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Consider the simple linear model:

$\vec Y=\beta_0+\beta_1 \vec X +\vec \varepsilon$

When I have read up on discussions about the different statistical tests that can be run on the model's slope coefficient $\beta_1$, I have exclusively seen the following null hypothesis / alternative hypothesis set up:

\begin{align} &H_0: \beta_1 =0 \\&H_a:\beta_1 \neq 0 \end{align}

With this particular test, there is then a $t$ statistic whose formula is given as $\beta_1$ divided by the standard error of $\beta_1$.

My question is what if I was instead interested in the following null hypothesis:

\begin{align} &H_0: \beta_1 \leq0 \\&H_a:\beta_1 \gt0 \end{align}

How exactly would I carry this out?

Is it possible to simply construct confidence intervals around $\beta_1$ (at a particular significance level) and if my resulting interval $(\beta_1-\delta,\beta_1+\delta)$ is strictly positive, I can then conclude that $\beta_1$ must be a positive number (as opposed to simply a non-zero number, which is what the prior hypothesis test would have provided us with)?

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    $\begingroup$ In Mathematical Statistics with Applications, by Wackerly, Mendenhall, and Scheaffer, Section 11.5, the authors lay out typical inferences such as $H_0: \beta_i=\beta_{i0}$ versus $H_a:\beta_i>\beta_{i0}$ or $H_a:\beta_i<\beta_{i0}$ or $H_a:\beta_i\not=\beta_{i0}.$ The test statistic is the same in all three cases, the rejection region changes. It's a fairly standard $t$ test in all three cases. Would one of these hypotheses match your research question? $\endgroup$
    – Adrian Keister
    Commented Apr 19, 2022 at 20:29
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    $\begingroup$ Completely coincidentally, I just posted an answer. You can find it in the middle of stats.stackexchange.com/a/572198/919. Look at the three short paragraphs beginning "The null hypothesis..." preceding the first figure. $\endgroup$
    – whuber ♦
    Commented Apr 19, 2022 at 20:29
  • $\begingroup$ @whuber Thank you for the input - I read your post but am having difficulties understanding how to implement your algorithm. AdrianKeister's response has given me further pause, as well. I was always under the impression that the null hypothesis and alternative hypothesis, when unioned, had to cover the whole space of possibilities. For example, $H_0: \beta_1 \leq 0$ and $H_a: \beta_1 \gt 0$ clearly cover all cases for $\beta_1 \in \mathbb R$. However, $H_0: \beta_1 =0$ and $H_a:\beta_1 \gt 0$ do not cover all cases. Is that a problem? $\endgroup$
    – S.C.
    Commented Apr 19, 2022 at 21:25
  • $\begingroup$ First, that's not relevant to my post, because I use the first formulation. Second, it's not a problem because the test in the second case is tantamount to the test in the first case. Read about the difference between two-sided and one-sided tests for more information; but intuitively, when $\beta_1\lt 0$ it's even less likely the data will reject the hypothesis that $\beta_1=0.$ As far as implementing the algorithm goes, it amounts to halving the reported p-value and optionally subtracting it from $1:$ which part is causing difficulties in implementation? $\endgroup$
    – whuber ♦
    Commented Apr 19, 2022 at 22:01

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