.000024 = 0,999976.

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